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# Catastrophe Theory for $\Gamma$-invariant Unfoldings with Applications to
Quantum Many-body Theory

K. Rodrigues Alves

###### Abstract

The theory of singularities and its broad ramifications, especially
catastrophe theory, have found fertile ground in some areas of physics (e.g.,
caustics, wave optics) for their applications. In the context of quantum many-
body theory, however, their results, despite being useful, are not generally
known by the scientific community and often require a non-trivial adaptation,
mainly due to the effects of symmetries in the associated physical system,
which are not encompassed by the original theory. In this article, we provide
an extension of the main results of René Thom’s catastrophe theory for the
case of germs and unfoldings possessing special symmetries. In a more
mathematically precise language, we provide a proof of the determinacy
theorems for germs that are invariant under the action of an arbitrary compact
Lie group, and of the transversality and stability theorems for the case of
invariant unfoldings. The results obtained can be seen as an extension and
adaptation of the works of [7] and [8] on singularity theory of
$\mathbb{R}^{n}$ to $\mathbb{R}^{n}$ mappings to the particular scenario of
catastrophe theory. Finally, we also provide a classification theorem for
unfoldings invariant under the action of $\mathbb{Z}_{2}$, and present some of
the possible applications of the theory to the study of phase transitions in
quantum many-body systems.

###### Contents

  1. 1 Introduction
  2. 2 $\Gamma$-invariant germs
    1. 2.1 Basic definitions
    2. 2.2 The tangent space of $\Gamma$-invariant germs
    3. 2.3 Determinacy of $\Gamma$-invariant germs
  3. 3 $\Gamma$-invariant unfoldings
    1. 3.1 The tangent space generated by unfoldings
    2. 3.2 Transversality theorem for $\Gamma$-invariant unfoldings
    3. 3.3 Stability theorem for $\Gamma$-invariant unfoldings
  4. 4 Classification theorem for $\mathbb{Z}_{2}$-invariant germs
  5. 5 Applications to Quantum Statistical Mechanics
  6. 6 Technical results
    1. 6.1 Adequately ordered systems of DA algebras

##  1 Introduction

During the years 1950s, the theory of classification of singularities of
smooth mappings has seen an outstanding progress. Such progress was mainly a
consequence of the seminal work developed by H. Whitney, who in 1952 published
an article on the classification of mappings from $\mathbb{R}^{2}$ to
$\mathbb{R}^{2}$ [12], that laid the basis for many subsequent works on the
study of singularities of mappings. One of these works, in particular, carried
out by the mathematician René Thom [10], now receives the name of catastrophe
theory. To first understand what kind of classification of singularities
Thom’s work is concerned with, we may begin by looking at mappings of
functions $f$ from $U\subset\mathbb{R}^{n}$ to $\mathbb{R}$. For a regular
point $x_{0}\in U$ of $f$, i.e., a point $x_{0}$ such that the gradient
$\vec{\nabla}f(x_{0})$ is non-degenerate, a consequence of the implicit
function theorem is that there exists a diffeomorphism $\phi$ such that, in
its coordinates, $f$ is given by

$(f\circ\phi)(x)=x_{1}.$ (1)

If, instead, $x_{0}$ is not a regular point but rather a non-degenerate
critical point, i.e., a point where the gradient $\vec{\nabla}f(x_{0})$ is
degenerate but the Hessian $\operatorname{Hess}(f)(x_{0})$ is not, then
similarly to the previous case, by the so-called Morse Lemma it follows that
there exists again a diffeomorphism $\phi$ such that, around $x_{0}$, $f$ is
given by

$(f\circ\phi)(x)=\pm x_{1}^{2}+\cdots\pm x_{n}^{2}.$ (2)

The qualitative analysis of the behaviour of $f$ around $x_{0}$ can be,
therefore, easily done in these two cases, since the diffeomorphism $\phi$
gives us a great deal of simplification. Any function near a regular or a non-
degenerate critical point follows into one of the examples above. However, the
situation changes if $x_{0}$ is now a degenerate critical point; a critical
point with a degenerate Hessian. Then, the behavior of $f$ around $x_{0}$
ceases to be trivial, and we need to enter more deeply in the theory of
singularities in order to understand it.

The brief analysis described above can be restated in a more general and
abstract way, as in the following consideration: we start with the space of
smooth real-valued functions on $\mathbb{R}^{n}$, which we simply denote by
$S$, and an equivalence relation associated with the (infinite-dimensional)
Lie group $G$ of smooth diffeomorphisms on $\mathbb{R}^{n}$ preserving the
origin, defined by $f\sim g$ iff there exists a $\phi\in G$ such that
$f=g\circ\phi$. The equivalence relation is supposed to preserve the
properties that we are interested in; therefore, given a function $f\in S$,
one of the goals is to find a function $g$, belonging to the same equivalence
class of $f$, such that $g$ is an element we can explicitly compute the
relevant properties of our system, which in turn will still hold true for $f$.
Another related important concept is that of stability: $f$ is stable when
there exists a neighborhood of $f$ (in a suitable topology) where every point
is equivalent to $f$ itself. A remarkable discovery of singularity theory is
that we can learn about the stability of $f$ by analyzing the differential of
the action of $G$ into $S$, i.e., the differential $Dr_{f}$ of the mapping
$r_{f}:G\rightarrow S$ given by $r_{f}(\phi)=f\circ\phi$. It follows that $f$
is stable if and only if $Dr_{f}$ is surjective. And this happens if and only
if $\vec{\nabla}f$ or $\operatorname{Hess}(f)$ are non-degenerate; in these
scenarios, $f$ falls into one of the two cases shown above.

Suppose now that $Dr_{f}$ is not surjective, i.e., that $f$ is not a stable
point. Then, we can consider a submanifold $F$ passing through $f$, such that
its tangent space $T_{F}$ at $f$ is a complement of the image of the
differential $Dr_{f}$. Note that $F$ can also be thought of as a parametric
family of functions in $S$. It follows that although $f$ is not stable, $F$,
by intersecting transversally the manifold generated by $r_{f}$, is itself
stable – for an associated equivalence relation. In fact, this relationship
between transversality and stability is one of the main theorems about
catastrophe theory. The number of extra dimensions that the differential
$Dr_{f}$ fails to generate is called the codimension of the function. This is
what is in the core of catastrophe theory: to show that such transversal
families of functions are stable, and equivalent to each other. Moreover, for
functions with codimension smaller than or equal to $6$, it can also be shown
that there exists only a finite number of equivalence classes, which in turn
implies the existence of only a finite number of equivalence classes of
parametric families of functions; a result that is famously known as Thom’s
Classification Theorem. The polynomial representatives of these equivalence
classes are usually called elementary catastrophes in the literature. 111The
first list of the elementary catastrophes went only up to codimension $4$.
Later, however, extensions to the initial classification were provided, and
from codimension $7$ onwards it was shown that the total number of functions
in the classification theorem already becomes infinite. To the author’s
knowledge, the largest classification table is exposed in [9]. There are,
however, certain limitations of the original theory regarding families of
functions $F$ possessing special symmetries. The symmetry of the functions in
$F$ can prevent its tangent space from generating the missing complementary
vectors, and hence, it would not be stable. However, if we in turn restrict
$G$ to only diffeomorphisms that preserve the desired symmetry (also called
equivariant diffeomorphisms), and rescrict the function space $S$ to just
functions that possess the assumed symmetry, thus limiting the perturbations
to only symmetric perturbations, one may then recover similar results for the
symmetric family $F$. Such extensions of the original theory of singularities
has been studied in a similar albeit different context by [7] for some kinds
of symmetry, and finally by [3] in more generality. Their studies, however,
did not focus on catastrophe theory, but rather to general mappings
$F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{k}$, with a different equivalence
relation and also a different symmetry: rather than being invariant under a
symmetry group, they are what is called equivariant. Here, we intend to
present a concise and complete exposure of these results adapted to the
context of catastrophe theory.

Thom’s original goal for developing catastrophe theory, introduced to the
scientific community in his book [10], was to provide a general theory for
classifying the behaviour of discontinuous phenomena coming from a continuous
parameter change. In his setup, given a system with $r$ control parameters,
the ”state” of the system (which will depend on the values of the parameters)
is described by the minima $x_{0}\in\mathbb{R}^{n}$ of a function
$F:(x,u)\mapsto F(x,u)\in\mathbb{R}$, i.e., the set of possible states of the
system for some fixed value of $u\in\mathbb{R}^{r}$ are described by the
points of the set

$S_{u}=\\{x_{0}\in\mathbb{R}^{n}\;|\;F(x_{0},u)=\min_{x}F(x,u)\\},\;\;u\in\mathbb{R}^{r}.$
(3)

As the parameter $u$ moves, so do the points in $S_{u}$, possibly leading to
an eventual sudden jump in the states of the system, i.e., a catastrophe.
Morover, as mentioned before, depending on the tangent space generated by $F$
at $u=0$, it is also possible to show that there is only a finite number of
equivalence classes of functions that $F$ must belong to; a result known as
Thom’s Classification Theorem. Thom’s works, together with Zeeman’s, had a
great impact at the time in areas outside of exact sciences, because of their
claims of being able to correctly model, with only a finite number of
functions, an incredible amount of different phenomena: from prison uprisings
to a dog’s aggressiveness. After this initial excitement, however, a stronger
criticism came [4], questioning the validity of modelling real-world phenomena
in such a way, and at last the theory fell into disuse outside of more
mathematical areas.

We, in particular, avoid this criticism here by considering the application of
catastrophe theory not as a generic theory for modelling discontinuous
processes, but as a direct application of singularity theory to the study of
phase transitions in classical and quantum mechanics, where the function $F$
does not appear as an a priori consideration of the physical system, but as a
direct consequence of the proposed model. The results coming from catastrophe
theory have relevant contributions, especially in proving the existence of
phase transitions and the stability of phase diagrams. As an example, we may
consider the BCS model of superconductivity. In the so-called strong-coupling
limit, and disregarding the kinetic energy of the electrons in the lattice, it
can be shown [1] that the thermodynamical pressure of the system converges, in
the infinite-volume limit, to

$P(u_{1},u_{2})=-\inf_{c\in\mathbb{R}}F(c,u_{1},u_{2}),\;\;F(c,u_{1},u_{2})=u_{1}c^{2}-\ln\left(1+e^{-u_{2}}\cosh\left(\sqrt{u_{2}^{2}+u_{1}^{2}c^{2}}\right)\right),$
(4)

where the parameters $u_{1},u_{2}$ are associated with the strengths of the
electron-electron interactions in our system, and the variable $c$ is
associated with different phases of our system. By a suitable change of
coordinates (that in particular must not mix the parameters and the variable),
one may show that the function $F$ is equivalent (in a suitable sense, via
diffeomorphisms that preserve its minima and critical points, as we shall
expose) to

$\tilde{F}(c,u_{1},u_{2})=c^{6}+u_{1}c^{2}+u_{2}c^{4},$ (5)

and hence, its set of bifurcations of states can be easily studied by the
analysis of the above polynomial function, giving rise to a phase diagram of
the system.

It follows, however, that the above example does not fall into the original
classification proposed by Thom; and the reason behind it is the existence of
symmetries in the family $F$. The symmetry that plays a role here is the
parity symmetry of $F(\cdot,u_{1},u_{2})$, which is associated with the $U(1)$
Gauge symmetry of the respective Hamiltonian of the quantum system. Families
of functions with some special symmetry in its variables are in general not
stable, since even an arbitrarily small non-symmetric perturbation can break
the symmetry of the function and, as a consequence, generate different
singularities and catastrophes that were not present in the original symmetric
case. However, they can be considered stable, in a slightly different sense,
if we restrict the perturbations only to those who also possess the same
symmetry of the functions in consideration. Then, we may recover the standard
results of catastrophe theory, but for functions possessing a special
symmetry, which are abundant in the context of physics. This is thus the aim
of this paper, to expose to a broader audience the features of catastrophe
theory that can be explored in a mathematically precise way, in contexts
encountered in quantum statistical mechanics, where the family $F$ has a
special symmetry, and hence the original theory cannot be applied. To the
author’s knowledge, there is no other work in quantum statistical mechanics
that uses this area of singularity theory as a powerful mathematical tool to
extract relevant qualitative behaviours of the system. In a further work, we
aim at using the results exposed here to prove the existence and stability of
phase transitions to a large class of mean-field models for fermionic systems.

The paper is divided as follows: In the second chapter, we prove the so-called
finite determinacy theorem (Theorem 2.3) for germs with special symmetries. In
the third chapter we show the transversality and stability theorems (Theorems
3.6 and 3.7, respectively) for families of functions possessing symmetries,
and finally in chapter $4$ we present a classification theorem (Theorem 4.4)
for the special case of $\mathbb{Z}_{2}$-symmetric families of functions.

##  2 $\Gamma$-invariant germs

###  2.1 Basic definitions

Here we start with the basic definitions necessary for the development of the
results. We note first that catastrophe theory deals with singularities
locally i.e., in a neighborhood of a point. Hence, the natural language for
the theory is the language of germs. We introduce now certain germ spaces that
will be used throughout the text. We keep a similar notation to the one of
[3].

######  Definition 2.1.

Let $U\subset\mathbb{R}^{n}$ be an open neighborhood of $0$. We define:

  1. i.)

$\mathcal{E}_{n}$ as the set of all germs at $0$ of functions in
$C^{\infty}(U,\mathbb{R})$, and $\vec{\mathcal{E}}_{n,m}$ as the set of all
germs at $0$ in $C^{\infty}(U,\mathbb{R}^{m})$,

  2. ii.)

$\mathcal{M}^{k}_{n}$ as the set of all germs at $0$ of functions in
$C^{\infty}(U,\mathbb{R})$ whose derivatives up to order $k-1$ vanish at $0$,
and $\vec{\mathcal{M}}_{n,m}^{k}$ as the set of all germs at $0$ in
$C^{\infty}(U,\mathbb{R}^{m})$ whose derivatives up to order $k-1$ vanish at
$0$,

  3. iii.)

$J_{n}^{k}$ as the set of all germs of polynomials in
$C^{\infty}(U,\mathbb{R})$ of order up to $k$, and $J_{0,n}^{k}\subset
J_{n}^{k}$ as the set of germs of polynomials in $J_{n}^{k}$ with no constant
term.

######  Remark.

We recall that, for an open $U\subset\mathbb{R}^{n}$ and a function $f\in
C^{\infty}(U,\mathbb{R})$ the germ of $f$ at $p\in U$ is the equivalence class
of all functions $g\in C^{\infty}(U,\mathbb{R})$ such that $g=f$ for some
neighborhood $V\subset U$ of $p$.

In order to include the symmetries in the germs, we, for now on, let $\Gamma$
be a compact Lie group, acting orthogonally on $\mathbb{R}^{n}$. Since we will
only be interested in the action of $\Gamma$, to simplify the notation, the
matrix representative in $GL(n)$ of any element $\gamma\in\Gamma$ will also be
denoted by the same letter $\gamma$.

######  Definition 2.2.

Let $\Gamma$ be a compact Lie group acting orthogonally on $\mathbb{R}^{n}$. A
germ $f\in\mathcal{E}_{n}$ satisfying

$f(\gamma x)=f(x)\quad\text{ for all }\gamma\in\Gamma$

will be called $\Gamma$-invariant. A germ $\phi\in\vec{\mathcal{E}}_{n}$
satisfying

$\phi(\gamma x)=\gamma\phi(x)\quad\text{ for all }\gamma\in\Gamma$

will be called $\Gamma$-equivariant. The set of all $\Gamma$-invariant and
$\Gamma$-equivariant germs will be denoted, respectively, by
$\mathcal{E}_{n}(\Gamma)$ and $\vec{\mathcal{E}}_{n}(\Gamma)$. In particular,
we denote the set of $\Gamma$-invariant polynomials by
$P_{n}(\Gamma)\subset\mathcal{E}_{n}(\Gamma)$ and the set of
$\Gamma$-equivariant germ diffeomorphisms preserving the origin by
$L_{n}(\Gamma)\subset\vec{\mathcal{E}}_{n}(\Gamma)$.

######  Remark.

One of the reasons to restrict the theory to compact Lie groups is to be able
to apply in that case the so-called Hilbert basis theorem, which states that
the sub-algebra $P_{n}(\Gamma)$ of $\Gamma$-invariant polynomial germs is
generated (as an algebra) by finitely many $\Gamma$-invariant polynomials. A
similar result also states that $\vec{\mathcal{E}}_{n}(\Gamma)$ is generated,
as an $\mathcal{E}_{n}(\Gamma)$-module, by finitely many homogeneous
$\Gamma$-equivariant polynomials.

The following definition gives us the relevant equivalence class of germs in
$\mathcal{E}_{n}(\Gamma)$, which will in turn give rise to the classification
theorem.

######  Definition 2.3.

Given $f,g\in\mathcal{E}_{n}(\Gamma)$, if there exists some $\phi\in
L_{n}(\Gamma)$ such that $g=f\circ\phi$, we say that $f$ and $g$ are
$\Gamma$-equivalent.

###  2.2 The tangent space of $\Gamma$-invariant germs

We now analyze the tangent space of the orbit of a germ $f$ with respect to
the action of $L_{n}(\Gamma)$ (recall Definition 2.2). As stated in the
introduction, the tangent space generated by the orbit of $f$ with respect to
$L_{n}(\Gamma)$ is associated with the stability of our germ $f$, as we shall
see later. Hence, let $f\in\mathcal{E}_{n}(\Gamma)$. The orbit of $f$ with
respect to the action of the (infinite-dimensional Lie) group $L_{n}(\Gamma)$
is denoted by

$\mathcal{O}_{f}\doteq\\{f\circ\phi\;|\;\phi\in L_{n}(\Gamma)\\}.$

Let $\mathbf{1}$ be the identity of $L_{n}(\Gamma)$. A tangent vector at
$\mathbf{1}$ is then of the type $(\mathbf{1}+t\phi)(x)$, where
$\phi\in\vec{\mathcal{M}}_{n}(\Gamma)$, since one must have $\phi(0)=0$.
Hence, an element of the tangent space $T_{\mathcal{O}_{f}}$ of
$\mathcal{O}_{f}$ is given by

$\frac{df(\mathbf{1}+t\phi)(x)}{dt}\bigg{|}_{t=0}=\sum_{i=1}^{n}\phi_{i}(x)\frac{\partial
f(x)}{\partial x_{i}}.$

The tangent space $T_{\mathcal{O}_{f}}$ is consequently given by

$T_{\mathcal{O}_{f}}=\\{\phi\cdot\vec{\nabla}f\;|\;\phi=(\phi_{1},\dots,\phi_{n})\in\vec{\mathcal{M}}_{n}(\Gamma)\\}.$

We formalize it in the next definition:

######  Definition 2.4.

Let $f\in\mathcal{E}_{n}(\Gamma)$. Its tangent space with respect to the
action of $L_{n}(\Gamma)$ is given by

$T_{\mathcal{O}_{f}}=\\{\phi\cdot\vec{\nabla}f\;|\;\phi=(\phi_{1},\dots,\phi_{n})\in\vec{\mathcal{M}}_{n}(\Gamma)\\}.$
(6)

We also define the extended tangent space by

$T^{\operatorname{ext}}_{\mathcal{O}_{f}}=\\{\phi\cdot\vec{\nabla}f\;|\;\phi=(\phi_{1},\dots,\phi_{n})\in\vec{\mathcal{E}}_{n}(\Gamma)\\}.$
(7)

######  Proposition 2.5.

For any $f\in\mathcal{E}_{n}(\Gamma)$,
$T_{\mathcal{O}_{f}}\subset\mathcal{M}_{n}(\Gamma)$ and
$T^{\operatorname{ext}}_{\mathcal{O}_{f}}\subset\mathcal{E}_{n}(\Gamma)$.

###### Proof.

The inclusions $T_{\mathcal{O}_{f}}\subset\mathcal{M}_{n}$ and
$T^{\operatorname{ext}}_{\mathcal{O}_{f}}\subset\mathcal{E}_{n}$ are obvious.
Furthermore, by the chain rule, note that

$\frac{\partial(f(\gamma x))}{\partial x_{i}}=\sum_{j=1}^{n}\frac{\partial
f(\gamma x)}{\partial x_{j}}\gamma_{j,i},$ (8)

where $\gamma_{i,j}$ are the matrix elements of $\gamma$ at row $i$ and column
$j$. I.e., one has

$\vec{\nabla}(f(\gamma x))=\gamma^{T}\vec{\nabla}f(\gamma x).$ (9)

Inverting the equation, making use of the fact that $\Gamma$ acts orthogonally
on $\mathbb{R}^{n}$, and that $f(\gamma x)=f(x)$, it follows that

$\vec{\nabla}f(\gamma x)=\gamma\vec{\nabla}f(x).$ (10)

Hence, if $\phi\in\vec{\mathcal{E}}_{x}(\Gamma)$, one has

$\phi(\gamma x)\cdot\vec{\nabla}f(\gamma
x)=((\gamma^{T}\gamma\phi(x))\cdot\vec{\nabla}f(x)=\phi(x)\cdot\vec{\nabla}f(x),$
(11)

i.e., $\phi\cdot\vec{\nabla}f$ is $\Gamma$-invariant. Hence, the propoition
follows.

######  Remark.

For a given family $F(x,u)$ with $f(x)=F(x,0)$, the number of parameters $u$
that $F$ needs for it to be stable is associated with the size of the tangent
space $T^{\operatorname{ext}}_{\mathcal{O}_{f}}$, this is why
$T^{\operatorname{ext}}_{\mathcal{O}_{f}}$ is so important. We note here that
the above tangent space is infinite-dimensional, and eventually we will need
to reduce it to a finite-dimensional manifold, by considering the orbit in the
jet space $J_{n}^{k}(\Gamma)$, where we then can make use of standard results
in the theory of euclidean manifolds.

###  2.3 Determinacy of $\Gamma$-invariant germs

To apply theorems about transversality, coming from standard differential
geometry, it will be required for us to work on a finite-dimensional manifold.
For this, we prove the so-called determinacy theorem, which allows us to work
on the jet space of functions up to some finite order $k$.

######  Definition 2.6.

Let $f\in\mathcal{E}_{n}(\Gamma)$. Then, $f$ is said to be
$\Gamma$-$k$-determined if $f$ is $\Gamma$-equivalent to any other
$g\in\mathcal{E}_{n}(\Gamma)$ such that $j^{k}f=j^{k}g$, where $j^{k}f$ is the
germ of the Taylor expansion of $f$ up to order $k$. Moreover, $f$ is said to
be $\Gamma$-finitely-determined if it is $\Gamma$-$k$-determined for some
$k<\infty$.

We note that if $f\in\mathcal{E}_{n}(\Gamma)$, then
$j^{k}f\in\mathcal{E}_{n}(\Gamma)$ for any $k\in\mathbb{N}$. To see this,
first recall that $j^{k}f$ is the unique polynomial that satisfies

$\lim_{\|x\|\rightarrow 0}\frac{|f(x)-j^{k}f(x)|}{\|x\|^{k}}=0.$

Since $\Gamma$ acts orthogonally on $\mathbb{R}^{n}$ and $f$ is
$\Gamma$-invariant, one also has

$\lim_{\|x\|\rightarrow 0}\frac{|f(x)-j^{k}f(\gamma
x)|}{\|x\|^{k}}=0,\quad\text{for all }\gamma\in\Gamma.$

In particular, by the uniqueness of $j^{k}f$, it follows that

$j^{k}f(x)=j^{k}f(\gamma x),\quad\text{for all }\gamma\in\Gamma,$

i.e., $j^{k}f$ is $\Gamma$-invariant.

######  Definition 2.7.

Let $f\in\mathcal{E}_{n}(\Gamma)$. The $\Gamma$-codimension of $f$ is defined
as

$\operatorname{cod}_{\Gamma}(f)\doteq\dim_{\mathbb{R}}\frac{\mathcal{M}_{n}(\Gamma)}{T^{\operatorname{ext}}_{\mathcal{O}_{f}}}.$
(12)

######  Remark.

Recall the definition of $T^{\operatorname{ext}}_{\mathcal{O}_{f}}$ (Equation
(2.4)). The codimension of a germ is related to the degree of degeneracy of
its critical point, and consequently to the number of parameters necessary to
have an unfolding of the germ stable under perturbations. In particular, only
germs with finite $\Gamma$-codimension admit stable unfoldings, as we shall
see later. To prove the main theorem about the determinacy of
$\Gamma$-invariant germs, which states that finite codimension implies finite
determinacy and vice-versa, we need a generalization for the scenario of
invariant germs of whats is known as Mather’s Lemma in the non-invariant case
[6], and such generalization is given in terms of DA-algebras.

######  Definition 2.8.

A differentiable algebra (DA-algebra) is an $\mathbb{R}$-algebra $A$, together
with a surjective homomorphism $\phi:\mathcal{E}_{m}\rightarrow A$, for some
$m\in\mathbb{N}$.

######  Theorem 2.9 (Schwarz theorem).

Let $\Gamma$ be a compact Lie group acting orthogonally on $\mathbb{R}^{n}$,
and let $\rho_{1},\dots,\rho_{m}$ be the generators of $P_{n}(\Gamma)$. Then,
it follows that, for any germ $f\in\mathcal{E}_{n}(\Gamma)$, there exists some
$h\in\mathcal{E}_{m}(\Gamma)$ such that

$f(x)=h(\rho(x)),\;\text{ where }\;\rho(x)=(\rho_{1}(x),\dots,\rho_{k}(x)).$
(13)

######  Remark.

We recall that for any compact Lie group $\Gamma$, the set $P_{n}(\Gamma)$ of
$\Gamma$-invariant polynomials has finite polynomial generators
$\rho_{1},\dots,\rho_{m}$, by Hilbert Basis theorem (see [8], Theorem 1).
Therefore, by choosing the homomorphism
$\rho^{*}:\mathcal{E}_{m}\rightarrow\mathcal{E}_{n}(\Gamma)$ defined as

$\rho^{*}(h)=h\circ\rho,\quad\rho(x)=(\rho_{1}(x),\dots,\rho_{k}(x)),$ (14)

it clearly follows, from Schwarz theorem stated above, that the pair
$(\mathcal{E}_{x}(\Gamma),\rho^{*})$ is a DA-algebra.

We now state the generalization of Mather’s lemma, whose proof we omit here
due to its technicality, but it can be found in [3, Lemma 7.3].

######  Theorem 2.10 (Generalized Mather’s Lemma).

Let $\mathfrak{R}=(R,\phi)$ be a DA-algebra, $a:N\rightarrow M$ be a
homomorphism of finitely generated $R$-modules, $\mathfrak{M}_{R}$ the maximal
ideal of $M$, and let $M_{0}$ be a submodule of finite codimension. Then:

  1. i.)

There is a number $\omega\in\mathbb{N}$ such that

$M_{0}\subset a(N)+\mathfrak{M}_{R}^{\omega}\cdot M_{0}$

implies

$M_{0}\subset a(N).$

  2. ii.)

If $\dim_{\mathbb{R}}M/a(N)<\infty$, then there exists a number
$\omega^{\prime}\in\mathbb{N}$ such that

$\mathfrak{M}_{R}^{\omega^{\prime}}\cdot M\subset a(N).$

Finally, we are able to prove the determinacy theorem for $\Gamma$-invariant
germs. But first we state a useful lemma, whose detailed proof we let for the
technical results section.

######  Lemma 2.11.

If $g\in\mathcal{E}_{n}(\Gamma)$ has finite codimension, then there exists
some $r\in\mathbb{N}$ such that for any $f\in\mathcal{E}_{n}(\Gamma)$
satisfying $j^{r}f=j^{r}g$, one has $\mathcal{M}_{n}^{r}(\Gamma)\subset
T_{\mathcal{O}_{f}}$.

######  Theorem 2.12 (Determinacy theorem).

Let $f\in\mathcal{E}_{n}(\Gamma)$. then, the conditions are equivalent:

  1. i.)

$f$ is $\Gamma$-finitely-determined,

  2. ii.)

$\mathcal{M}_{n}^{k}(\Gamma)\subset T^{\operatorname{ext}}_{\mathcal{O}_{f}}$
for some $k\in\mathbb{N}$,

  3. iii.)

$\operatorname{cod}_{\Gamma}(f)$ is finite.

###### Proof.

$ii.)\implies iii.)$. This implication is trivial, since the dimension of
$\mathcal{M}_{n}(\Gamma)/\mathcal{M}_{n}^{k}(\Gamma)$ is always finite for any
$k$. Recall Definitions 2.1 and 2.7.

$iii.)\implies ii.)$. Let
$a:\vec{\mathcal{E}}_{n}(\Gamma)\rightarrow\mathcal{E}_{n}(\Gamma)$ be the map
given by $a(\phi)=\phi\cdot\vec{\nabla}f$. Now, we apply Theorem 2.10 with
$R=\mathcal{E}_{n}(\Gamma)$, $N=\vec{\mathcal{E}}_{n}(\Gamma)$ and
$M=\mathcal{E}_{n}(\Gamma)$ (recall that $\vec{\mathcal{E}}_{n}(\Gamma)$ is
finitely generated as an $\mathcal{E}_{n}(\Gamma)$-module). Since the maximal
ideal of $\mathcal{E}_{n}(\Gamma)$ is $\mathcal{M}_{n}(\Gamma)$ and
$a(\vec{\mathcal{E}}_{n}(\Gamma))=T^{\operatorname{ext}}_{\mathcal{O}_{f}}$
(see Definition (2.4)), the implication directly follows from item $(b)$ of
Theorem 2.10.

$ii.)\implies i.)$. Let $g\in\mathcal{E}_{n}(\Gamma)$ be such that
$j^{r}f=j^{r}g$, for the respective $r$ of Lemma 2.11. The idea of the proof
is the following: let

$F_{s}=sg(x)+(1-s)f(x),\quad s\in[0,1].$ (15)

To prove condition $i.)$, we must show that $F_{0}$ is equivalent to $F_{1}$.
The goal will be then to prove that, for every $s\in[0,1]$, there exits an
interval $I_{s}$ around $s$ such that for any $s^{\prime}\in I_{s}$,
$F_{s^{\prime}}$ is $\Gamma$-equivalent to $F_{s}$. Since
$\cup_{s\in[0,1]}I_{s}$ is an open cover for $[0,1]$ and $[0,1]$ is compact,
it follows that there exists a finite sequence $(s_{1},\dots,s_{M})$ such that

$F_{0}\sim F_{s_{1}}\sim\dots\sim F_{s_{M}}\sim F_{1},$ (16)

which implies that $i.)$ holds. Hence, define for $s\in[0,1]$,

$F_{s}(x,t)=(s+t)g(x)+(1-s-t)f(x).$ (17)

We must prove that there exists a neighborhood of $0\in\mathbb{R}$ and a map
$h(x,t)$ such that, if $t$ belongs to that neighborhood, we have
$h(\cdot,t)\in L_{x}(\Gamma)$ and $F_{s}(h(x,t),t)=F_{s}(x,0)$. Since
$j^{r}F_{s}(\cdot,t)=j^{r}g$ for any $t$ and
$g-f\in\mathcal{M}^{r}_{n}(\Gamma)$, by Lemma 2.11, it follows that there
exists some $\vec{\xi}\in\vec{\mathcal{M}}_{n+1}$ with
$\vec{\xi}(\cdot,t)\in\vec{\mathcal{M}}_{n}(\Gamma)$, such that for $t$ in
some sufficiently small neighborhood $I$ of $0$,

$\vec{\nabla}_{x}F_{s}(x,t)\cdot\vec{\xi}(x,t)+(g-f)(x)=0.$ (18)

Now, take $h(x,t)$ as the (unique) solution to:

$\begin{cases}\frac{\partial h(x,t)}{\partial t}=\vec{\xi}(h(x,t),t),\quad
t\in I,\\\ h(x,0)=x.\end{cases}$ (19)

By the $\Gamma$-equivariance of $\vec{\xi}(\cdot,t)$ and the uniqueness of the
solution of Equation (19), it follows that $h(\cdot,t)$ is also
$\Gamma$-equivariant, and by Equation (18) one easily checks that

$\frac{\partial F_{s}(h(x,t),t)}{\partial t}=0,\quad t\in I$

i.e., $F_{s}(h(x,t),t)=F_{s}(x,0)$, for $t\in I$.

$i.)\implies ii.)$. Let $f$ be $\Gamma$-$k$-determined, and define

$D_{f}=\\{g\in\mathcal{E}_{n}(\Gamma)\;|\;j^{k}g=j^{k}f\\}.$ (20)

$D_{f}$ is a linear subspace of $\mathcal{E}_{n}(\Gamma)$, and a tangent
vector to $D_{f}$ at $f$ is of the type $f+th$, for any
$h\in\mathcal{M}^{k+1}_{n}(\Gamma)$. Hence, it follows that its tangent space
$T_{D_{f}}$ at $f$ is given by

$T_{D_{f}}=\mathcal{M}^{k+1}_{n}(\Gamma).$ (21)

But since $f$ is $\Gamma$-k-determined, it follows that $D_{f}$ is a subset of
the orbit of $f$ with respect to $L_{n}(\Gamma)$; in particular, one also has
the inclusion of the tangent spaces, i.e., $T_{D_{f}}\subset
T^{\operatorname{ext}}_{\mathcal{O}_{f}}$.

######  Remark.

The determinacy theorem is of central importance, mainly because it connects
the size of the tangent space of $f$, with $f$ being $\Gamma$-finitely-
determined. As a consequence, this theorem will allow us further to work in
finite dimensional jet spaces for germs which are $\Gamma$-finitely-
determined, and hence enabling to use important tools that comes from finite-
dimensional differential geometry.

##  3 $\Gamma$-invariant unfoldings

In this section we properly define the families of germs that will be relevant
for the theory here proposed. In particular, we will be interested in families
– also called ufoldings – whose elements are $\Gamma$-invariant.

######  Definition 3.1.

A $\Gamma$-invariant $r$-unfolding of a germ $f\in\mathcal{E}_{n}(\Gamma)$ is
a germ $F\in\mathcal{E}_{n+r}$ such that $F(x,0)=f(x)$ and, for some
neighborhood $W$ of $0\in\mathbb{R}^{r}$, one has $x\mapsto
F(x,u)\in\mathcal{E}_{n}(\Gamma)$ for all $u\in W$. The set of all
$\Gamma$-invariant $r$-unfoldings will be from now on denoted by
$\mathcal{U}_{n,r}(\Gamma)$.

The following definition is of central importance for the applications of
catastrophe theory that we are concerned, since it states what classes of
unfoldings we will call equivalent to each other. For our purposes, we need
equivalent unfoldings preserving the minima and the critical points of our
family while also preserving the assumed symmetry. Therefore, we allow for a
$\Gamma$-equivariant diffeomorphism that connects the variables, another that
connects the parameters, and a possibly third one which represents only a
constant shift.

######  Definition 3.2.

Let $f\in\mathcal{E}_{n}(\Gamma)$, $F$ be a $\Gamma$-invariant $r$-unfolding
of $f$ and $G$ be a $\Gamma$-invariant $d$-unfolding of $f$. We say that $G$
is induced from $F$ if there exists:

  1. i.)

a germ $\phi\in\vec{\mathcal{M}}_{n+d,n}$, where
$x\mapsto\phi(x,u)\in\vec{\mathcal{E}}_{n}(\Gamma)$ for $u$ in some
neighborhood of $0\in\mathbb{R}^{d}$.

  2. ii.)

a germ $\psi\in\vec{\mathcal{M}}_{d,r}$,

  3. iii.)

a germ $h\in\mathcal{M}_{d}$,

such that

$G(x,u)=F(\phi(x,u),\psi(u))+h(u).$

If $r=d$ and $\psi$ is a germ diffeomorphism, $F$ and $G$ are said to be
$\Gamma$-equivalent. Moreover, if every $\Gamma$-invariant unfolding of $f$ is
induced from $F$, $F$ is said to be versal.

######  Proposition 3.3.

Let $F_{1},F_{2},F_{3}$ be $\Gamma$-invariant $r_{1}$-, $r_{2}$-,
$r_{3}$-unfoldings, respectively, such that $F_{1}$ is induced from $F_{2}$
and $F_{2}$ is induced from $F_{3}$. Then, $F_{1}$ is induced from $F_{3}$.

###### Proof.

Let $\phi_{i}$, $\psi_{i}$ and $h_{i}$, $i=1,2$, be the corresponding germs of
Definition 3.2, satisfying, for $u_{i}\in\mathbb{R}^{r_{i}}$,

$F_{i}(x,u_{i})=F_{i+1}(\phi_{i}(x,u_{i})),\psi_{i}(u_{i}))+h_{i}(u_{i}),\quad
i=1,2.$

Defining

$\displaystyle\phi_{3}(x,u_{1})=\phi_{2}(\phi_{1}(x,u_{1}),\psi_{1}(u_{1})),\quad\psi_{3}(u_{1})=\psi_{2}(\psi_{1}(u_{1})),\;\text{
and}$ $\displaystyle h_{3}(u_{1})=h_{1}(u_{1})+h_{2}(\psi(u_{1})),$

it follows that $\phi_{3},\psi_{3},h_{3}$ satisfy the conditions (a), (b) and
(c), of Definition 3.2, with $d=r_{1}$ and $r=r_{3}$. Moreover, one has

$F_{1}(x,u_{1})=F_{3}(\phi_{3}(x,u_{1}),\psi_{3}(u_{1}))+h_{3}(u_{1}).$

Hence, $F_{1}$ is induced from $F_{3}$.

######  Remark.

Note that, by the transitivity just stated above, it follows that the set of
unfoldings that are $\Gamma$-equivalent to each other, according to Definition
3.2, defines an equivalence class in $\mathcal{U}_{n,r}(\Gamma)$.

###  3.1 The tangent space generated by unfoldings

We now look at the unfoldings of a germ $f\in\mathcal{M}_{n}(\Gamma)$ as a
parametric family of germs in $\mathcal{M}_{n}(\Gamma)$, and analize its
generated tangent space. Let $F\in\mathcal{U}_{n,r}(\Gamma)$ unfold
$f\in\mathcal{M}_{n}(\Gamma)$, and Let $\mathbb{R}_{\Gamma}^{n}$ be the set of
all points invariant under the action of $\Gamma$, i.e.,

$\mathbb{R}^{n}_{\Gamma}=\\{x\in\mathbb{R}^{n}\;|\;\gamma\cdot x=x\text{ for
all }\gamma\in\Gamma\\}.$

Consider the mapping
$S_{F}:\mathbb{R}^{n}_{\Gamma}\times\mathbb{R}^{r}\rightarrow\mathcal{M}_{n}(\Gamma)$
defined by

$S_{F}(x,u)(x^{\prime})\doteq F(x+x^{\prime},u)-F(x,u).$

Let $(v_{1},v_{2})$ be arbitrary vectors in
$\mathbb{R}^{n}_{\Gamma}\times\mathbb{R}^{r}$. Then, the tangent vector of the
differential of $S_{F}$ with respect to the curve
$(\eta_{1}(t),\eta_{2}(t))=(tv_{1},tv_{2})$ is given by

$\frac{\partial S_{F}(\eta_{1}(t),\eta_{2}(t))(x^{\prime})}{\partial
t}|_{t=0}=v_{1}\cdot(\vec{\nabla}f(x^{\prime})-\vec{\nabla}f(0))+v_{2}\cdot(\vec{\nabla}_{u}F(x^{\prime},0)-\vec{\nabla}_{u}F(0,0)).$

Hence, the tangent space associated with the differential of $S_{F}$ is then
given by

$\\{v_{1}\cdot(\vec{\nabla}f(x)-\vec{\nabla}f(0))+v_{2}\cdot(\vec{\nabla}_{u}F(x,0)-\vec{\nabla}_{u}F(0,0))\;|\;(v_{1},v_{2})\in\mathbb{R}^{n}_{\Gamma}\times\mathbb{R}^{r}\\}\in\mathcal{M}_{n}(\Gamma).$

######  Definition 3.4.

Let $F\in\mathcal{U}_{n,r}(\Gamma)$ be a $\Gamma$-invariant unfolding of some
$f\in\mathcal{M}_{n}(\Gamma)$.

$T_{S_{F}}\doteq\\{v_{1}\cdot(\vec{\nabla}f(x)-\vec{\nabla}f(0))+v_{2}\cdot(\vec{\nabla}_{u}F(x,0)-\vec{\nabla}_{u}F(0,0))\;|\;(v_{1},v_{2})\in\mathbb{R}^{n}_{\Gamma}\times\mathbb{R}^{r}\\},$
(22)

where

$\mathbb{R}^{n}_{\Gamma}=\\{x\in\mathbb{R}^{n}\;|\;\gamma\cdot x=x\text{ for
all }\gamma\in\Gamma\\}.$ (23)

Moreover, we say that $F$ is $\Gamma$-transversal if $T_{S_{F}}$ is
transversal to $T_{\mathcal{O}_{f}}$ in $\mathcal{M}_{n}(\Gamma)$, i.e., if

$\mathcal{M}_{n}(\Gamma)=T_{\mathcal{O}_{f}}+T_{S_{F}}.$ (24)

We now introduce the following notation:

######  Definition 3.5.

Let $R$ be a ring and $M$ an $R$-module. Given $v_{1},\dots,v_{n}\in M$, We
denote by

$\langle v_{1},\dots,v_{n}\rangle_{R}$ (25)

the $R$-sub-module of $M$ generated by $v_{1},\dots,v_{n}$.

######  Proposition 3.6.

Let $F\in\mathcal{U}_{n,r}(\Gamma)$ be a $\Gamma$-invariant unfolding of some
$f\in\mathcal{M}_{n}(\Gamma)$. Then, $F$ is $\Gamma$-transversal if and only
if

$\mathcal{E}_{n}(\Gamma)=T^{\operatorname{ext}}_{\mathcal{O}_{f}}+\langle
1,\alpha_{1}(F),\dots,\alpha_{r}(F)\rangle_{\mathbb{R}},$ (26)

where

$\alpha_{i}(F)\doteq\frac{\partial F(x,0)}{\partial u_{i}}.$ (27)

###### Proof.

Note that $\mathcal{E}_{n}(\Gamma)\cong\mathcal{M}_{n}(\Gamma)+\mathbb{R}$,
and

$\\{v\cdot\vec{\nabla}f\;|\;v\in\mathbb{R}^{n}_{\Gamma}\\}+T_{\mathcal{O}_{f}}=\\{\phi\cdot\vec{\nabla}f\;|\;\phi\in\vec{\mathcal{E}}_{n}(\Gamma)\\}=T^{\operatorname{ext}}_{\mathcal{O}_{f}}.$
(28)

Hence, adding $\mathbb{R}$ to both sides of Equation (24), since we have the
inclusions
$\\{v\cdot\vec{\nabla}f(0)\;|\;v\in\mathbb{R}^{n}_{\Gamma}\\}\subset\mathbb{R}$
and
$\\{v\cdot\vec{\nabla}_{u}F(0,0)\;|\;v\in\mathbb{R}^{n}\\}\subset\mathbb{R}$,
it is not hard to see that one may arrive at

$\mathcal{E}_{n}(\Gamma)=T^{\operatorname{ext}}_{\mathcal{O}_{f}}+\langle
1,\alpha_{1}(F),\dots,\alpha_{r}(F)\rangle_{\mathbb{R}}.$ (29)

###  3.2 Transversality theorem for $\Gamma$-invariant unfoldings

Here we present the main result of the section: namely the transversality
theorem for symmetric unfoldings. Such theorem is of fundamental importance,
since it connects the property of an unfolding being versal, i.e., of inducing
any other unfolding of the same germ, to the algebraic condition related to
the transversality (see Proposition 3.6), which in many cases can be
analytically checked.

######  Definition 3.7.

Let $F,G\in\mathcal{U}_{n,r}(\Gamma)$ be two $\Gamma$-transversal unfoldings
of the same germ $f\in\mathcal{M}_{n}(\Gamma)$. $F$ and $G$ are said to be
elementary $\Gamma$-homotopic when, for all $s\in[0,1]$, the unfolding

$H_{s}=sG+(1-s)F$

is $\Gamma$-transversal. Moreover, $F$ and $G$ are said to be
$\Gamma$-homotopic when there exists a finite sequence of $\Gamma$-invariant
unfoldings $F=F_{0},F_{1},\dots,F_{n-1},F_{n}=G$ such that $F_{i-1}$ and
$F_{i}$ are $\Gamma$-elementary homotopic for all $i=1,\dots,n$.

######  Theorem 3.8.

Let $F,G\in\mathcal{U}_{n,r}(\Gamma)$ be two $\Gamma$-transversal
$r$-unfoldings of the same germ $\eta\in\mathcal{M}_{n}(\Gamma)$. Then, $F$
and $G$ are $\Gamma$-equivalent.

###### Proof.

Recall Definition 3.2. First, it shall be shown that it is enough to prove the
theorem for the special case where $F$ and $G$ are $\Gamma$-elementary
homotopic. Let $k\in\mathbb{N}$ be such that
$\mathcal{M}^{k}_{n}(\Gamma)\subset T^{\operatorname{ext}}_{\mathcal{O}_{f}}$
(such $k$ always exists by Theorem 2.12). Define in $J_{n}^{k}(\Gamma)$ the
subspace

$A=j_{k}\left(T^{\operatorname{ext}}_{\mathcal{O}_{f}}\right).$ (30)

Note that $J_{n}^{k}(\Gamma)$ is isomorphic to an Euclidean space. Hence, let
$A^{\perp}$ be the orthogonal subspace of $A$, and let
$\\{z_{1},\dots,z_{q}\\}$ be a basis of $A^{\perp}$ (later we discuss the case
of $A^{\perp}=\\{0\\}$). Since $F$ is $\Gamma$-transversal by hypothesis,
there exists some $a\in A$ and some constants $a_{1},\dots,a_{r}\in\mathbb{R}$
such that

$z_{1}=a+\sum_{j=1}^{r}a_{j}j^{k}(\alpha_{j}(F)).$ (31)

Let $j_{1}$ be such that $a_{j_{1}}\neq 0$ in the above equation. Hence,
define a new unfolding $F_{1}$ given by

$F_{1}(x,u)=f(x)+u_{1}\alpha_{1}(F)+\dots+u_{j_{1}}z_{1}+\dots+u_{r}\alpha_{r}(F),$

if $a_{j_{1}}>0$, or

$F_{1}(x,u)=f(x)+u_{1}\alpha_{1}(F)+\dots-
u_{j_{1}}z_{1}+\dots+u_{r}\alpha_{r}(F),$

if $a_{j_{1}}<0$. Note that $F_{1}$ is also a $\Gamma$-invariant $r$-unfolding
of $f$. Moreover, $F$ and $F_{1}$ are $\Gamma$-elementary homotopic. Indeed,
for an unfolding $H_{s}=sF_{1}+(1-s)F$ where $s\in[0,1]$,
$j^{k}(\alpha_{j}(H_{s}))=j^{k}(\alpha_{j}(F))$ for all $j\neq j_{1}$, and

$j^{k}(\alpha_{j_{1}}(H_{s}))=sj^{k}(\alpha_{j_{1}}(F))+sgn(a_{j_{1}})(1-s)z_{1}.$

By Equation (31), $j^{k}(\alpha_{j_{1}}(F))$ can be expressed as

$\displaystyle j^{k}(\alpha_{j_{1}}(F))=$
$\displaystyle\frac{j^{k}(\alpha_{j_{1}}(H_{s}))-sgn(a_{j_{1}})\left(a+\sum_{\begin{subarray}{c}j=1\\\
j\neq
j_{1}\end{subarray}}^{r}a_{j_{1}}j_{k}(\alpha_{j}(F))\right)}{s+(1-s)|a_{j_{1}}|}$
$\displaystyle=$
$\displaystyle\frac{j^{k}(\alpha_{j_{1}}(H_{s}))-sgn(a_{j_{1}})\left(a+\sum_{\begin{subarray}{c}j=1\\\
j\neq
j_{1}\end{subarray}}^{r}a_{j_{1}}j^{k}(\alpha_{j}(H_{s}))\right)}{s+(1-s)|a_{j_{1}}|}.$

Hence, since $F$ is $\Gamma$-transversal, it is easy to see that that $H_{s}$
will also be $\Gamma$-transversal. Moreover, by a diffeomorphic change of
parameters $e\in\epsilon(r,r)$ given by

$e(u_{1},\dots,u_{j_{1}},\dots,u_{r})=(u_{j_{1}},u_{2},\dots,sgn(a_{j_{1}})u_{1},u_{j_{1}+1},\dots,u_{r}),$

one can obtain another $\Gamma$-transversal $r$-unfolding $\tilde{F}_{1}$ of
$f$ given by

$\displaystyle\tilde{F}_{1}(x,u)=F_{1}(x,e(u))=$ $\displaystyle
f(x)+u_{1}z_{1}+u_{2}\alpha_{2}(F)+\dots+u_{j_{1}}\alpha_{1}(F)+u_{j_{1}+1}\alpha_{j_{1}+1}(F)$
$\displaystyle+\dots+u_{r}\alpha_{r}(F),$

where $\tilde{F}_{1}$ is clearly $\Gamma$-equivalent to $F_{1}$. Now, consider
a $\Gamma$-transversal $r$-unfolding $F_{m}$ of $f$, where $m<q$, such that
$\alpha_{j}(F_{m})=z_{j}$ for $j\leq m$, and $\alpha_{j}(F_{m})=\alpha_{j}(F)$
for $j>m$. Then, since $F_{m}$ is $\Gamma$-transversal, it follows that,
similar to the above case, there exists some $a\in A$ and some constants
$a_{1},\dots,a_{r}\in\mathbb{R}$ such that

$z_{m+1}=a+\sum_{j=1}^{m}a_{j}z_{m}+\sum_{j=m+1}^{r}a_{j}j^{k}(\alpha_{j}(F)).$

Again, clearly $a_{j^{\prime}}\neq 0$ for some $j^{\prime}>m$. Defining a new
unfolding $F_{m+1}$ of $f$ by

$\displaystyle F_{m+1}(x,u)=$ $\displaystyle
f(x)+u_{1}z_{1}+\dots+u_{m}z_{m}+u_{m+1}\alpha_{m+1}(F_{m})+\dots+sgn(a_{j^{\prime}})u_{j^{\prime}}z_{m+1}$
$\displaystyle+u_{j^{\prime}+1}\alpha_{j^{\prime}+1}(F_{m})+\dots+u_{r}\alpha_{r}(F_{m}),$

and proceeding in a similar way to the above discussion, it is not hard to see
that $F_{m+1}$ is also $\Gamma$-transversal and elementary $\Gamma$-homotopic
to $F_{m}$. Moreover, as before, by the germ diffeomorphism
$e\in\epsilon(r,r)$

$e(u_{1},\dots,u_{r})=(u_{1},\dots,u_{m},u_{j^{\prime}},u_{m+2},\dots,u_{j^{\prime}-1},sgn(a_{j^{\prime}})u_{m+1},u_{j^{\prime}+1},\dots,u_{r}),$

one can find another unfolding $\tilde{F}_{m+1}$ of $f$ given by

$\displaystyle\tilde{F}_{m+1}(x,u)=F_{m+1}(x,e(u))=$ $\displaystyle
f(x)+u_{1}z_{1}+\dots+u_{m+1}z_{m+1}+u_{m+2}\alpha_{m+2}(F_{m})$
$\displaystyle+\dots+$ $\displaystyle
u_{j^{\prime}}\alpha_{m+1}(F_{m})+u_{j^{\prime}+1}\alpha_{j^{\prime}+1}(F_{m})+\dots+u_{r}\alpha_{r}(F_{m}),$

where $\tilde{F}_{m+1}$ is $\Gamma$-equivalent to $F_{m+1}$. By induction, it
follows that there exists a finite sequence of $\Gamma$-transversal unfoldings
of $f$, $F,F_{1},\tilde{F}_{q},\dots,F_{q},\tilde{F}_{n}$, where every
unfolding in the sequence is either $\Gamma$-equivalent or elementary
$\Gamma$-homotopic to the next, and $\tilde{F}_{q}$ is given by

$\tilde{F}_{n}(x,u)=f(x)+u_{1}z_{1}+\dots+u_{q}z_{q}+h(x,u_{q+1},\dots,u_{r}).$

By a straightforward calculation it also follows that $\tilde{F}_{q}$ is
elementary $\Gamma$-homotopic to the unfolding

$Y(x,u)=f(x)+u_{1}z_{1}+\dots+u_{q}z_{q}.$

Hence, for any $\Gamma$-transversal unfolding $F,G$ of $f$ with $r$
parameters, there exists a finite sequence of unfoldings
$F,F_{1},\tilde{F}_{1},\dots,F_{q},\tilde{F}_{q},Y,\tilde{G}_{q},G_{q},\dots,\tilde{G}_{1},G_{1},G$
such that every unfolding in the sequence is either $\Gamma$-equivalent or
elementary $\Gamma$-homotopic to the next. Consequently, If one proves that
elementary $\Gamma$-homotopic unfoldings are also $\Gamma$-equivalent, one
proves that $F$ is $\Gamma$-equivalent to $G$.

######  Remark.

If $A^{\perp}=\\{0\\}$ i.e., $\operatorname{cod}_{\Gamma}(f)=0$, then we take
$Y$ to be the trivial unfolding $Y(x,u)=f(x)$.

The idea of the proof is to show that, when $F$ and $G$ are elementary
$\Gamma$-homotopic, then for every $s\in[0,1]$, there is an open interval
$I_{s}$ of $[0,1]$ with $s\in I_{s}$ such that for all $s^{\prime}\in I_{s}$,
the unfolding

$H_{s^{\prime}}=s^{\prime}G+(1-s^{\prime})F$

is isomorphic to $H_{s}$. Assuming that this holds, then the union
$\cup_{s\in[0,1]}I_{s}$ is an open cover of $[0,1]$. But since $[0,1]$ is
compact, it follows that one may find a finite sequence
$\\{s_{0},s_{1},\dots,s_{M}\\}$ with $s_{0}=0$ and $s_{M}=1$ such that
$H_{0},H_{s_{1}},\dots,H_{s_{M-1}},H_{1}$ are all equivalent. But since
$H_{0}=F$ and $H_{1}=G$, this implies that $F$ and $G$ are equivalent. Hence,
let $F,G$ be two elementary homotopic even unfoldings of $f$, and $s\in(0,1)$.
Define

$H(x,u,t)=(s+t)G+(1-s-t)F.$

The goal is to show that for some sufficiently small $t\in\mathbb{R}$,
$H(x,u,t)$ is equivalent to $H(x,u,0)$. Note that there exits a neighborhood
$V$ of $0\in\mathbb{R}$ such that if $t\in V$, $(x,u)\mapsto H(x,u,t)$ is
$\Gamma$-transversal, since by hypothesis $F$ and $G$ are elementary
$\Gamma$-homotopic. Therefore, Corollary 6.6 applies, and it follows that

$\mathcal{U}_{n,r+1}(\Gamma)=\tilde{T}_{H}+\left\langle 1,\frac{\partial
H}{\partial u_{1}},\dots,\frac{\partial H}{\partial
u_{r}}\right\rangle_{\mathcal{E}_{r}}.$ (32)

Now, note that $H(x,u,t)\in\mathcal{U}_{n,r}(\Gamma)$ and

$\frac{\partial H(x,0,t)}{\partial t}=\frac{\partial(f(x))}{\partial t}=0.$
(33)

Hence, by lemma 6.2,

$\frac{\partial H}{\partial
t}\in\mathcal{M}_{r}\cdot\mathcal{U}_{n,r+1}(\Gamma).$ (34)

Multiplying Equation (32) by $\mathcal{M}_{r}$, it follows that there exists
some germs:

$\displaystyle\xi(x,u,t)\in\mathcal{M}_{r}\cdot\vec{\mathcal{E}}_{n+r+1,n},\;\text{
with }x\mapsto\xi(x,u,t)\in\vec{\mathcal{E}}_{n}(\Gamma)\text{ for $(u,t)$ in
some neighborhood of $0\in\mathbb{R}^{r+1}$},$
$\displaystyle\chi(u,t)\in\mathcal{M}_{r}\cdot\vec{\mathcal{E}}_{r+1,r},\;\text{
and}$ $\displaystyle\chi_{0}(u,t)\in\mathcal{M}_{r}\cdot\mathcal{E}_{r+1}.$

Such that

$\frac{\partial H}{\partial
t}(x,u,t)=\xi(x,u,t)\cdot\vec{\nabla}_{x}H(x,u,t)+\chi(u,t)\cdot\vec{\nabla}_{u}H(x,u,t)+\chi_{0}(u,t).$
(35)

Applying the geometric lemma (Lemma 6.1) for
$H\in\mathcal{U}_{n,r+1}(\Gamma)$, it follows that there exists some germs

$\displaystyle\phi(x,u,t)\in\vec{\mathcal{E}}_{n+r+1,n},\;\text{ with
}x\mapsto\xi(x,u,t)\in\vec{\mathcal{E}}_{n}(\Gamma)\text{ for $(u,t)$ in some
neighborhood of $0\in\mathbb{R}^{r+1}$},$
$\displaystyle\psi(u,t)\in\vec{\mathcal{E}}_{r+1,r},\;\text{ and}$
$\displaystyle h(u,t)\in\mathcal{E}_{r+1}$

such that

$H(x,u,0)=H(\phi(x,u,t),\psi(u,t),t)+h(u,t)$ (36)

for small enough $t\in\mathbb{R}$. Note also that, by definition of the germs
$\phi,\psi$ and $h$, they in particular satisfy the following system of
O.D.Es. (for more details see the proof of Lemma 6.1)

$\begin{cases}\frac{\partial\phi(x,0,t)}{\partial
t}=-\xi(\phi(x,0,t),0,t),\quad\phi(x,0,0)=x,\\\
\frac{\partial\psi(0,t)}{\partial
t}=-\chi(\psi(0,t),t),\quad\quad\quad\>\psi(0,0)=0,\\\ \frac{\partial
h(0,t)}{\partial
t}=-\chi_{0}(0,t),\quad\quad\quad\quad\quad\,h(0,0)=0,\end{cases}$ (37)

for small enough $t\in\mathbb{R}$. Since
$\xi(x,0,t)=\chi(0,t)=\chi_{0}(0,t)=0$, the unique solution of the above
system of O.D.Es. is

$\phi(x,0,t)=x,\;\;\psi(0,t)=0,\;\;h(0,t)=0.$ (38)

In particular, this and ¡equation (35) together imply that $H(x,u,t)$ is
equivalent to $H(x,u,0)$ for small enough $t\in\mathbb{R}$.

######  Theorem 3.9 (Transversality theorem for $\Gamma$-invariant
unfoldings).

Let $F\in\mathcal{U}_{n,r}(\Gamma)$ be a $\Gamma$-equivariant $r$-unfolding of
$f\in\mathcal{M}_{x}(\Gamma)$. Then, the following are equivalent:

  1. i.)

$F$ is versal,

  2. ii.)

$F$ is $\Gamma$-transversal, i.e.,
$\mathcal{E}_{x}(\Gamma)=T^{\operatorname{ext}}_{\mathcal{O}_{f}}+\left\langle
1,\alpha_{1}(F),\dots,\alpha_{r}(F)\right\rangle_{\mathbb{R}}$.

###### Proof.

$i)\implies ii)$. Let $p\in\mathcal{E}_{x}(\Gamma)$ be an arbitrary
$\Gamma$-invariant germ, and consider the $1$-unfolding $G(x,t)=f(x)+tp(x)$.
Clearly $G$ is a $\Gamma$-invariant unfolding of $f$, and if $F$ is versal,
then by hypothesis there exists some germs $\phi,\psi$ and $h$ such that

$G(x,t)=f(x)+tp(x)=F(\phi(x,t),\psi(t))+h(t).$

Differentiating with respect to $t$ and taking $t=0$, we arrive at

$p(x)=\frac{\partial\phi(x,0)}{\partial
t}\cdot\vec{\nabla}_{x}f(x)+\sum_{i}\frac{\partial F(x,0)}{\partial
u_{i}}\frac{\partial\psi_{i}(0)}{\partial t}+\frac{\partial h(0)}{\partial
t}.$

By the assumed $\Gamma$-equivariance of $t\mapsto\phi(x,t)$ in a suitable
neighborhood of $0\in\mathbb{R}$, one has

$\frac{\partial\phi(x,0)}{\partial t}\cdot\vec{\nabla}_{x}f(x)\in
T_{\mathcal{O}_{f}},$ (39)

In particular,

$p\in T^{\operatorname{ext}}_{\mathcal{O}_{f}}+\left\langle
1,\alpha_{1}(F),\dots,\alpha_{r}(F)\right\rangle_{\mathbb{R}},$ (40)

hence $ii)$ follows.

$ii)\implies i)$. Let $F$ be a $\Gamma$-transversal $r$-unfolding of $f$, and
let $G$ be an arbitrary $\Gamma$-invariant $d$-unfolding of $f$. Define two
$(r+d)$-unfoldings of $f$ by

$H(x,u,v)=F(x,u)+G(x,v)-f(x),\;\text{and }\;K(x,u,v)=F(x,u).$

Clearly, $H$ and $K$ are $\Gamma$-transversal, as $F$ is $\Gamma$-transversal.
Hence, by Proposition 3.8, $H$ and $K$ are $\Gamma$-equivalent. Moreover, $G$
is induced from $H$, by the germs $\phi(x,u)=x$, $\psi(u,v)=u$,
$\gamma(u,v)=0$, and $K$ is induced from $F$, by the germs $\phi(x,u)=x$,
$\psi(v)=(0,v)$, $\gamma(v)=0$. Therefore, by Proposition 3.3, $G$ is induced
from $F$, i.e., $F$ is versal.

######  Remark.

In view of the definition of $\Gamma$-codimension (Definition 2.7) and the
algebraic condition of transversality (Equation (26)), it clearly follows that
the minimum possible number of parameters that a $\Gamma$-invariant
$r$-unfolding $F$ of a germ $f$ needs to have in order to be
$\Gamma$-transversal is $r=\operatorname{cod}_{\Gamma}(f)$. By
thetTransversality theorem, one may also state that this is also the minimum
possible number of parameters for $F$ to be versal. Such unfoldings which are
versal/transversal while having the minimum number of parameters
$r=\operatorname{cod}_{\Gamma}(f)$ are usually referred to in the literature
as universal unfoldings.

###  3.3 Stability theorem for $\Gamma$-invariant unfoldings

We now move on to the proof of the stability theorem. In short, the theorem
states that any perturbation added to a transversal unfolding, if small enough
(in a suitable topology), does not break its transversality, and in fact the
perturbed unfolding is equivalent to the original unfolding. For the case of
$\Gamma$-invariant unfoldings, we must be cautious with the kind of
perturbations that we allow. In general, transversal $\Gamma$-invariant
unfoldings are not stable when the perturbation breaks the assumed symmetry of
the unfolding. Therefore, here we analize the situation for symmetry-
preserving perturbations. First, we start by defining a suitable topology for
the perturbations.

######  Definition 3.10.

Let $U$ be a neighborhood of $0\in\mathbb{R}^{n}\times\mathbb{R}^{r}$. Define
$C_{\Gamma}^{\infty}(U,\mathbb{R})$ as the set of all infinitely
differentiable functions $F:U\rightarrow\mathbb{R}$ such that $x\mapsto
F(x,u)$ is $\Gamma$-invariant for all $u$ such that $(0,u)\in U$. The topology
considered in $C_{\Gamma}^{\infty}(U,\mathbb{R})$ is the so-called
$C^{\infty}$-topology, whose local basis at $0$ is the union of all sets

$V(L,k,\epsilon)=\left\\{F\in
C_{\Gamma}^{\infty}(U,\mathbb{R})\;|\;\sum_{|\alpha|\leq
k}\frac{1}{\alpha!}\sup_{(x,u)\in
L}\left|\frac{\partial^{|\alpha|}(F)(x,u)}{\partial
x_{1}^{\alpha_{1}}\cdots\partial
x_{n}^{\alpha_{n}}}\right|<\epsilon\right\\},$

where $L\subset U$ is compact, $k\in\mathbb{N}$ and $\epsilon>0$.

######  Remark.

Note that in such topology, by definition, for any $k\in\mathbb{N}$, there
exists a neighborhood $V$ of $0$ such that the derivatives of all of the
elements in $V$ up to order $k$ are as small as one wants. This property will
be very important in the proof of the stability theorem, as we will need to
control all the derivatives of the perturbations up to some finite order $k$.

An arbitrary perturbation added to an unfolding may also have the effect of
dislocating the point of singularity of the initial germ. Therefore, We need
to modify our equivalence relation, to include also diffeomorphisms that does
not preserve the origin.

######  Definition 3.11.

Let $U,V$ be open subsets of $\mathbb{R}^{n+r}$, $G:U\rightarrow\mathbb{R}$,
$F:V\rightarrow\mathbb{R}$ two smooth functions, and $x,y\in\mathbb{R}^{n}$
such that

$x=\gamma x\text{ for all }\gamma\in\Gamma,\quad\text{and}\quad y=\gamma
y\text{ for all }\gamma\in\Gamma.$ (41)

Then, $G$ at $(x,u)\in U$ is equivalent to $F$ at $(y,v)\in V$ if, for some
neighborhood $U_{1}$ of $\mathbb{R}^{n}$ and some neighborhood $U_{2}$ of
$\mathbb{R}^{r}$, such that $U_{1}\times U_{2}\subset U$, there exist smooth
functions

  1. i.)

$\phi:U_{1}\times U_{2}\rightarrow\mathbb{R}^{n}$, such that
$\phi(0,0)=x^{\prime}$ and $x\mapsto\phi(x,u)$ is a diffeomorphism for all
$u\in U_{2}$,

  2. ii.)

$\psi:U_{2}\rightarrow\mathbb{R}^{r}$, such that $\psi(0)=u^{\prime}$ and
$\psi$ is a diffeomprhism,

  3. iii.)

$h:U_{2}\rightarrow\mathbb{R}$,

such that, for all $(x^{\prime},u^{\prime})\in U_{1}\times U_{2}$, one has

$G(x^{\prime},u^{\prime})=F(\phi(x^{\prime},u^{\prime}),\psi(u^{\prime}))+\gamma(u^{\prime}).$

######  Definition 3.12.

Let $F\in\mathcal{U}_{n,r}(\Gamma)$, with representative $\tilde{F}$. $F$ is
said to be stable if for every open neighborhood $U$ of
$0\in\mathbb{R}^{n+r}$, there exists a neighborhood $W$ of $\tilde{F}$ in the
$C^{\infty}$-topology of $C^{\infty}_{\Gamma}(\mathbb{R}^{n+r},\mathbb{R})$
such that for every $\tilde{F}^{\prime}\in W$, there exists some
$(x^{\prime},u^{\prime})\in U^{\Gamma}$, where $U^{\Gamma}=\\{(x,u)\in
U\;|\;x=\gamma x\text{ for all }\gamma\in\Gamma\\}$, such that $F$ is
equivalent to $\tilde{F}^{\prime}$ at $(x^{\prime},u^{\prime})$; i.e., there
exists some neighbhood $U_{1}$ of $0\in\mathbb{R}^{n}$, some neighborhood
$U_{2}$ of $0\in\mathbb{R}^{r}$, and some smooth functions

  1. i.)

$\phi:U_{1}\times U_{2}\rightarrow\mathbb{R}^{n}$, such that
$\phi(0,0)=x^{\prime}$ and $x\mapsto\phi(x,u)$ is a diffeomorphism for all
$u\in U_{2}$,

  2. ii.)

$\psi:U_{2}\rightarrow\mathbb{R}^{r}$, such that $\psi(0)=u^{\prime}$ and
$\psi$ is a diffeomprhism,

  3. iii.)

$h:U_{2}\rightarrow\mathbb{R}$,

such that

$G(x,u)=\tilde{G}^{\prime}(\phi(x,u),\psi(u))+h(u).$

######  Definition 3.13.

Let $U=V\times W\subset\mathbb{R}^{n+r}$ be an open neighborhood of $0$ and
let $F\in C^{\infty}_{\Gamma}(U,\mathbb{R})$. For $k\in\mathbb{N}$, define
$j_{1}^{k}F:\mathbb{R}^{n}_{\Gamma}\times\mathbb{R}^{r}\rightarrow
J_{n}^{k}(\Gamma)$ by

$j_{1}^{k}F(x,u)(z)=\sum_{|\alpha|\leq k,\alpha\neq
0}\frac{1}{\alpha!}\frac{\partial^{|\alpha|}F(x,u)}{\partial
x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}z_{1}^{\alpha_{1}}\dots
z_{n}^{\alpha_{n}},$

i.e., $j_{1}^{k}F(x,u)(z)=j^{k}(F(x+z,u))-F(x,u)$.

######  Remark.

We recall that

$\mathbb{R}_{\Gamma}^{n}=\\{x\in\mathbb{R}^{n}\;|\;x=\gamma x\text{ for all
}\gamma\in\Gamma\\}.$ (42)

######  Theorem 3.14 (Stability theorem for $\Gamma$-invariant unfoldings).

Let $f\in\mathcal{M}_{n}(\Gamma)$, and let $F\in\mathcal{U}_{n,r}(\Gamma)$ be
a $\Gamma$-transversal unfolding of $f$. Then, $F$ is $\Gamma$-stable.

###### Proof.

Let $V,W$ be an open neighborhood of $0\in\mathbb{R}^{n}$, $\mathbb{R}^{r}$,
respectively, and $F^{\prime}$ a representative of $F$ defined on $U=V\times
W$. For $g\in C^{\infty}(U,\mathbb{R})$ and $w\in W$, define $\Xi:W\times
C^{\infty}(U,\mathbb{R})\rightarrow C^{\infty}(U,\mathbb{R})$ by

$\Xi(w,g)(x,u)=g(x,w+u)-g(0,w).$

Define also a function $\zeta:W\times C^{\infty}(U,\mathbb{R})\rightarrow
C^{\infty}(U,\mathbb{R})$ by

$\zeta(w,g)(x)=g(x,w)-g(0,w)=\Xi(w,g)(x,0).$

Now, note that if $G\in\mathcal{U}_{n,r}(\Gamma)$ is a $\Gamma$-invariant
unfolding of a germ $\mu\in\mathcal{M}_{n}(\Gamma)$ that is $\Gamma$-finitely-
determinate, then it follows that $G$ is $\Gamma$-transversal if and only if
$J_{n}^{k}(\Gamma)$ is generated by the elements:
$1,j^{k}(\alpha_{1}(G)),\dots,j^{k}(\alpha_{r}(G))$ and by a basis of
$j^{k}(T^{\operatorname{ext}}_{\mathcal{O}_{\eta}})$, for some
$k\in\mathbb{N}$ coming from item $ii.)$ of Theorem 2.12. Denote every element
in the above list, when taking the unfolding $F$, by
$\beta_{1}(F),\dots,\beta_{p}(F)$, in a way such that
$j^{k}(\beta_{1}(F)),\dots,j^{k}(\beta_{q}(F))$ $(q\leq p)$ form a basis of
$J_{n}^{k}(\Gamma)$. Then, choose a compact neighborhood $K$ of
$0\in\mathbb{R}^{r}$ such that for all $w\in K$:

  1. i.)

$j^{k}\zeta(w,F^{\prime})$ $\Gamma$-$k$-determinate, and

  2. ii.)

$j^{k}\beta_{1}(\Xi(w,F^{\prime})),\dots,j^{k}\beta_{q}(\Xi(w,F^{\prime}))$
form a basis of $J_{n}^{k}(\Gamma)$.

Let $c>0$ be such that $[-c,c]^{r}\subset K$, and define
$S\subset\mathbb{R}^{q}$ as the $q-1$ sphere of radius $c$ centered at $0$.
Then, by the continuity of the mapping $\mathfrak{i}:K\times S\rightarrow
J_{n}^{k}(\Gamma)$ given by

$\mathfrak{i}(w,s_{1},\dots,s_{q})=\sum_{i=1}^{q}s_{i}j^{k}\beta_{i}(\Xi(w,F^{\prime})),$

and the compactness of $K\times S$, it follows that there exists some
$\delta>0$ such that

$\left\|\sum_{i=1}^{q}s_{i}j^{k}\beta_{i}(\Xi(w,F^{\prime}))\right\|\geq\delta$

for all $w\in K$ and $(s_{1},\dots,s_{q})\in S$. Moreover, for any $h\in
C_{\Gamma}^{\infty}(U,\mathbb{R})$ with derivatives up to order $k+1$ small
enough in $\\{0\\}\times K$, it also follows that

$\displaystyle\left\|\sum_{i=1}^{q}s_{i}j^{k}\beta_{i}(\Xi(w,F^{\prime}+th))\right\|\geq\frac{\delta}{2}$
(43)

for all $w\in W$, $(s_{1},\dots,s_{q})\in S$ and $t\in[0,1]$. For $h\in
C_{\Gamma}^{\infty}(U,\mathbb{R})$ with derivatives up to order $k+1$ small
enough in $\\{0\\}\times K$, one also has

$\|j^{k}h(\cdot,w)\|<\frac{\delta}{2}\;\text{ for all }w\in K.$ (44)

Hence, suppose that $h$ is in a convenient neighborhood of $0$ in
$C_{\Gamma}^{\infty}(U,\mathbb{R})$ such that Equations (43) and (44) are
satisfied. For $w\in K$ and $a\in[0,1]$, define
$H\in\mathcal{U}_{n,r+1}(\Gamma)$ by

$H(x,u,t)=(F^{\prime}+(a+t)h)(x,w+u)-(F^{\prime}+ah)(0,w).$

The strategy of the proof is very similar to the one of Theorem 3.8, i.e., we
will prove that $H(x,u,0)$ is $\Gamma$-equivalent to $H(x,u,t)$ for $t$ in
some neighborhood of $0\in\mathbb{R}$, and this concludes the theorem. Note
that $H(x,0,0)=f=\zeta(w,F^{\prime}+ah)$ and $H(x,u,0)=\Xi(w,F^{\prime}+ah)$.
Therefore, since Equation (43) implies that the set
$\\{j^{k}\beta_{1}(\Xi(w,F^{\prime}+ah)),\dots,j^{k}\beta_{q}(\Xi(w,F^{\prime}+ah))\\}$
is linearly iindependent (and hence forms a basis of $J_{n}^{k}(\Gamma)$), it
follows that $H$ is a $\Gamma$-transversal unfolding of $f$. in particular,
one has, by Corollary 6.6

$\mathcal{U}_{n,r+1}(\Gamma)=\tilde{T}_{H}+\left\langle 1,\frac{\partial
H}{\partial u_{1}},\dots,\frac{\partial H}{\partial
u_{r}}\right\rangle_{\mathcal{E}_{r}}.$ (45)

Now note that $\frac{\partial H(x,u,t)}{\partial t}=h(x,w+u)$, and hence
$\frac{\partial H(x,0,0)}{\partial t}=h(x,w)$. Since
$\\{j^{k}\beta_{1}(\Xi(w,F^{\prime}+ah)),\dots,j^{k}{k}\beta_{q}(\Xi(w,F^{\prime}+ah))\\}$
is a basis of $J_{n}^{k}(\Gamma)$, one has

$\frac{\partial H(x,0,0)}{\partial
t}=\sum_{i=1}^{q}c_{i}\beta_{i}(\Xi(w,F^{\prime}+ah))+g,$ (46)

where $g\in\mathcal{M}_{n}^{k+1}(\Gamma)$. Assume that $|c_{j}|\geq c$ for
some $j$. Then, define $z=\frac{c^{2}}{c_{1}^{2}+\dots+c_{q}^{2}}<1$. Note
that $(zc_{1})^{2}+\dots+(zc_{q})^{2}=c^{2}$, i.e., $(zc_{1},\dots,zc_{q})\in
S$. But by Equation (43) it follows that

$\left\|\sum_{i=1}^{q}c_{i}j^{k}(\beta_{i}(\Xi(w,F^{\prime}+ah)))\right\|=\frac{1}{z}\left\|\sum_{i=1}^{q}zc_{i}j^{k}(\beta_{i}(\Xi(w,F^{\prime}+ah)))\right\|\geq\frac{\delta}{2},$

which contradicts Equation (44). Hence, $|c_{i}|<c$ for all $i$. Therefore, by
Equation (46), one has (since
$\alpha_{i}(\Xi(w,F^{\prime}+ah))=\alpha_{i}(H)$)

$\frac{\partial H(x,0,0)}{\partial t}=\sum_{i=1}^{n}\frac{\partial f}{\partial
x_{i}}\xi_{i}+\sum_{i=1}^{r}b_{i}\alpha_{i}(H)+b_{0}+g,$

for some $b_{0},\dots,b_{r}\in\mathbb{R}$ and some germ
$\xi=(\xi_{1},\dots,\xi_{n})\in\vec{\mathcal{E}}_{n}(\Gamma)$, where $\xi_{i}$
is odd in $x_{i}$ and even in the other variables. Moreover, they also satisfy
$|b_{0}|,\dots,|b_{r}|<c$ and $|\xi_{i}(0)|<c$. Since
$\mathcal{M}_{n}^{k}(\Gamma)\subset T^{\operatorname{ext}}_{\mathcal{O}_{f}}$,
we have

$g=\sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}\eta_{i},$

where $\eta\in\vec{\mathcal{E}}_{n}(\Gamma)$. Therefore,

$\frac{\partial H(x,0,0)}{\partial t}=\sum_{i=1}^{n}\frac{\partial f}{\partial
x_{i}}\tilde{\xi}_{i}+\sum_{i=1}^{r}b_{i}\alpha_{i}(H)+b_{0},$

where $\tilde{\xi}_{i}=\xi_{i}+g_{i}$. Now, define

$\mu=\frac{\partial H}{\partial t}-\sum_{i=1}^{n}\tilde{\xi}_{i}\frac{\partial
H}{\partial x_{i}}-\sum_{i=1}^{r}b_{i}\frac{\partial H}{\partial
u_{i}}-b_{0}.$ (47)

Clearly $\mu(x,0,0)=0$ and hence
$\mu\in\mathcal{M}_{r+1}\cdot\mathcal{U}_{n,r+1}(\Gamma)$. Therefore, by
Equation (45), it follows that

$\mu=\sum_{i=1}^{n}\xi^{\prime}_{i}\frac{\partial H}{\partial
x_{i}}+\sum_{i=1}^{r}\chi_{i}\frac{\partial H}{\partial
u_{i}}+\chi_{r+1}\frac{\partial H}{\partial t}+\chi_{0}.$ (48)

From Equations (47) and (48) one arrives at

$\frac{\partial H}{\partial t}=\sum_{i=1}^{n}\rho^{\prime}_{i}\frac{\partial
H}{\partial x_{i}}+\sum_{i=1}^{r}\chi_{i}^{\prime}\frac{\partial H}{\partial
u_{i}}+\chi_{0}^{\prime},$

where $\rho^{\prime}_{i}=(\tilde{\xi}_{i}+\xi^{\prime}_{i})/(1-\chi_{r+1})$
and $\chi_{i}^{\prime}=(b_{i}+\chi_{i})/(1-\chi_{r+1})$. Moreover, it follows
that $|\gamma_{i}^{\prime}(0)|<c$, $|\chi_{i}^{\prime}(0)|<c$. By Lemma 6.1,
it follows that there are some germs $\phi$, $\psi$ and $\lambda$ such that

$\begin{cases}\frac{\partial\phi(x,0,t)}{\partial
t}=-\rho^{\prime}(\phi(x,0,t),0,t),\quad\phi(x,u,0)=x,\\\
\frac{\partial\psi(0,t)}{\partial
t}=-\chi^{\prime}(\psi(0,t),t),\quad\quad\quad\>\psi(u,0)=u,\\\
\frac{\partial\lambda(0,t)}{\partial
t}=-\chi^{\prime}_{0}(0,t),\quad\quad\quad\quad\quad\,\lambda(u,0)=0,\end{cases}$
(49)

Satisfying

$H(\phi(x,u,t),\psi(x,u,t),t)+\lambda(u,t)=H(x,u,0).$ (50)

I.e., $f^{\prime}+ah$ at $(0,w)$ is equivalent to $f^{\prime}+(a+t)h$ at
$(0,w+\psi_{t}(0))$.

##  4 Classification theorem for $\mathbb{Z}_{2}$-invariant germs

In this section we prove a classification theorem for
$\mathbb{Z}_{2}$-invariant unfoldings, similar to Thom’s classification
theorem for the usual catastrophes. We have already shown that
$\Gamma$-transversal unfoldings of the same germ and with the same number of
parameters are $\Gamma$-equivalent. The classification theorem, in turn,
states that $\Gamma$-transversal unfoldings are equivalent to certain
polynomial unfoldings, which can be more easily analyzed. One of the most
important results to prove the classification theorem is the following version
of Morse lemma, stating the existence of a symmetry-preserving diffeomorphism
that reduces the $\Gamma$-invariant functions near a non-degenarate critical
point to a quadratic polynomial.

######  Theorem 4.1 (Morse Lemma for $\Gamma$-invariant unfoldings).

Let $f\in\mathcal{E}_{n}(\Gamma)$ have a nondegenerate critical point at the
origin, with $f(0)=0$. Then, $f$ is equivalent to its quadratic part.

###### Proof.

Let $f(x)=h(x)+g(x)$, where $h$ is its quadratic part and $g$ is of third
order. Our strategy is to prove the existence of an homotopy between $f$ and
$h$; we will prove the existence of a mapping $t\mapsto\phi_{t}\in
L_{n}(\Gamma)$ with $\phi_{0}(x)=x$ such that

$(h+tg)(\phi_{t}(x))=h(x),\;t\in[0,1].$

Applying the classical Morse lemma to $f+tg$, it follows that $f+tg$ is
equivalent (via a possibly non-$\Gamma$-equivariant family of diffeomorphism
smooth in $t$) to $\pm x_{1}^{2}\dots\pm x_{n}^{2}$, and consequently it
follows that

$\mathcal{M}_{n}^{2}=\\{\phi\cdot\vec{\nabla}_{x}(h+tg)\;|\;\phi\in\mathcal{M}_{n}\\}.$
(51)

In particular, since $g$ is of third order, there exists some
$\xi_{t}\in\mathcal{M}_{n}$ satisfying

$\xi_{t}(x)\cdot\vec{\nabla}\left(h(x)+tg(x)\right)=-g(x),$ (52)

Now, define

$\eta_{t}(x)\doteq\int_{\Gamma}\gamma\xi_{t}(\gamma^{-1}x)\mathrm{d}\gamma,$
(53)

where the above integral is with respect to the Haar measure of $\Gamma$.
Then, it follows that $\eta_{t}$ is $\Gamma$-equivariant and $\eta_{t}(0)=0$.
Hence, taking $\phi_{t}$ as the solution of

$\begin{cases}\partial_{t}\phi_{t}(x)=\eta_{t}(\phi_{t}(x)),\\\
\phi_{0}(x)=x,\end{cases}$ (54)

the lemma follows.

######  Proposition 4.2.

Let $F\in\mathcal{U}_{n,r}(\Gamma)$ be a $\Gamma$-transversal unfolding of a
germ $f\in\mathcal{M}_{n}(\Gamma)$, and $\phi\in L_{n}(\Gamma)$. Then, the
unfolding $G(x,u)=F(\phi(x),u)$ is a $\Gamma$-transversal unfolding of the
germ $g=f\circ\phi$.

###### Proof.

Let $\phi^{*}:\mathcal{E}_{n}(\Gamma)\rightarrow\mathcal{E}_{n}(\Gamma)$ be
given by

$\phi^{*}h=h\circ\phi.$ (55)

Note that, since $\phi$ is a diffeomorphism, $\phi^{*}$ is an isomorphism; in
particular, $\phi^{*}(\mathcal{E}_{n}(\Gamma))=\mathcal{E}_{n}(\Gamma)$.
Hence, one has

$\mathcal{E}_{n}(\Gamma)=\phi^{*}\left(T^{\operatorname{ext}}_{\mathcal{O}_{f}}+\langle
1,\alpha_{1}(F),\dots,\alpha_{r}(F)\rangle_{\mathbb{R}}\right)=T^{\operatorname{ext}}_{\mathcal{O}_{f\circ\phi}}+\langle
1,\alpha_{1}(F\circ\phi),\dots,\alpha_{r}(F\circ\phi)\rangle_{\mathbb{R}},$

i.e., $G(x,u)=F(\phi(x),u)$ is $\Gamma$-transversal.

######  Remark.

From now on, we consider $\Gamma=\mathbb{Z}_{2}$, where $\mathbb{Z}_{2}$ acts
on $\mathbb{R}^{n}$ in the following way:

$\gamma.x=(\gamma_{1}x_{1},\dots,\gamma_{n}x_{n}),\quad\gamma_{i}=\pm 1.$

We note that the germs which are invariant under the action of
$\mathbb{Z}_{2}$ are the ones possessing parity symmetry.

######  Lemma 4.3 (Splitting Lemma for even functions).

Let $f\in\mathcal{M}_{n}^{2}(\mathbb{Z}_{2})$, with Hessian of corank $r$ at
$0$. Then, there exists some $\phi\in L_{n}(\mathbb{Z}_{2})$ such that

$f\circ\phi(x)=\hat{f}(x_{1},\dots,x_{r})\pm x_{r+1}^{2}\pm\dots\pm
x_{n}^{2},$

where
$\hat{f}(x_{1},\dots,x_{r})=f(x_{1},\dots,x_{r},0)\in\mathcal{M}_{r}^{4}(\mathbb{Z}_{2})$.

###### Proof.

Since $f\in\mathcal{M}_{n}(\mathbb{Z}_{2})$ and its Hessian at $0$ has corank
$r$, then without loss of generality, we can assume that in a suitable
neighborhood $V$ of $0\in\mathbb{R}^{r}$, for every $x\in V$ the Hessian of
$f$ at $(x,0)$ is a matrix given by

$Hf(x,0)=\begin{pmatrix}\begin{matrix}\\\
&\mbox{\Large$H_{x,x}f(x,0)$}\end{matrix}&\vline&\mbox{\Large$0$}\\\
\hline\cr\mbox{\Large$0$}&\vline&\begin{matrix}\pm a_{1}(x)&\cdots&0\\\
\vdots&\ddots&\vdots\\\ 0&\cdots&\pm a_{n-r}(x)\end{matrix}\end{pmatrix},$

where $H_{x,x}f(0,0)$ is denegerate, $a_{i}$ is even with respect to
$x_{1},\dots,x_{r}$ for all $i$ and $a_{i}(x)>0$ for all $i$ and all $x\in V$.
In particular, by Taylor’s theorem, $f$ can be written locally as

$f(x,y)=f(x,0)\pm y_{1}^{2}(a_{1}(x)+g_{1}(x,y))\pm\dots\pm
y_{n-r}^{2}(a_{n-r}(x)+g_{n-r}(x,y)),\;\;x\in\mathbb{R}^{r},y\in\mathbb{R}^{n-r},$

where $g_{i}$ is even in all variables for all $i$ and $g_{i}(0,0)=0$ for all
$i$. Therefore, defining

$\tilde{\phi}(x,y)=(x,y_{1}\sqrt{a_{1}(x)+g_{1}(x,y)},\dots,y_{n-r}\sqrt{a_{n-r}(x)+g_{n-r}(x,y)})\in
L_{n}(\mathbb{Z}_{2}),$

one has

$f\circ\phi^{-1}(x,y)=f(x,0)\pm y_{1}^{2}\pm\dots\pm y_{n-r}^{2},$

and hence, by applying Theorem 4.1, the result follows.

Now we are in position to prove the classification theorems for the case of
symmetric germs of funtions. We start with the classification theorem for the
initial germs, associated with the equivalence relation that Definition 2.3
gives rise to. In order to prove it, we take advantage of the classification
theorem exposed in [9] for germs without symmetry.

######  Theorem 4.4.

Let $f\in\mathcal{M}_{n}^{2}(\mathbb{Z}_{2})$, with
$\operatorname{cod}(f)<\infty$ and Hessian of corank $r$ at $0$. Then,
$\operatorname{cod}(f)>8$ or there exists some $\phi\in L_{n}(\mathbb{Z}_{2})$
such that

$f\circ\phi(x)=\hat{f}(x_{1},\dots,x_{r})\pm x_{r+1}^{2}\pm\dots\pm
x_{n}^{2},$

where $\hat{f}\in\mathcal{M}_{r}^{4}(\mathbb{Z}_{2})$ is one of the following
germs:

corank | $\hat{f}$ | $\operatorname{cod}(f)$ | $\sigma(f)$
---|---|---|---
$0$ | $0$ | $0$ | $2$
$1$ | $\pm x^{2k},\;k\in\mathbb{N}$ | $2k-2$ | $2k$
| $\pm(x_{1}^{2}\pm x_{2}^{2})(x_{1}^{2}+\alpha x_{2}^{2}),\;\alpha\neq 0,\pm 1$ | $8$ | $4$
$2$ | $\pm(x^{4}+\beta x^{2}y^{2}+y^{4}),\;\beta\neq\pm 2$ | $8$ | $4$
| $x^{4}-y^{4}$ | $8$ | $4$
Table 1: Classification of even germs of codimension $\leq 8$.

Moreover, $\alpha,\beta$ are local invariants i.e., for any $\alpha\neq 0,\pm
1$ there exists some neighborhood $V$ of $\alpha$ such that the germs
$\\{\pm(x_{1}^{2}\pm
x_{2}^{2})(x_{1}^{2}+\alpha^{\prime}x_{2}^{2})\;|\;\alpha^{\prime}\in V\\}$
are all non-equivalent to each other, and for any $\beta\neq\pm 2$ there
exists some neighborhood $V$ of $\beta$ such that the germs
$\\{\pm(x_{1}^{4}+\beta^{\prime}x_{1}^{2}x_{2}^{2}+x_{2}^{4})\;|\;\beta^{\prime}\in
V\\}$ are all non-equivalent to each other.

###### Proof.

First, note that if $f\in\mathcal{M}_{n}^{2}(\mathbb{Z}_{2})$ then

$f(x)=a_{1}x_{1}^{2}+\dots+a_{n}x_{n}^{2}+h(x),$

where $O(h)\geq 4$. From this, it is not hard to see that if the corank $r$ of
the Hessian of $f$ is $\geq 3$, eg., $a_{1}=a_{2}=a_{3}=0$, then
$T^{\operatorname{ext}}_{\mathcal{O}_{f}}$ does not generate
$x_{1},x_{2},x_{3},x_{1}^{2},x_{1}x_{2},x_{1}x_{3},x_{2}^{2},x_{2}x_{3},x_{3}^{2}$,
i.e., $\operatorname{cod}(f)\geq 9$. Hence, $r\leq 2$. If $r=2$, then by the
splitting lemma there exists some $\phi\in L_{n}(\mathbb{Z}_{2})$ such that

$f(x)=\hat{f}(x_{1},x_{2})\pm x_{3}^{2}\pm\dots\pm x_{n}^{2},$ (56)

where
$\hat{f}(x_{1},x_{2})=f(x_{1},x_{2},0)\in\mathcal{M}_{r}^{4}(\mathbb{Z}_{2})$.
Moreover, since $j^{3}\hat{f}=0$, we are under the hypothesis of [9, Theorem
4.6]. note that $j^{k}\hat{f}(x)$ can be always written as

$j^{k}\hat{f}(x)=ax^{4}+bx^{2}y^{2}+cy^{4},$

for some $a,b,c\in\mathbb{R}$. However, note that, if $a=0$ or $b=0$, we would
be in case 2, 3, 5 or 6 of [9, Theorem 4.6]. And hence, by [9, Theorems 4.6,
4.8], it would follow that $\operatorname{cod}{\hat{f}}\geq 9$. Therefore, we
may restrict the analysis to the case where $a,c\neq 0$. Then, without loss of
generality, $j^{4}\hat{f}$ can be assume to be

$j^{4}\hat{f}(x)=\pm(x^{4}+bx^{2}y^{2}\pm y^{4}).$ (57)

Case 1: $j^{4}\hat{f}(x)=\pm(x^{4}+bx^{2}y^{2}+y^{4})$ and $b>2$. By a change
of variables of the type

$\phi(x_{1},x_{2})=(x_{1},\beta x_{2}),\beta>0,$

it can be shown that $j^{4}\tilde{f}$ is equivalent to
$\pm(x^{2}+y^{2})(x^{2}+\alpha y^{2})$ for some $\alpha>0$, $\alpha\neq 1$.

Case 2: $j^{4}\hat{f}(x)=\pm(x^{4}+bx^{2}y^{2}+y^{4})$ and $b<2$. Again by a
change of variables of the type

$\phi(x_{1},x_{2})=(x_{1},\beta x_{2}),\beta>0,$

it can be shown that $j^{4}\tilde{f}$ is equivalent to
$\pm(x^{2}-y^{2})(x^{2}+\alpha y^{2})$ for some $\alpha<0$, $\alpha\neq-1$.

Note also that, if $b=\pm 2$, we are in case 3 of [9, Theorem 4.6], and hence
$\operatorname{cod}(\tilde{f})\geq 10$. Finally, if $r=1$, the result is
straightforward to check.

######  Proposition 4.5.

Let $F$ be a $\mathbb{Z}_{2}$-invariant $r$-unfolding of
$f\in\mathcal{M}_{n}(\mathbb{Z}_{2})$, let $\phi\in L_{n}(\mathbb{Z}_{2})$ and
define the unfolding $G(x,u)=F(\phi(x),u)$. Then, $F$ is universal if and only
if $G$ is universal.

###### Proof.

Let $F$ be versal, and let $H$ be some $\mathbb{Z}_{2}$-invariant versal
$d$-unfolding $H$ of $\phi(f)$. By Theorem 3.8 and Proposition 4.2,
$H(\phi^{-1}(x),v)$ is a versal $d$-unfolding of $f$, and since $F$ is
universal, one must have $r\leq d$. Moreover, note also that by the same
propostions, $G$ is a versal $r$-unfolding of $\phi(f)$. Hence, it follows
that $G$ is universal. By a similar argument, the converse also holds.

We conclude with the classification theorem for even unfoldings.

######  Theorem 4.6 (Classification theorem for even unfoldings).

Let $F$ be an universal unfolding of a germ
$f\in\mathcal{M}_{n}(\mathbb{Z}_{2})$ and Hessian of corank $r$ at $0$. Then,
$\operatorname{cod}_{\mathbb{Z}_{2}}(f)\geq 4$ or $F$ is equivalent to an
unfolding $G$ of the form

$G(x,u)=g(x_{1},\dots,x_{r},u)\pm x_{r+1}^{2}\pm\dots\pm x_{n}^{2},$

where $g(x,u)$ is one of the following catastrophes:

corank | $g(x_{1},\dots,x_{r},u)$ | $\operatorname{cod}_{\mathbb{Z}_{2}}(f)$ | $\sigma(f)$
---|---|---|---
$0$ | $0$ | $0$ | $2$
$1$ | $\pm x^{2k}+u_{1}x^{2}+\dots+u_{k-1}x^{2k-2},\;k\in\mathbb{N}$ | $k-1$ | $2k$
| $\pm(x_{1}^{2}\pm x_{2}^{2})(x_{1}^{2}+\alpha x_{2}^{2})+u_{1}x^{2}+u_{2}y^{2}+u_{3}x^{2}y^{2},\;\alpha\neq 0,\pm 1$ | $3$ | $4$
$2$ | $\pm(x^{4}+\beta x^{2}y^{2}+y^{4})+u_{1}x^{2}+u_{2}y^{2}+u_{3}x^{2}y^{2},\;\beta\neq\pm 2$ | $3$ | $4$
| $x^{4}-y^{4}+u_{1}x^{2}+u_{2}y^{2}+u_{3}x^{2}y^{2}$ | $3$ | $4$
Table 2: Classification of $\mathbb{Z}_{2}$-invariant unfoldings.

###### Proof.

First note that, by direct computations, it can be shown that all of the
possible unfoldings $G(x,u)$ of the theorem are universal. By Theorem 4.4, it
follows that there exists some $\phi\in L_{e}(n)$ such that
$f\circ\phi=\hat{f}(x_{1},\dots,x_{r})\pm x_{r+1}^{2}\pm x_{n}^{2}$, where
$\hat{f}$ is one of the germs in table 1. Therefore, by proposition 4.5,
$F(\phi(f),u)$ is an universal unfolding of $\hat{f}(x_{1},\dots,x_{r})\pm
x_{r+1}^{2}\pm x_{n}^{2}$, and hence, it will be equivalent to the universal
unfolding of table 2 associated with the germ $\hat{f}(x_{1},\dots,x_{r})$.

##  5 Applications to Quantum Statistical Mechanics

In the field of statistical physics, be it classical or quantum, one of the
relevant goals is to find treatable analytical models that are able to exhibit
and mimic the behavior of phase transitions encountered in nature. However, it
follows that for most realistic physical models, which generally possess
interactions that are short-ranged, it is extremely difficult to prove the
existence of phase transitions. A mathematical approach to overcome such
difficulty is to find an approximation of such interactions by what is called
mean-field models. A mean-field model is a model where the forces of particle-
particle interaction are constant across distance. Although they seem
physically unreasonable, such forces, however, may be seen as a limit of
short-range interactions whose interaction range diverges, while its strength
goes to $0$ [5]. Its greatest advantage is that the mathematical analysis of
the thermodynamics of mean-field models can be carried out analytically in a
series of physically relevant scenarios: e.g., the Curie-Weiss model of
ferromagnetism, the BCS model of superconductivity, etc. Moreover, the results
of catastrophe theory just exposed above can be a powerful mathematical tool
to carry out the analysis of its phase transitions. Below, we provide an
example of such application in the context of Quantum Statistical Mechanics.
We note that in the following, we will only present results, mostly ommiting
the proofs. The full power of the catastrophe theory exposed in this paper to
the study of phase transitions in Q.S.M with its detailed analysis will be the
subject of a subsequent paper.

Fermionic systems over a lattice: We let $\mathbb{Z}^{d}$ be the
$d$-dimensional cubic lattice, and $\mathfrak{A}$ be the CAR algebra generated
by the creation and annihilation elements
$\\{a_{x,s}\;|\;x\in\mathbb{Z}^{d},\;s\in\\{\uparrow,\downarrow\\}\\}$, i.e.,
the unique (up to $*$-isomorphism) $C^{*}$-algebra that satisfies the
relations

$\displaystyle\\{a_{x,s},a_{y,t}\\}=$ $\displaystyle 0,$
$\displaystyle\\{a_{x,s},a_{y,t}^{*}\\}=$
$\displaystyle\delta_{x,y}\delta_{s,t}\mathds{1},\quad
x,y\in\mathbb{Z}_{d},\;s,t\in\\{\uparrow,\downarrow\\},$ (58)

where $\\{A,B\\}\doteq AB+BA$ denotes the anti-commutator. From now on,
$\Lambda_{l}\subset\mathbb{Z}^{d}$ will denote the cubic box of size $2l+1$
centered at the origin of $\mathbb{Z}^{d}$; i.e.,

$\Lambda_{l}=\\{(x_{1},\dots,x_{d})\in\mathbb{Z}^{d}\;|\;|x_{i}|\leq
l,\;i=1,\dots,d\\}.$ (59)

Mean-field model: We consider now a finite-dimensional Hamiltonian on the box
$\Lambda_{l}$ given by

$\displaystyle H_{l}=$
$\displaystyle\sum_{x\in\Lambda_{l}}(-\mu(n_{x,\uparrow}+n_{x,\downarrow}-h(n_{x,\uparrow}-n_{x,\downarrow})+2\lambda
n_{x,\uparrow}n_{x,\downarrow})-\frac{\gamma}{|\Lambda_{l}|}\sum_{x,y\in\Lambda_{l}}a_{x,\uparrow}^{*}a_{x,\downarrow}^{*}a_{y,\downarrow}a_{y,\uparrow}$
$\displaystyle-\frac{\delta}{|\Lambda_{l}|}\sum_{x,y\in\Lambda_{l};s,t\in\\{\uparrow,\downarrow\\}}a_{x,s}^{*}a_{x,s}a_{y,t}^{*}a_{y,t},$
(60)

where $n_{x,s}=a^{*}_{x,s}a_{x,s}$. The second and third terms of the above
Hamiltonian are what is called mean-field interactions. Note that the
interaction strength is constant across distances. Altough they seem at first
non-physical, these kind of interactions are in fact physically relevant,
specially because they can model with good precision the behavior of
macroscopic materials, for example superconductors and ferromagnets. It is
well known that the thermodynamics such systems with mean-field interactions
may be analyzed by a simpler Hamiltonian, with no mean-field terms, usucally
called in the literature as the approximating Hamiltonian (see [2] for more
details).

######  Theorem 5.1.

The limiting thermodynamical pressure $\mathrm{P}$ of $H_{l}$, defined as

$\mathrm{P}\doteq\lim_{l\rightarrow\infty}p_{l},$ (61)

where $p_{l}$ is the pressure of the system restricted to the finite box
$\Lambda_{l}$, is equal to

$\mathrm{P}=\inf_{c_{1},c_{2}\in\mathbb{R}}\\{\gamma c_{1}^{2}+\delta
c_{2}^{2}-\mathrm{P}_{0}(c_{1},c_{2})\\},$ (62)

and $\mathrm{P}_{0}(c_{1},c_{2})$ is the thermodynamical pressure of the
system with Hamiltonian given by

$\displaystyle H_{l}(c_{1},c_{2})$
$\displaystyle=\sum_{x\in\Lambda_{l}}(-\mu(n_{x,\uparrow}+n_{x,\downarrow})-h(n_{x,\uparrow}-n_{x,\downarrow})+2\lambda
n_{x,\uparrow}n_{x,\downarrow})-\gamma\overline{c_{1}}\sum_{x\in\Lambda_{l}}a_{y,\downarrow}a_{y,\uparrow}$
$\displaystyle-\gamma
c_{1}\sum_{x\in\Lambda_{l}}a_{x,\uparrow}^{*}a_{x,\downarrow}^{*}-\delta\overline{c_{2}}\sum_{x\in\Lambda_{l};s\in\\{\uparrow,\downarrow\\}}a_{x,s}^{*}a_{x,s}-\delta
c_{2}\sum_{x\in\Lambda_{l};s\in\\{\uparrow,\downarrow\\}}a_{x,s}^{*}a_{x,s}.$
(63)

Morover, the absolute value of $c_{1}$ where the infimum is achieved
corresponds to the cooper pair density of the system, and the absolute value
of $c_{2}$ where the infimum is achieved corresponds to the electron density
of the system.

Because of the permutation invariance of the system, the pressure associated
with $H_{l}(c_{1},c_{2})$ can be analitically computed, and is given by

$\mathrm{P}_{0}(c_{1},c_{2})=-\beta_{-1}\ln 2+(\mu+2\delta
c_{2})+\beta^{-1}\ln\left[\cosh(\beta
h)+e^{-\lambda\beta}\cosh\left(\beta\sqrt{(\mu+2\delta
c_{2}-\lambda)^{2}+\gamma^{2}c_{1}^{2}}\right)\right].$ (64)

Therefore, in view of theorem 5.1, one can study the thermodynamics and the
different phases of the system by means of Equation (62). And for this, the
catastrophe theory exposed here can be of great use, since its equivalent
relations preserve the minima of functions, and hence, it preserves the
information of the phase diagram of our system. With the help of numerical
computations, one can show that for a particular choice of the parameters
$\beta,h,\mu,\gamma,\lambda,\delta$, the pressure $P_{0}$ possesses a
degenerate critical point at $(0,0)$, and it is moreover transversal with
respect to the parameters $\gamma,\lambda,\delta$. Therefore, we may apply the
classification theorem 4.6.

######  Theorem 5.2.

For $\beta=1,h=0,\mu=2$, it follows that $\gamma c_{1}^{2}+\delta
c_{2}^{2}-\mathrm{P}_{0}(c_{1},c_{2})$ is equivalent (in the sense of
definition 3.2) to the unfolding

$x^{4}+\alpha
x^{2}y^{2}+y^{4}+u_{1}x^{2}+u_{2}y^{2}+u_{3}x^{2}y^{2},\quad\alpha>2.$ (65)

Idea of the proof: From Equation (64), we calculate the Taylor expansion of
the pressure at its degenerate critical point with respect to the variables
$(c_{1},c_{2})$, and we see that it is equivalent to the germ

$f(x,y)=x^{4}+\alpha x^{2}y^{2}+y^{4}$ (66)

for some $\alpha\neq\pm 0,1$. Then, we show that $\gamma c_{1}^{2}+\delta
c_{2}^{2}-\mathrm{P}_{0}(c_{1},c_{2})$ is $\Gamma$-transversal by means of
Proposition 3.6, and hence , by the classification table 2, the result
follows.

######  Remark.

Note that because of the fact that our initial function $\gamma
c_{1}^{2}+\delta c_{2}^{2}-\mathrm{P}_{0}(c_{1},c_{2})$ possesses a symmetry,
we cannot not apply the standard results of catastrophe theory; we instead
have to resort to the invariant version of catastrophe theory that was exposed
here. We also point out that the symmetry of the pressure function above is
linked with physical properties of our interaction; in special, to the so-
called gauge invariance of $H_{l}$.

In view of Theorem 5.2, one concludes that it is now analytically feasible to
evaluate the phase diagram of our system apart from diffeomorphisms.
Furthermore, not only the golbal minima also the local minima of the pressure
mapping give us information about the system; is this case, about the so-
called metastable states. To conclude, we shortly discuss about the stability
of the phase diagrams of our model above. We recall that the perturbations
allowed for the unfoldings to be said stable are perturbations that have their
derivative controlled up to an arbitrary order $k$. And it follows that not
every ”small” interaction added in $H_{l}$ would in turn correspond to a small
derivative of any order. Nevertheless, by means of the so-called cluster
expansion of the partition function [11], one is able to find a class of
interactions whose derivatives can be adequately controlled to the application
of catastrophe theory. One, in particular, corresponds to adding a kinetic
interaction into the system.

######  Theorem 5.3.

Let

$K_{l,\epsilon}=\sum_{\begin{subarray}{c}x,y\in\Lambda_{l}\;|\;|x-y|=1,\\\
s\in\\{\uparrow,\downarrow\\}\end{subarray}}\epsilon a^{*}_{x,s}a_{y,s}$ (67)

be a kinetic interaction. Then, it follows that, for $\epsilon>0$ small
enough, the pressure of the system $H_{l}+K_{l,\epsilon}$ is equivalent to the
unperturbed pressure. In particular, the phase diagram is preserved.

Idea of the proof: For this result, one first needs to show that the the
perturbation $K_{l,\epsilon}$ is well behaved in the sense that for $\epsilon$
small enough, the perturbation of $\mathrm{P}_{0}$ introduced by the addition
of $K_{l,\epsilon}$ belong to some neighborhood of $\mathrm{P}_{0}$ in the
$C^{\infty}$-topology of Definition 3.10. We can do that by techniques coming
from the cluster expansions of the partition function studied in [11]. After
that, the result follows from Theorem 3.14.

##  6 Technical results

We dedicate this section to the proof of more technical results that are
necessary to prove the main theorems of the paper, but unimportant to the
comprehension of the paper.

Proof of Lemma 2.11: Let $e\in\mathbb{N}$ be such that
$\mathcal{M}_{n}^{e}(\Gamma)\subset T_{0,g}$. By item $i)$ of Theorem 2.10,
taking now $N=\mathcal{M}_{n}(\Gamma)$ and
$M_{0}=\mathcal{M}_{n}^{e}(\Gamma)$, it follows that there exists some
$\omega\in\mathbb{N}$ such that, if $\mathcal{M}_{n}^{e}(\Gamma)\subset
T_{\mathcal{O}_{f}}+\mathcal{M}_{n}^{\omega+e}(\Gamma)$, then
$\mathcal{M}_{n}^{e}(\Gamma)\subset T_{\mathcal{O}_{f}}$. Choose
$r=e+\omega+1$. If $j^{r}f=j^{r}g$ it is straightforward to check that

$T_{0,g}\subset T_{\mathcal{O}_{f}}+\mathcal{M}_{n}^{r-1}(\Gamma),$

i.e., one has

$\mathcal{M}_{n}^{e}(\Gamma)\subset T_{0,g}\subset
T_{\mathcal{O}_{f}}+\mathcal{M}_{n}^{r-1}(\Gamma)=T_{\mathcal{O}_{f}}+\mathcal{M}_{n}^{\omega+e}(\Gamma),$
(68)

which, by $i)$ of Theorem 2.10, implies

$\mathcal{M}_{n}^{r}(\Gamma)\subset\mathcal{M}_{n}^{e}(\Gamma)\subset
T_{\mathcal{O}_{f}}.$ (69)

######  Lemma 6.1 (Geometric lemma for $\Gamma$-invariant unfoldings).

Let $H(x,u,t)\in\mathcal{U}_{n,r+1}(\Gamma)$ satisfy the following
differential equation

$\frac{\partial H(x,u,t)}{\partial
t}=\xi(x,u,t)\cdot\vec{\nabla}_{x}H(x,u,t)+\chi(u,t)\cdot\vec{\nabla}_{u}H(x,u,t)+\chi_{0}(u,t),$
(70)

where:

$\displaystyle\xi(x,u,t)\in\vec{\mathcal{E}}_{n+r+1,n},\;\text{ with
}x\mapsto\xi(x,u,t)\in\vec{\mathcal{E}}_{n}(\Gamma)\text{ for $(u,t)$ in some
small neighborhood of $0\in\mathbb{R}^{r+1}$},$
$\displaystyle\chi(u,t)\in\vec{\mathcal{E}}_{r+1,r},\;\text{ and}$
$\displaystyle\chi_{0}(u,t)\in\mathcal{E}_{r+1}.$

Then, it follows that for small enough $t\in\mathbb{R}$, $H(x,u,t)$ is
$\Gamma$-equivalent to $H(x,u,0)$, i.e., there exist some germs

$\displaystyle\phi(x,u,t)\in\vec{\mathcal{E}}_{n+r+1,n},\;\text{ with
}x\mapsto\xi(x,u,t)\in\vec{\mathcal{E}}_{n}(\Gamma)\text{ for $(u,t)$ in some
small neighborhood of $0\in\mathbb{R}^{r+1}$},$
$\displaystyle\psi(u,t)\in\vec{\mathcal{E}}_{r+1,r},\;\text{ and}$
$\displaystyle h(u,t)\in\mathcal{E}_{r+1}$

such that

$H(x,u,0)=H(\phi(x,u,t),\psi(u,t),t)+h(u,t)$ (71)

for small enough $t\in\mathbb{R}$.

###### Proof.

We define $\phi(x,u,t)\in\vec{\mathcal{E}}_{n+r+1,n}$,
$\psi(u,t)\in\vec{\mathcal{E}}_{r+1,r}$ and
$\mathfrak{h}(u,t)\in\mathcal{E}_{r+1}$ to be the (unique) germs sattisfiying
the following system of ODEs:

$\displaystyle\frac{\partial\phi(x,u,t)}{\partial
t}=-\xi(\phi(x,u,t),\psi(u,t),t),$
$\displaystyle\frac{\partial\psi(x,u,t)}{\partial
t}=-\chi(\psi(u,t),t),\;\text{ and}$ $\displaystyle\frac{\partial
h(u,t)}{\partial t}=-\chi_{0}(\psi(u,t),t),$

with initial conditions

$\phi(x,u,0)=x,\quad\psi(u,0)=u,\quad h(u,0)=0.$ (72)

First, note that since $xi$ is $\Gamma$-invariant, for any $\gamma\in\Gamma$
it follows that $\gamma^{-1}\phi(\gamma x,u,t)$ is also a solution of the ODE
with the same initial condition. Hence, by the uniqueness of the solution, one
has

$\gamma^{-1}\phi(\gamma x,u,t)=\phi(x,u,t),$ (73)

i.e., the mapping $x\mapsto\phi(x,u,t)$ is $\Gamma$-invariant for $(u,t)$ in
some small neighborhood of $0\in\mathbb{R}^{r+1}$. Now we proceed to show the
equality in Equation (71). Differentiating the r.h.s.of Equation (71) with
respect to $t$, we obtain

$\displaystyle\frac{\partial}{\partial t}(H(\phi(x,u,t),\psi(u,t),t)+h(u,t))=$
$\displaystyle-\xi(\phi(x,u,t),\psi(u,t),t)\cdot\vec{\nabla}_{x}H(\phi(x,u,t),\psi(u,t),t)$
$\displaystyle-\chi(\psi(u,t),t)\cdot\vec{\nabla}_{u}H(\phi(x,u,t),\psi(u,t),t)$
$\displaystyle-\chi_{0}(\psi(u,t),t))+\frac{\partial H}{\partial
t}(\phi(x,u,t),\psi(u,t),t).$ (74)

It follows that the r.h.s. of Equation (6) is $0$, and hence, by the initial
conditions, one clearly has, for small enough $t\in\mathbb{R}$,

$H(\phi(x,u,t),\psi(u,t),t)+h(u,t)=H(\phi(x,u,0),\psi(u,0),0)+h(u,0)=H(x,u,0).$
(75)

######  Lemma 6.2.

Let $f\in\mathcal{U}_{e}(n,r)$ be such that, for some $k\leq r$, $f$ satisfies

$f(x,0,\dots,0,u_{k+1},\dots,u_{r})=0,$

for all $x\in\mathbb{R}^{n}$ in some neighborhood of $0$ and all
$(u_{k+1},\dots,u_{r})\in\mathbb{R}^{r-k}$ in some neighborhood of $0$. Then,
there exists germs $g_{1},\dots,g_{k}\in\mathcal{U}_{e}(n,r)$ such that

$f(x,u)=\sum_{i=1}^{k}u_{i}g_{i}(x,u).$

###### Proof.

Note that

$f(x,u)=\int_{0}^{1}\frac{d}{dt}(f(x,tu_{1},\dots,tu_{k},u_{k+1},\dots,u_{r}))dt.$

Computing the derivative on the right side of the equality, one has

$\displaystyle f(x,u)=$
$\displaystyle\int_{0}^{1}\sum_{i=1}^{k}\left.\frac{\partial f}{\partial
u_{i}}\right|_{(x,tu_{1},\dots,tu_{k},u_{k+1},\dots,u_{r})}\cdot u_{i}dt$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{k}\int_{0}^{1}\left.\frac{\partial
f}{\partial u_{i}}\right|_{(x,tu_{1},\dots,tu_{k},u_{k+1},\dots,u_{r})}dt\cdot
u_{i}.$

For each $i=1,\dots,k$, define

$g_{i}(x,u)\doteq\int_{0}^{1}\left.\frac{\partial f}{\partial
u_{i}}\right|_{(x,tu_{1},\dots,tu_{k},u_{k+1},\dots,u_{r})}dt.$

Since $f\in\mathcal{U}_{e}(n,r)$, it follows that
$g_{i}\in\mathcal{U}_{e}(n,r)$ for all $i=1,\dots,k$, and hence, the lemma
follows.

###  6.1 Adequately ordered systems of DA algebras

In order to prove the main theorem on the equivalence of transveral
unfoldings, we need a technical result of DA algebras. The reader may find it
in [3, Corollary 6.16], in a more general scenario.

######  Definition 6.3.

Let $R$ be a ring and a DA-algebra, $I$ a Jacobson ideal in $R$. Then, $(R,I)$
is said to be adequate if, for any homomorphism $\Psi:N\rightarrow M$ where
$M$ is finitely generated such that

$\Psi(N)+I\cdot M=M,$ (76)

one also has

$\Psi(N)=M\quad\text{and}\quad\Psi(I\cdot N)=I\cdot M.$ (77)

######  Remark.

Recall that a Jacobson ideal in a ring $R$ is an ideal $I$ such that
$\mathbf{1}+a$ is invertible for any $a\in I$.

######  Theorem 6.4 (Preparation theorem for $\Gamma$-invariant unfoldings).

The pair $(\mathcal{U}_{n,r}(\Gamma),\mathcal{M}_{r})$ is adequate, in the
sense of Definition 6.3.

######  Definition 6.5.

Let $F\in\mathcal{U}_{n,r}(\Gamma)$. We define

$\tilde{T}_{F}\doteq\\{\Phi\cdot\vec{\nabla}_{x}F\;|\;\Phi\in\vec{\mathcal{E}}_{n+r+1,n}\;\text{
and }x\mapsto\Phi(x,u,t)\in\vec{\mathcal{E}}_{n}(\Gamma)\\}.$ (78)

######  Corollary 6.6.

Let $F\in\mathcal{U}_{n,r}(\Gamma)$ unfold $f\in\mathcal{M}_{n}(\Gamma)$. The
following conditions are equivalent

  1. i.)

$F$ is $\Gamma$-transversal, i.e.,
$\mathcal{E}_{n}(\Gamma)=T^{\operatorname{ext}}_{\mathcal{O}_{f}}+\langle
1,\alpha_{1}(F),\dots,\alpha_{r}(F)\rangle_{\mathbb{R}}$,

  2. ii.)

$\mathcal{U}_{n,r}(\Gamma)=\tilde{T}_{F}+\left\langle 1,\frac{\partial
F}{\partial u_{1}},\dots,\frac{\partial F}{\partial
u_{r}}\right\rangle_{\mathcal{E}_{r}}$.

###### Proof.

$ii.)\implies i.)$. This is obvious, by taking $u=0$.

$i.)\implies ii.)$. We first make the following initial observation. For any
$\Gamma$-invariant unfolding $G\in\mathcal{U}_{n,r}(\Gamma)$ of a germ
$g\in\mathcal{E}_{n}(\Gamma)$, by Taylor’s theorem we can always write it as

$G(x,u)=g(x)+\sum_{i=1}^{r}u_{i}G_{i}(x,u),$ (79)

for some $G_{i}(x,u)\in\mathcal{U}_{n,r}(\Gamma)$, $i=1,\dots,r$, i.e.,

$\mathcal{U}_{n,r}(\Gamma)\cong\mathcal{E}_{n}(\Gamma)\oplus\mathcal{M}_{r}\cdot\mathcal{U}_{n,r}(\Gamma).$
(80)

Similarly, we note that, also by Taylor expansion, one has for any element of
$\tilde{T}_{F}$,

$\displaystyle\Phi(x,u)\cdot\vec{\nabla}_{x}F(x,u)=\phi(x)\cdot\vec{\nabla}_{x}f(x)+\sum_{i=1}^{r}u_{i}G_{i}(x,u),\;\text{
where }\;\phi(x)\cdot\vec{\nabla}_{x}f(x)\in
T^{\operatorname{ext}}_{\mathcal{O}_{f}},$
$\displaystyle\sum_{i=1}^{r}u_{i}G_{i}(x,u)\in\mathcal{M}_{r}\cdot\mathcal{U}_{n,r}(\Gamma),$
(81)

and for any element of $\left\langle 1,\frac{\partial F}{\partial
u_{1}},\dots,\frac{\partial F}{\partial
u_{r}}\right\rangle_{\mathcal{E}_{r}}$,

$\displaystyle\chi_{0}(u)+\sum_{i=1}^{r}\frac{\partial F(x,u)}{\partial
u_{i}}\chi_{i}(u)=\chi_{0}(0)+\sum_{i=1}^{r}\frac{\partial F(x,0)}{\partial
u_{i}}\chi_{i}(0)+\sum_{i=1}^{r}u_{i}G_{i}(x,u),\;\text{ where}$
$\displaystyle\chi_{0}(0)+\sum_{i=1}^{r}\frac{\partial F(x,0)}{\partial
u_{i}}\chi_{i}(0)\in\langle
1,\alpha_{1}(F),\dots,\alpha_{r}(F)\rangle_{\mathbb{R}}\;\text{ and
}\;\sum_{i=1}^{r}u_{i}G_{i}(x,u)\in\mathcal{M}_{r}\cdot\mathcal{U}_{n,r}(\Gamma).$
(82)

Now, let $N,\,M$ be given by

$N=\\{\Phi\in\vec{\mathcal{E}}_{n+r,n}\;|\;x\mapsto\Phi(x,u)\in\vec{\mathcal{E}}_{n}(\Gamma)\\}\oplus\vec{\mathcal{E}}_{r,r+1},\quad
M=\mathcal{U}_{n,r}(\Gamma),$ (83)

and define $\Psi:N\rightarrow M$ by

$\Psi(\Phi,(\chi_{0},\dots,\chi_{r}))(x,u)=\Phi(x,u)\cdot
F(x,u)+\chi_{0}(u)+\sum_{i=1}^{r}\chi_{i}(u)\frac{\partial F(x,u)}{\partial
u_{i}}.$

Clearly, $\Phi(N)=\tilde{T}_{F}+\left\langle 1,\frac{\partial F}{\partial
u_{1}},\dots,\frac{\partial F}{\partial
u_{r}}\right\rangle_{\mathcal{E}_{r}}$. Hence, condition $ii.)$ is equivalent
to stating that $\Psi$ is surjective. Since the pair
$(\mathcal{U}_{n,r}(\Gamma),\mathcal{M}_{r})$ is adequate, by Theorem 6.4 it
follows that in order to prove the surjectivity of $\Psi$, it is enough to
prove

$\Psi(N)+\mathcal{M}_{r}\cdot\mathcal{U}_{n,r}(\Gamma)=\mathcal{U}_{n,r}(\Gamma).$
(84)

But from Equations (80), (6.1) and (6.1), one has that the equality (84) holds
if and only if

$T^{\operatorname{ext}}_{\mathcal{O}_{f}}+\langle
1,\alpha_{1}(F),\dots,\alpha_{r}(F)\rangle_{\mathbb{R}}=\mathcal{E}_{n}(\Gamma),$
(85)

which is exactly contition $i.)$.

Acknowledgements: The author would like to thank Prof. Javier Bobadilla, Prof.
Maria Aparecida Ruas, and Prof. Miriam Garcia Manoel for the helpful
discussions.

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