$Z(x,y)=\int\mathcal{D}a_{\mu}\mathcal{D}\phi\mathcal{D}\phi^{*}e^{i\sum_{pk}\eta_{p}z_{pk}\phi_{pk}^{*}\cdot\phi_{pk}}e^{-\sum{}^{{}^{\prime}}_{\mu}\frac{1}{2\sigma^{2}T_{\mu}}{\rm
tr}a_{\mu}^{2}+\sum_{pkql}\eta_{p}\eta_{q}\sum{}^{{}^{\prime}}_{\mu}a_{\mu}^{pk;ql}i\frac{1}{\sqrt{D}}\phi_{pk}^{*}\cdot
X^{U}_{\mu}\cdot\phi_{ql}}.$ (168)

We can now perform the Gaussian integral over $\phi$ and $\phi^{*}$ resulting
in the partition function

$Z(x,y)=\int\mathcal{D}a_{\mu}e^{-\sum{}^{{}^{\prime}}_{\mu}\frac{1}{2\sigma^{2}T_{\mu}}{\rm
tr}a_{\mu}^{2}-\textrm{Tr}\log(z+\sum{}^{{}^{\prime}}_{\mu}\frac{1}{\sqrt{D}}a_{\mu}X^{U}_{\mu})}.$
(169)

A rescaling of the integration variable $a_{\mu}\to\sigma a_{\mu}$ gives
equation (32). This is the partition function obtained by Altland and Bagrets
Altland-ml-2017eao .

Note that the $\mu=0$ term in this sum is exactly the GUE result. This
derivation raises an important concern. The $\mu=0$ term in (165) consists of
$4n^{2}D^{2}{N\choose q}$ terms of the form
$\phi^{i*}_{pk}\phi_{ql}^{i}\phi^{j*}_{ql}\phi_{pk}^{j}$ each with weight
$\sigma^{2}/2D$. However, the expression before the Fierz transformation (159)
has only $4n^{2}D{N/2\choose q}$ such terms from the diagonal
$\Gamma^{\alpha}$ (there are ${N/2\choose q}$ of them) and
$4n^{2}D\left[{N\choose q}-{N/2\choose q}\right]$ such terms from the off-
diagonal $\Gamma^{\alpha}$, each with weight $\sigma^{2}/2$. Note that all
$\Gamma^{\alpha}$ have only one nonzero matrix element in each column and in
each row in the representation we are working in. So in total we have
$4n^{2}D{N\choose q}$ terms of this form in equation (159) which is an
exponentially smaller number than in equation (165), and most terms of the
form $\phi^{i*}_{pk}\phi_{ql}^{i}\phi^{j*}_{ql}\phi_{pk}^{j}$ in (165) are
actually canceled by the other terms obtained after the Fierz transformation.
Indeed, the dominance of the GUE contribution has to break down for $q=2$,
where the eigenvalues are uncorrelated. Yet, there is no qualitative
difference between the $\sigma$-model for $q=2$ and $q=4$.

For $N\mod 8=0$ the $\gamma$ matrices are real and the spectral correlations
are in the universality class of the Gaussian Orthogonal Ensemble (GOE). In
this case, the term in (159)

$\phi^{*}_{pk}\cdot\left(\Gamma^{U}_{\alpha}\right)\cdot\phi_{pk}$ (170)

can be written as

$\frac{1}{2}\left(\phi^{*}_{pk}\cdot\left(\Gamma^{U}_{\alpha}\right)\cdot\phi_{pk}+\phi_{pk}\cdot\left(\Gamma^{U}_{\alpha}\right)\cdot\phi^{*}_{pk}\right).$
(171)

After applying the Fierz transformation, we obtain additional terms of the
form

$\sum_{pkql}\sum{}^{{}^{\prime}}_{\mu}T_{\mu}\phi^{*}_{pk}\cdot
X^{U}_{\mu}\cdot\phi^{*}_{ql}\phi_{ql}\cdot X^{U}_{\mu}\cdot\phi_{pk},$ (172)

which is the so-called Cooperon contribution to the GOE result. Note that in
order to get a $\beta$-independent spectral density for the Wigner-Dyson
ensemble, we have to scale the variance of the random matrix Hamiltonian as
$1/\beta$. For the SYK model we do not have such rescaling and we expect an
additional $\sqrt{\beta}$ dependence in the saddle-point equation and the
corresponding resolvent and semi-circular spectral density.

##  Appendix B Some combinatorial identities

One can easily prove the identity

$\frac{1}{D^{2}}\sum_{|\mu|=0}^{N}{N\choose|\mu|}T_{\mu}^{2}=4{N\choose
q}^{-1}.$ (173)

For arbitrary $p$ the sum behaves as

$\frac{1}{D^{2}}\sum_{|\mu|=0}^{N}{N\choose|\mu|}T_{\mu}^{p}\sim N^{-qp/2}.$
(174)

If $\nu$ is summed over with $|\nu|$ fixed we have

$\frac{1}{D}\sum_{\nu,|\nu|{\rm
fixed}}\textrm{Tr}X_{\mu}X_{\nu}X_{\mu}X_{\nu}=\sum_{s}(-1)^{s}{N\choose|\mu|}{N-|\mu|\choose|\nu|-s}{|\mu|\choose
s}$ (175)

Using this we obtain the identity

$\frac{1}{D^{3}}\sum_{|\mu|=0}^{N}{N\choose|\mu|}\sum_{|\nu|=0}^{N}{N\choose|\nu|}\sum_{\nu}T_{\mu}^{p}T_{\nu}\textrm{Tr}X_{\mu}X_{\nu}X_{\mu}X_{\nu}=4\eta^{p},$
(176)

which can also be shown by applying the Fierz transformation to
$T_{\nu}X_{\nu}X_{\nu}$. To obtain the sixth moment from the $\sigma$-model
calculation we need the identity

$\displaystyle\sum_{m_{1}=0}^{N}\sum_{m_{2}=0}^{N}\sum_{s=0}^{m_{1}}{N\choose
m_{1}}{m_{1}\choose s}{N-m_{1}\choose
m_{2}-s}(-1)^{s}T_{m_{1}}T_{m_{2}}T_{m_{1}+m_{2}-2s}$
$\displaystyle=4{N\choose
q}^{-2}\sum_{k=0}^{q}\sum_{m=0}^{q}(-1)^{q-k-m}{N-2k\choose q-m}{2k\choose
m}{N-q\choose k}{q\choose k}.$ $\displaystyle\equiv 4T_{6}.$ (177)

Here $T_{6}$ denotes the value of the single-trace diagram with three chords
all intersecting with each other, see the footnote near the end of section
5.1.

##  Appendix C Replica limit of the GUE partition function

###  C.1 One-point function

The GUE partition function for $n$ replicas is given by

$Z(z)=\int d\sigma\frac{1}{\det^{N}(z+\sigma)}e^{-\frac{N}{2}\sigma^{2}},$
(178)

where the integral is over $n\times n$ Hermitian matrices $\sigma$. Using that
the replica limit of the partition function is equal to one, we find that the
resolvent is given by

$G(z)=-\lim_{n\to 0}\frac{1}{n}\frac{1}{N}\frac{d}{dz}\log Z(z)=\lim_{n\to
0}\frac{1}{n}\int d\tilde{\sigma}{\rm
tr}\tilde{\sigma}e^{-\frac{N}{2}\tilde{\sigma}^{2}-N{\rm
tr}\log(z+\tilde{\sigma})},$ (179)

where we have expressed the derivative with respect $z$ in terms of a
derivative with respect to the $\tilde{\sigma}_{kk}$ and partially integrated
these variables. We evaluate the resolvent in powers of $1/z$ and in powers of
$1/N$. The saddle point equation is given by

$\tilde{\sigma}+\frac{1}{\tilde{\sigma}+z}=0.$ (180)

Expanding around the physical solution $\bar{\sigma}$ that asymptotes as $1/z$
for large $z$,

$\tilde{\sigma}=\bar{\sigma}+\sigma,$ (181)

and using the saddle point equation we obtain the expansion

$\displaystyle G(z)$ $\displaystyle=\bar{G}(z)+\lim_{n\to 0}\frac{1}{n}\int
d\sigma{\rm tr}\sigma
e^{-\frac{N}{2}(1-\bar{\sigma}^{2})\sigma^{2}+\sum_{k=3}^{\infty}\frac{1}{k}{\rm
tr}\bar{\sigma}^{k}\sigma^{k}}$ $\displaystyle=\bar{G}(z)+\lim_{n\to
0}\frac{1}{n}\\!\left(\\!\frac{N}{3}\left\langle{\rm tr}\sigma{\rm
tr}(\bar{\sigma}\sigma)^{3}\right\rangle+\frac{N}{5}\left\langle{\rm
tr}\sigma{\rm
tr}(\bar{\sigma}\sigma)^{5}\right\rangle+\frac{N}{7}\left\langle{\rm
tr}\sigma{\rm
tr}(\bar{\sigma}\sigma)^{7}\right\rangle+\frac{N}{9}\left\langle{\rm
tr}\sigma{\rm tr}(\bar{\sigma}\sigma)^{9}\right\rangle\right.$
$\displaystyle\quad+\frac{N^{2}}{12}\left\langle{\rm tr}\sigma{\rm
tr}(\bar{\sigma}\sigma)^{3}{\rm
tr}(\bar{\sigma}\sigma)^{4}\right\rangle+\frac{N^{2}}{18}\left\langle{\rm
tr}\sigma{\rm tr}(\bar{\sigma}\sigma)^{3}{\rm
tr}(\bar{\sigma}\sigma)^{6}\right\rangle+\frac{N^{2}}{20}\left\langle{\rm
tr}\sigma{\rm tr}(\bar{\sigma}\sigma)^{4}{\rm
tr}(\bar{\sigma}\sigma)^{5}\right\rangle$
$\displaystyle\quad\left.+\frac{N^{3}}{162}\left\langle{\rm tr}\sigma({\rm
tr}(\bar{\sigma}\sigma)^{3})^{3}\right\rangle\right)+\cdots,$ (182)

where the expectation values are given by the sum over all corresponding Wick
contractions. The terms with an odd number of factors of $\sigma$ vanish and
we have not included them in the above. One can easily show that the following
terms vanish in the replica limit

$\displaystyle\lim_{n\to 0}\frac{1}{n}\left\langle{\rm tr}\sigma{\rm
tr}\sigma^{3}\right\rangle$ $\displaystyle=0,$ $\displaystyle\lim_{n\to
0}\frac{1}{n}\left\langle{\rm tr}\sigma{\rm tr}\sigma^{7}\right\rangle$
$\displaystyle=0,$ $\displaystyle\lim_{n\to 0}\frac{1}{n}\left\langle{\rm
tr}\sigma{\rm tr}\sigma^{3}{\rm tr}\sigma^{6}\right\rangle$ $\displaystyle=0,$
$\displaystyle\lim_{n\to 0}\frac{1}{n}\left\langle{\rm tr}\sigma{\rm
tr}\sigma^{4}{\rm tr}\sigma^{5}\right\rangle$ $\displaystyle=0.$ (183)

This leaves us with the following nonvanishing contributions

$\displaystyle A_{1}$ $\displaystyle=\lim_{n\to
0}\frac{1}{n}\frac{N}{5}\left\langle{\rm tr}\sigma{\rm
tr}(\bar{\sigma}\sigma)^{5}\right\rangle=\frac{1}{N^{2}}\frac{\bar{\sigma}^{5}}{(1-\bar{\sigma}^{2})^{3}}=\frac{1}{N^{2}}\left[\frac{1}{z^{5}}+8\frac{1}{z^{7}}+47\frac{1}{z^{9}}+\cdots\right],$
$\displaystyle A_{2}$ $\displaystyle=\lim_{n\to
0}\frac{1}{n}\frac{N}{9}\left\langle{\rm tr}\sigma{\rm
tr}(\bar{\sigma}\sigma)^{9}\right\rangle=\frac{21}{N^{4}}\frac{\bar{\sigma}^{9}}{(1-\bar{\sigma}^{2})^{5}}=\frac{21}{N^{4}}\frac{1}{z^{9}},$
$\displaystyle A_{3}$ $\displaystyle=\lim_{n\to
0}\frac{1}{n}\frac{N^{2}}{12}\left\langle{\rm tr}\sigma{\rm
tr}(\bar{\sigma}^{3}\sigma)^{3}\textrm{Tr}(\bar{\sigma}\sigma)^{4}\right\rangle=\frac{1}{N^{2}}\frac{\bar{\sigma}^{7}}{(1-\bar{\sigma}^{2})^{4}}=\frac{1}{N^{2}}\left[\frac{1}{z^{7}}+11\frac{1}{z^{9}}+\cdots\right],$
$\displaystyle A_{4}$ $\displaystyle=\lim_{n\to
0}\frac{1}{n}\frac{N^{2}}{12}\left\langle{\rm tr}\sigma{\rm
tr}(\bar{\sigma}\sigma)^{3}\textrm{Tr}(\bar{\sigma}\sigma)^{4}\right\rangle=\frac{1}{N^{2}}\frac{\bar{\sigma}^{7}}{(1-\bar{\sigma}^{2})^{3}}=\frac{1}{N^{2}}\left[\frac{1}{z^{7}}+11\frac{1}{z^{9}}+\cdots\right],$
$\displaystyle A_{6}$ $\displaystyle=\lim_{n\to
0}\frac{1}{n}\frac{N^{3}}{162}\left\langle{\rm tr}\sigma({\rm
tr}(\bar{\sigma}\sigma)^{3})^{3}\right\rangle=\frac{1}{N^{2}}\frac{\bar{\sigma}^{9}}{(1-\bar{\sigma}^{2})^{5}}=\frac{1}{N^{2}}\left[\frac{1}{z^{9}}+\cdots\right].$
(184)

A few comments on the evaluation of the contractions are in order. In total
$9\times 105$ different diagrams contribute to $\langle{\rm tr}\sigma{\rm
tr}\sigma^{9}\rangle$ but only $9\times 21$ of them do not vanish in the
replica limit. The expression for $A_{3}$ and $A_{4}$ correspond to two
different contraction patterns, namely,

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2.15277pt}}{\rm tr}\sigma{\rm tr}\sigma\sigma\sigma{\rm
tr}\sigma\sigma\sigma\sigma$ (185)

and

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2.15277pt}}{\rm tr}\sigma{\rm tr}\sigma\sigma\sigma{\rm
tr}\sigma\sigma\sigma\sigma.$ (186)

This result (184) for the moments is in agreement with the general formula
obtained by Mehta Mehta-2004 (see also Witte-ml-2013cea )

$M_{2p}=(2p-1)!!\sum_{j=0}^{p}{p\choose j}{N\choose j+1}2^{j},$ (187)

by comparing to $M_{2p}/(NM_{2}^{p})$.

###  C.2 Two-point function

The generating function for the two-point function of the GUE is given by

$Z(z)=\int d\sigma\frac{1}{\det^{N}(z+\sigma)}e^{-\frac{N}{2}\sigma^{2}},$
(188)

with $\sigma$ a Hermitian $2n\times 2n$ matrix and
$z=\underbrace{(x,\cdots,x}_{n},\underbrace{y,\cdots,y}_{n})$. The
corresponding blocks of the $\sigma$ matrix are referred to as the 11 block,
the 12 block, the 21 block and the 22 block. For $x=y$ and both with the same
infinitesimal increment, this becomes the generating function for the one-
point function but now with $2n$ replicas. Note that for the two-point
function, $x$ and $y$ have an opposite infinitesimal imaginary part, while for
the one-point function, the imaginary parts of all $z$ have the same sign.

The two-point correlation function is given by

$\displaystyle C(x,y)$ $\displaystyle=\left.\lim_{n\to
0}\frac{1}{n^{2}N^{2}}\frac{d}{dx}\frac{d}{dy}\log Z(z)\right|_{\rm
connected},$ $\displaystyle=\lim_{n\to 0}\frac{1}{n^{2}}\int d\sigma{\rm
tr}P_{11}\sigma{\rm
tr}P_{22}\sigma\frac{1}{\det^{N}(z+\sigma)}e^{-\frac{N}{2}\sigma^{2}}$ (189)

with $P_{kk}$ the projection on the $kk$ block. The differentiation gives
other contributions but they do not contribute to the connected two-point
function. We evaluate the integral by a saddle point approximation. The saddle
point equation is given by

$\frac{1}{z+\sigma}+\sigma=0.$ (190)

In a 12 block notation, the solution with $\sigma_{12}=0$ has the replica-
diagonal form

$\bar{\sigma}=\left(\begin{array}[]{cc}\bar{\sigma}(x)&0\\\
0&\bar{\sigma}(y)\end{array}\right).$ (191)

The propagators and the vertices follow from the expansion of the logarithm

$-{\rm
tr}\log(z+\sigma)=\sum_{k=1}^{\infty}\frac{1}{k}(\bar{\sigma}\sigma)^{k}.$
(192)

This results in the following quadratic part of the action

$-\frac{N}{2}\left[(1-\bar{\sigma}(x)^{2}){\rm
tr}\sigma_{11}^{2}+(1-\bar{\sigma}(y)^{2})\textrm{Tr}\sigma_{22}^{2}+2(1-\bar{\sigma}(x)\bar{\sigma}(y)){\rm
tr}\sigma_{12}\sigma_{21}\right].$ (193)

To leading order in $1/N^{2}$ two diagrams contribute to the two-point
function:

$\lim_{n\to 0}\frac{1}{n^{2}N^{2}}\left\langle{\rm tr}P_{11}\sigma{\rm
tr}P_{22}\sigma\frac{1}{4}{\rm tr}(\bar{\sigma}\sigma)^{4}\right\rangle,$
(194)

and

$\lim_{n\to 0}\frac{1}{n^{2}N^{2}}\left\langle{\rm tr}P_{11}\sigma{\rm
tr}P_{22}\sigma\frac{1}{18}{\rm tr}^{2}(\bar{\sigma}\sigma)^{3}\right\rangle.$
(195)

The connected part of the first diagram is given by

$\displaystyle\lim_{n\to 0}\frac{1}{n^{2}N^{2}}\left\langle{\rm
tr}\sigma_{11}{\rm tr}\sigma_{22}{\rm
tr}\sigma_{11}\sigma_{12}\sigma_{22}\sigma_{21}\right\rangle_{c}$
$\displaystyle=\frac{1}{N^{2}}\frac{(\bar{\sigma}(x)\bar{\sigma}(y))^{2}}{(1-\bar{\sigma}(x)\bar{\sigma}(y))(1-\bar{\sigma}^{2}(x))(1-\bar{\sigma}^{2}(x))}$
$\displaystyle=\frac{1}{N^{2}}\left(\frac{1}{x^{2}y^{2}}+\frac{1}{x^{3}y^{3}}+\cdots\right).$
(196)

The connect part of the second diagram can be evaluated as

$\displaystyle\lim_{n\to 0}\frac{1}{n^{2}N^{2}}\left\langle{\rm
tr}\sigma_{11}{\rm tr}\sigma_{22}{\rm tr}\sigma_{11}\sigma_{12}\sigma_{21}{\rm
tr}\sigma_{22}\sigma_{21}\sigma_{12}\right\rangle_{c}$
$\displaystyle=\frac{1}{N^{2}}\frac{(\bar{\sigma}(x)\bar{\sigma}(y))^{3}}{(1-\bar{\sigma}(x)\bar{\sigma}(y))^{2}(1-\bar{\sigma}^{2}(x))(1-\bar{\sigma}^{2}(x))}$
$\displaystyle=\frac{1}{N^{2}}\left(\frac{1}{x^{3}y^{3}}+\cdots\right).$ (197)

The sum of the two contributions is equal to

$\displaystyle=\frac{1}{N^{2}}\frac{(\bar{\sigma}(x)\bar{\sigma}(y))^{2}}{(1-\bar{\sigma}(x)\bar{\sigma}(y))^{2}(1-\bar{\sigma}^{2}(x))(1-\bar{\sigma}^{2}(x))}$
$\displaystyle=\frac{1}{N^{2}}\left(\frac{1}{x^{2}y^{2}}+\frac{2}{x^{3}y^{3}}+\cdots\right),$
(198)

which gives the correct result for two-point correlator to order $1/N^{2}$
(see Verbaarschot1984 ) and the $M_{1,1}$ and $M_{2,2}$ moments. Note that in
the normalization of this appendix, $\bar{\sigma}(x)\bar{\sigma}(y)\to 1$ for
$x\to y$, so that the two-point function behaves as $1/(N^{2}(x-y)^{2})$ in
this limit.

##  Appendix D Illustration of (103)–(106) by 3-cross-linked examples

To illustrate the general prescription for the evaluation double-trace
diagrams (see section 6.1) we work out two double-trace diagrams with three
cross-links in this appendix.

The starting point is the fact that
$2^{-N/2}\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\ldots\Gamma_{a_{m}})$ can
only be $0$ or $\pm 1$. This implies

$2^{-N}\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\ldots\Gamma_{a_{m}})\textrm{Tr}(\Gamma_{a_{m}}\Gamma_{a_{m-1}}\ldots\Gamma_{a_{1}})=2^{-N}|\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\ldots\Gamma_{a_{m}})|^{2}=0\text{
or }1.$ (199)

This particular simplicity motivates us to shuffle the $\Gamma$’s to the above
“canonical” ordering. The shuffling will introduce phase factors due to the
relation

$\Gamma_{\alpha}\Gamma_{\beta}=(-1)^{q+c_{\alpha\beta}}\Gamma_{\beta}\Gamma_{\alpha},$
(200)

where $c_{\alpha\beta}=|\alpha\cap\beta|$ is the number of common elements in
sets $\alpha$ and $\beta$. For example let us consider the contraction
$\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{3}}\Gamma_{a_{2}})$,
the shuffling (see figure 18) will introduce a phase factor of
$(-1)^{c_{a_{1}a_{2}}+c_{a_{1}a_{3}}}$, and hence

$\sum_{a_{1},a_{2},a_{3}}\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{3}}\Gamma_{a_{2}})=\sum_{a_{1},a_{2},a_{3}}(-1)^{c_{a_{1}a_{2}}+c_{a_{1}a_{3}}}|\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})|^{2}.$
(201)

In fact in this particular case we can get rid of the phase factor by
cyclically permuting the $\Gamma$’s in the second trace, so we must have

$\sum_{a_{1},a_{2},a_{3}}(-1)^{c_{a_{1}a_{2}}+c_{a_{1}a_{3}}}|\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})|^{2}=\sum_{a_{1},a_{2},a_{3}}|\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})|^{2}.$

However we want to illustrate a general point beyond this simple example, so
we keep the phase factors in our discussion. We see in general each
intersection among the chords introduces a phase factor of
$(-1)^{q+c_{\alpha\beta}}$. We would still like to get rid of the trace in our
equation (201) in favor of a purely combinatorial term, so that it can be
effectively handled by computers. The key question is, when is
$2^{-N}|\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})|^{2}$ equal to
$0$ and when is it equal to $1$?

$a_{1}$$a_{2}$$a_{3}$$a_{1}$$a_{3}$$a_{2}$$=$$\times(-1)^{c_{a_{1}a_{2}}+c_{a_{1}a_{3}}}$$a_{1}$$a_{2}$$a_{3}$$a_{3}$$a_{2}$$a_{1}$
Figure 18: Shuffling a double trace to its canonical ordering (subscripts not
summed over).

Recall that each $\Gamma$ is a product of $q$ different Dirac matrices. The
necessary and sufficient condition for
$\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})$ to be nonvanishing
is that every one of the $N$ Dirac matrices occurs exactly even number
(including zero) of times in the totality of $\Gamma_{a_{1}},\Gamma_{a_{2}}$
and $\Gamma_{a_{3}}$.

This suggests that a useful perspective will be provided by the $d$-variables:
$d_{a_{i_{1}}a_{i_{2}}\ldots a_{i_{k}}}$ is the number of elements common and
only common to the sets $a_{i_{1}},a_{i_{2}},\ldots,a_{i_{k}}$ (naturally,
$i_{1},\ldots,i_{k}$ are all different from each other in this definition). In
the case of three index sets ($k=3$), we have the $d$-variables
$\\{d_{a_{1}a_{2}},d_{a_{1}a_{3}},d_{a_{2}a_{3}},d_{a_{1}a_{2}a_{3}}\\}$.
There are also $\\{d_{a_{1}},d_{a_{2}},d_{a_{3}}\\}$ which count the number of
subscripts that occur exactly once in only $a_{1},a_{2}$ or $a_{3}$, but they
are not independent variables due to the constraint that each index set has
$q$ elements.999For example see equation (205). Both the $c$-variables
($c_{a_{i_{1}}a_{i_{2}}}$) and the $d$-variables have played an important role
in calculating single-trace contractions, and the relations between them were
discussed in Garcia-Garcia-ml-2018kvh . Here we cite one diagram that makes
their relation clear, see figure 19:

$\begin{split}c_{a_{1}a_{2}}&=d_{a_{1}a_{2}}+d_{a_{1}a_{2}a_{3}};\\\
c_{a_{1}a_{3}}&=d_{a_{1}a_{3}}+d_{a_{1}a_{2}a_{3}};\\\
c_{a_{2}a_{3}}&=d_{a_{2}a_{3}}+d_{a_{1}a_{2}a_{3}}.\end{split}$ (202)

In general,

$c_{a_{i_{1}}a_{i_{2}}}=d_{a_{i_{1}}a_{i_{2}}}+\sum_{i_{3}\notin\\{i_{1},i_{2}\\}}d_{a_{i_{1}}a_{i_{2}}a_{i_{3}}}+\sum_{i_{3},i_{4}\notin\\{i_{1},i_{2}\\}}d_{a_{i_{1}}a_{i_{2}}a_{i_{3}}a_{i_{4}}}+\cdots+\sum_{i_{3},\ldots,i_{k}\notin\\{i_{1},i_{2}\\}}d_{a_{i_{1}}a_{i_{2}}a_{i_{3}}\ldots
a_{i_{k}}}.$ (203)

We come back to the case of
$|\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})|^{2}$. Here
$d_{a_{1}a_{2}}$ counts the number of Dirac matrices of which the subscripts
appear exactly in sets $a_{1}$ and $a_{2}$, no less and no more. So there the
Dirac matrices appear exactly twice in the totality of
$\Gamma_{a_{1}},\Gamma_{a_{2}}$ and $\Gamma_{a_{3}}$.

$a_{1}$$a_{2}$$a_{3}$Red region has cardinality
$c_{a_{2}a_{3}}$$a_{1}$$a_{2}$$a_{3}$Red region has cardinality
$d_{a_{2}a_{3}}$

Figure 19: Venn diagrams with three index sets. Each index set is represented
by a circle, containing $q$ elements. The box is the set of all possible
values an index can take, which has cardinality $N$. The box is partitioned
into eight regions. (Taken from Garcia-Garcia-ml-2018kvh .)

The same can be said about $d_{a_{1}a_{3}}$ and $d_{a_{2}a_{3}}$. The total
number of Dirac matrices that appear exactly twice in the totality of
$\Gamma_{a_{1}},\Gamma_{a_{2}},\Gamma_{a_{3}}$ is thus

$d_{2}=d_{a_{1}a_{2}}+d_{a_{1}a_{3}}+d_{a_{2}a_{3}}.$ (204)

On the other hand $d_{a_{1}a_{2}a_{3}}$ counts the number of Dirac matrices
that appear exactly three times in the totality of
$\Gamma_{a_{1}},\Gamma_{a_{2}},\Gamma_{a_{3}}$. Hence if
$d_{a_{1}a_{2}a_{3}}\neq 0$,
$|\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})|^{2}=0$. We also
want to know how many Dirac matrices appear exactly once:

$\begin{split}\text{Exactly once in $a_{1}:$}&\
d_{a_{1}}=q-d_{a_{1}a_{2}}-d_{a_{1}a_{3}}-d_{a_{1}a_{2}a_{3}};\\\
\text{Exactly once in $a_{2}:$}&\
d_{a_{2}}=q-d_{a_{1}a_{2}}-d_{a_{2}a_{3}}-d_{a_{1}a_{2}a_{3}};\\\
\text{Exactly once in $a_{3}:$}&\
d_{a_{3}}=q-d_{a_{1}a_{3}}-d_{a_{2}a_{3}}-d_{a_{1}a_{2}a_{3}}.\end{split}$
(205)

And $|\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})|^{2}=0$ also if
any of the $\\{d_{a_{1}},d_{a_{2}},d_{a_{3}}\\}$ is nonzero. Synthesizing
everything discussed so far, we arrive at the formula

$\begin{split}&2^{-N}\sum_{a_{1},a_{2},a_{3}}\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{3}}\Gamma_{a_{2}})\\\
&=2^{-N}\sum_{a_{1},a_{2},a_{3}}(-1)^{c_{a_{1}a_{2}}+c_{a_{1}a_{3}}}|\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})|^{2}\\\
&=\sum_{d_{a_{1}a_{2}}=0}^{q}\sum_{d_{a_{1}a_{3}}=0}^{q}\sum_{d_{a_{2}a_{3}}=0}^{q}(-1)^{d_{a_{1}a_{2}}+d_{a_{1}a_{3}}}\frac{N!}{(N-3q+d_{2})!d_{a_{1}a_{2}}!d_{a_{1}a_{3}}!d_{a_{2}a_{3}}!}\\\
&\quad\qquad\qquad\qquad\times\delta(q-d_{a_{1}a_{2}}-d_{a_{1}a_{3}})\delta(q-d_{a_{1}a_{2}}-d_{a_{2}a_{3}})\delta(q-d_{a_{1}a_{3}}-d_{a_{2}a_{3}}).\end{split}$
(206)

Note for the second equality we replaced the sum over index sets
$a_{1},a_{2},a_{3}$ by a sum over numbers
$d_{a_{1}a_{2}},d_{a_{1}a_{3}},d_{a_{2}a_{3}}$. In principle there is also a
sum over $d_{a_{1}a_{2}a_{3}}$, but we have argued only the
$d_{a_{1}a_{2}a_{3}}=0$ case contributes. It is clear that in this case there
is only one scenario where the three Kronecker $\delta$ constraint is
satisfied:

$d_{a_{1}a_{2}}=d_{a_{1}a_{3}}=d_{a_{2}a_{3}}=\frac{q}{2},$ (207)

which implies

$2^{-N}\sum_{a_{1},a_{2},a_{3}}(-1)^{c_{a_{1}a_{2}}+c_{a_{1}a_{3}}}|\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})|^{2}=\begin{cases}0\
\,\quad\qquad\qquad\qquad\text{for odd }q,\\\
\frac{N!}{\left(N-\frac{3q}{2}\right)!\left(\left(\frac{q}{2}\right)!\right)^{3}}\qquad\text{for
even }q.\end{cases}$ (208)

In more general cases no such drastic simplification can be expected.

$\alpha_{1}$$\alpha_{2}$$\alpha_{3}$$\alpha_{1}$$\alpha_{3}$$\alpha_{2}$
Figure 20: The single trace chord diagram that has the same intersection
structure as that of figure 18, note here all chords are attached to a single
backbone as opposed to two in figure 18.

It is instructive to contrast equation (206) with the single trace contraction
that has the same intersection structure (figure 20), using the results of
Garcia-Garcia-ml-2018kvh :

$\displaystyle
2^{-N}\sum_{\alpha_{1},\alpha_{2},\alpha_{3}}\textrm{Tr}(\Gamma_{\alpha_{1}}\Gamma_{\alpha_{2}}\Gamma_{\alpha_{3}}\Gamma_{\alpha_{1}}\Gamma_{\alpha_{3}}\Gamma_{\alpha_{2}})$
$\displaystyle=\sum_{\alpha_{1},\alpha_{2},\alpha_{3}}(-1)^{c_{\alpha_{1}\alpha_{2}}+c_{\alpha_{1}\alpha_{3}}}$
$\displaystyle=\sum_{d_{\alpha_{1}\alpha_{2}\alpha_{3}}}\sum_{d_{\alpha_{1}\alpha_{2}}}\sum_{d_{\alpha_{1}\alpha_{3}}}\sum_{d_{\alpha_{2}\alpha_{3}}}(-1)^{d_{\alpha_{1}\alpha_{2}}+d_{\alpha_{1}\alpha_{3}}}\frac{N!}{(N-3q+d_{2}+2d_{\alpha_{1}\alpha_{2}\alpha_{3}})!d_{\alpha_{1}\alpha_{2}}!d_{\alpha_{1}\alpha_{3}}!d_{\alpha_{2}\alpha_{3}}!}$
$\displaystyle\quad\qquad\qquad\times\frac{1}{d_{\alpha_{1}\alpha_{2}\alpha_{3}}!}\frac{1}{(q-d_{\alpha_{1}\alpha_{2}}-d_{\alpha_{1}\alpha_{3}}-d_{\alpha_{1}\alpha_{2}\alpha_{3}})!}\frac{1}{(q-d_{\alpha_{1}\alpha_{2}}-d_{\alpha_{2}\alpha_{3}}-d_{\alpha_{1}\alpha_{2}\alpha_{3}})!}$
$\displaystyle\quad\qquad\qquad\times\frac{1}{(q-d_{\alpha_{1}\alpha_{3}}-d_{\alpha_{2}\alpha_{3}}-d_{\alpha_{1}\alpha_{2}\alpha_{3}})!}.$
(209)

How about cases with both single-trace links and cross links? Let us consider
the example in figure 21, we would need to calculate

$\begin{split}&2^{-N}\sum_{a_{1},a_{2},a_{3},\beta}\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{3}}\Gamma_{\beta}\Gamma_{a_{2}}\Gamma_{\beta})\\\
&=2^{-N}\sum_{a_{1},a_{2},a_{3},\beta}(-1)^{q+c_{a_{1}a_{2}}+c_{a_{1}a_{3}}+c_{a_{2}\beta}}|\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})|^{2}.\end{split}$
(210)

Now the $d$-variables are

$\\{d_{a_{1}a_{2}},d_{a_{1}a_{3}},d_{a_{2}a_{3}},d_{a_{1}\beta},d_{a_{2}\beta},d_{a_{3}\beta},d_{a_{1}a_{2}a_{3}},d_{a_{1}a_{2}\beta},d_{a_{1}a_{3}\beta},d_{a_{2}a_{3}\beta},d_{a_{1}a_{2}a_{3}\beta}\\}.$
(211)

However the constraint of the sum on the right-hand side is still given by the
Kronecker deltas arising from
$|\textrm{Tr}(\Gamma_{a_{1}}\Gamma_{a_{2}}\Gamma_{a_{3}})|^{2}$, just as in
equation (206), so we may say similar things about the conditions on the
contributing summands:

$a_{1}$$a_{2}$$a_{3}$$a_{1}$$a_{3}$$\beta$$a_{2}$$\beta$ Figure 21: A chord
diagram with three cross links and one single-trace link.

the traces with Dirac matrices whose subscripts appear odd number of times in
the totality of $a_{1},a_{2},a_{3}$ are vanishing. This means

$d_{a_{1}\beta},d_{a_{2}\beta},d_{a_{3}\beta},d_{a_{1}a_{2}a_{3}},d_{a_{1}a_{2}a_{3}\beta}$

must all be zero. So the list (211) is shortened to

$\\{d_{a_{1}a_{2}},d_{a_{1}a_{3}},d_{a_{2}a_{3}},d_{a_{1}a_{2}\beta},d_{a_{1}a_{3}\beta},d_{a_{2}a_{3}\beta}\\}.$
(212)

We still need to take into account the Dirac matrices whose subscripts only
appear once exclusively in either $a_{1}$, $a_{2}$ or $a_{3}$, and they are
counted respectively by

$\displaystyle\begin{split}d_{a_{1}}&=q-d_{a_{1}a_{2}}-d_{a_{1}a_{3}}-d_{a_{1}a_{2}\beta}-d_{a_{1}a_{3}\beta},\\\
d_{a_{2}}&=q-d_{a_{1}a_{2}}-d_{a_{2}a_{3}}-d_{a_{1}a_{2}\beta}-d_{a_{2}a_{3}\beta},\\\
d_{a_{3}}&=q-d_{a_{1}a_{3}}-d_{a_{2}a_{3}}-d_{a_{2}a_{3}\beta}-d_{a_{1}a_{3}\beta}.\end{split}$
(213)

So these must also be zero. We also have

$\begin{split}c_{a_{1}a_{2}}&=d_{a_{1}a_{2}}+d_{a_{1}a_{2}\beta},\\\
c_{a_{1}a_{3}}&=d_{a_{1}a_{3}}+d_{a_{1}a_{3}\beta},\\\
c_{a_{2}\beta}&=d_{a_{1}a_{2}\beta}+d_{a_{2}a_{3}\beta}.\end{split}$ (214)

We conclude then that equation (210) is equal to

$\begin{split}&\sum_{d_{a_{1}a_{2}}}\sum_{d_{a_{1}a_{3}}}\sum_{d_{a_{2}a_{3}}}\sum_{d_{a_{1}a_{2}\beta}}\sum_{d_{a_{1}a_{3}\beta}}\sum_{d_{a_{2}a_{3}\beta}}(-1)^{q+d_{a_{1}a_{2}}+d_{a_{1}a_{3}}+d_{a_{1}a_{3}\beta}+d_{a_{2}a_{3}\beta}}\\\
&\qquad\qquad\times\frac{N!}{(N-4q+d_{2}+2d_{3})!}\frac{1}{d_{a_{1}a_{2}}!d_{a_{1}a_{3}}!d_{a_{2}a_{3}}!d_{a_{1}a_{2}\beta}!d_{a_{1}a_{3}\beta}!d_{a_{2}a_{3}\beta}!}\\\
&\qquad\qquad\times\delta(q-d_{a_{1}a_{2}}-d_{a_{1}a_{3}}-d_{a_{1}a_{2}\beta}-d_{a_{1}a_{3}\beta})\\\
&\qquad\qquad\times\delta(q-d_{a_{1}a_{2}}-d_{a_{2}a_{3}}-d_{a_{1}a_{2}\beta}-d_{a_{2}a_{3}\beta})\\\
&\qquad\qquad\times\delta(q-d_{a_{1}a_{3}}-d_{a_{2}a_{3}}-d_{a_{2}a_{3}\beta}-d_{a_{1}a_{3}\beta}),\end{split}$
(215)

where

$\begin{split}d_{2}&=d_{a_{1}a_{2}}+d_{a_{1}a_{3}}+d_{a_{2}a_{3}},\\\
d_{3}&=d_{a_{1}a_{2}\beta}+d_{a_{1}a_{3}\beta}+d_{a_{2}a_{3}\beta}.\end{split}$
(216)

These examples enable us to see how the general situation can be handled,
which is summarized in equations (103)–(106).

##  Appendix E Low-order double-trace moments

Using the properties of double traces discussed so far, we can sort
out101010The sorting is mostly based on the dihedral action property 109
discussed in section 6.2. There are a few groups of diagrams which are not
related by the dihedral action but they have the same intersection structures,
hence the same values. the following few nontrivial low-order double-trace
moments:

$\displaystyle\tilde{M}_{3,3}$
$\displaystyle=3\left(1+(-1)^{q(q-1)/2}\right)t_{123|321},$ (217)
$\displaystyle\tilde{M}_{4,4}$
$\displaystyle=\tilde{M}_{4}^{2}+8\binom{N}{q}^{-1}\tilde{M}_{4}^{2}+8t_{1234|4321}+16t_{1234|3421},$
(218) $\displaystyle\tilde{M}_{3,5}$
$\displaystyle=15(1+\eta)\left(1+(-1)^{q(q-1)/2}\right)t_{123|321},$ (219)
$\displaystyle\tilde{M}_{4,6}$
$\displaystyle=\tilde{M}_{4}\tilde{M}_{6}+12\binom{N}{q}^{-1}\tilde{M}_{4}\tilde{M}_{6}+6(1+\eta)\left(8t_{1234|4321}+16t_{1234|3421}\right)$
$\displaystyle\quad+24\left(t_{1234|43\alpha 21\alpha}+t_{1234|34\alpha
21\alpha}+t_{1234|42\alpha 31\alpha}\right),$ (220)
$\displaystyle\tilde{M}_{5,5}$
$\displaystyle=75\left(1+(-1)^{q(q-1)/2}\right)(1+\eta)^{2}t_{123|321}+5\left(1+(-1)^{q(q-1)/2}\right)$
$\displaystyle\quad\qquad\quad\times\left(t_{12345|54321}+5t_{12345|45321}+5t_{12345|35421}+t_{12345|42531}\right),$
(221) $\displaystyle\tilde{M}_{6,6}$
$\displaystyle=\tilde{M}_{6}^{2}+18\binom{N}{q}^{-1}\tilde{M}_{6}^{2}+36(1+\eta)^{2}\left(8t_{1234|4321}+16t_{1234|3421}\right)$
$\displaystyle\quad+36\times 8(1+\eta)\left(t_{1234|43\alpha
21\alpha}+t_{1234|34\alpha 21\alpha}+t_{1234|42\alpha 31\alpha}\right)$
$\displaystyle\quad+36\left(t_{\alpha 12\alpha 34|43\beta 21\beta}+t_{\alpha
12\alpha 34|34\beta 21\beta}+t_{\alpha 12\alpha 34|42\beta 31\beta}\right.$
$\displaystyle\quad\qquad\qquad\qquad\qquad\quad\left.+2t_{\alpha 12\alpha
34|24\beta 31\beta}+t_{\alpha 12\alpha 34|23\beta 41\beta}\right)$
$\displaystyle\quad+12\left(t_{123456|654321}+6t_{123456|564321}+12t_{123456|465321}\right.$
$\displaystyle\quad\left.\qquad\qquad+3t_{123456|456321}+9t_{123456|563421}+12t_{123456|536421}\right.$
$\displaystyle\quad\left.\qquad\qquad\qquad+12t_{123456|356421}+2t_{123456|436521}+3t_{123456|462531}\right).$
(222)

The $t$-variables with subscripts denote chord diagrams, whose meaning should
be evident from the two examples in figure 22.

$1$$2$$3$$4$$3$$4$$2$$1$$t_{1234|3421}$

$1$$2$$3$$4$$3$$4$$\alpha$$2$$1$$\alpha$$t_{1234|34\alpha 21\alpha}$

Figure 22: Definition of the $t$-variables.

We are interested in the case of $N=32$ and $q=4$ for comparison with the
numerical results. Every term in these low-order double-trace moments can be
calculated using the formulas developed in section 6.1. We only present the
final values of these terms in the next subsection of this appendix.

###  E.1 Values for low-order double-trace contractions ($N=32,q=4$)

In this appendix we list the explicit values of the terms that appeared in the
low-order double -trace moments. Some of the contributing terms are purely
single-trace ($\eta,M_{4},M_{6}$), and are thoroughly discussed in Garcia-
Garcia-ml-2018kvh :

$\begin{split}\eta&=1191/4495,\\\ \tilde{M}_{4}&=10181/4495,\\\
\tilde{M}_{6}&=137227959/20205025.\end{split}$ (223)

The rest of the terms are the connected double-trace contractions that can be
calculated by the method laid out in section 6.1:

$\begin{split}t_{123|321}&=567/323280400,\\\
t_{1234|4321}&=7168277/23250326368000,\\\
t_{1234|3421}&=-555243/23250326368000,\\\ t_{1234|43\alpha
21\alpha}&=5928671067/104510217024160000,\\\ t_{1234|34\alpha
21\alpha}&=518354267/104510217024160000,\\\ t_{1234|42\alpha
31\alpha}&=-711893157/104510217024160000,\\\
t_{12345|54321}&=8116630557/104510217024160000,\\\
t_{12345|45321}&=-462480963/104510217024160000,\\\
t_{12345|35421}&=160002717/104510217024160000,\\\
t_{12345|42531}&=-6933507/104510217024160000,\\\ t_{\alpha 12\alpha 34|43\beta
21\beta}&=7519992447797/469773425523599200000,\\\ t_{\alpha 12\alpha
34|34\beta 21\beta}&=520655935797/469773425523599200000,\\\ t_{\alpha 12\alpha
34|42\beta 31\beta}&=-368559700939/469773425523599200000,\\\ t_{\alpha
12\alpha 34|24\beta 31\beta}&=120146887221/469773425523599200000,\\\ t_{\alpha
12\alpha 34|23\beta 41\beta}&=4043901987381/469773425523599200000,\\\
t_{123456|654321}&=6135081997081/234886712761799600000,\\\
t_{123456|564321}&=-60665945079/234886712761799600000,\\\
t_{123456|465321}&=-687382239/234886712761799600000,\\\
t_{123456|456321}&=31234591/7576990734251600000,\\\
t_{123456|563421}&=89686592361/234886712761799600000,\\\
t_{123456|536421}&=3173529/234886712761799600000,\\\
t_{123456|356421}&=-19709513199/234886712761799600000,\\\
t_{123456|436521}&=5189715001/234886712761799600000,\\\
t_{123456|462531}&=589707393/234886712761799600000.\end{split}$ (224)

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# When all points are generic for ergodic measures

Tomasz Downarowicz and Benjamin Weiss  Tomasz Downarowicz  Faculty of Pure and
Applied Mathematics, Wrocław University of Technology, Wrocław, Poland
<EMAIL_ADDRESS>Benjamin Weiss  Einstein Institute of Mathematics, The
Hebrew University of Jerusalem, Jerusalem, Israel<EMAIL_ADDRESS>
###### Abstract.

We establish connections between several properties of topological dynamical
systems, such as:
– every point is generic for an ergodic measure,
– the map sending points to the measures they generate is continuous,
– the system splits into uniquely (alternatively, strictly) ergodic
subsystems,
– the map sending ergodic measures to their topological supports is
continuous,
– the Cesàro means of every continuous function converge uniformly.

##  1\. Introduction

It follows from the pointwise ergodic theorem that for a topological dynamical
system $(X,T)$ with an ergodic invariant measure $\mu$, a.e. point $x\in X$ is
generic for $\mu$, i.e. for any continuous function $f:X\to\mathbb{R}$, the
Cesàro means

$\frac{1}{n}\sum_{i=0}^{n-1}f(T^{i}x)$

converge to $\int f\,d\mu$. In 1952, Oxtoby [O] showed that in a uniquely
ergodic topological dynamical system $(X,T)$ every point $x\in X$ is generic
for the unique invariant measure $\mu$ carried by $X$, and moreover the
convergence, mentioned above, is uniform. This implies, in particular, that
any topological system $(X,T)$ which splits as a disjoint union of uniquely
ergodic subsystems, has the property that every point $x\in X$ is generic for
an ergodic measure (we will say that $(X,T)$ is _pointwise ergodic_). The
converse need not hold: in 1981, Y. Katznelson and B. Weiss [KW], provided an
example of a pointwise ergodic system with uncountably many ergodic measures,
but just one minimal subset (this makes splitting into uniquely ergodic
subsystems impossible).

In 1970, W. Krieger, following R. Jewett, established that every ergodic
measure-preserving system has a strictly ergodic (i.e. uniquely ergodic and
minimal) topological model. It is worth noticing, that once a uniquely ergodic
model is found, a strictly ergodic model is easily obtained by taking the
(unique) minimal subsystem of that model. Four years later, G. Hansel
generalized the Jewett–Krieger Theorem to nonergodic systems as follows: every
measure-preserving system has a topological model $(X,T)$ which splits as a
union of strictly ergodic subsystems, moreover, the Cesáro means of any
continuous function converge uniformly on $X$ (he called the latter property
_uniformity_). It has to be pointed out, that in the nonergodic case, finding
a model which splits into uniquely ergodic subsystems is by far insufficient:
the minimal subsystems of the uniquely ergodic components need not unite to a
closed set, and the uniformity, which holds individually on each uniquely
ergodic component, need not coordinate to a global uniformity on $X$. Thus, a
model which splits into uniquely ergodic subsystems neither automatically
provides a model which splits into strictly ergodic subsystems, nor one which
is uniform. This is why Hansel’s result can be considered much stronger than
just a straightforward generalization of the Jewett–Krieger Theorem.

We are interested in studying these properties of non-uniquely ergodic
topological dynamical system which in the uniquely (alternatively, strictly)
ergodic case are automatic:

  * •

pointwise ergodicity,

  * •

splitting into uniquely ergodic subsystems,

  * •

splitting into strictly ergodic subsystems,

  * •

uniformity.

Two more properties emerge naturally in the context of pointwise ergodicity
(both of them hold trivially in uniquely ergodic systems):

  * •

continuity of the mapping that associates to each point $x\in X$ the ergodic
measure for which $x$ is generic, and

  * •

continuity of the set-valued function that associates to each ergodic measure
its topological support.

In this paper we establish a diagram of implications (and equivalences)
between these properties (and some of their conjunctions). Our culminating
result is proving a characterization of uniformity in terms of the topological
organization of the uniquely ergodic components.

##  2\. Some terminological conventions

Throughout this paper, by a _topological dynamical system_ we will mean a pair
$(X,T)$, where $X$ is a compact metric space and $T:X\to X$ is a
homeomorphism. By $\mathcal{M}(X)$ and $\mathcal{M}_{T}(X)$ we will denote the
collection of all Borel probability measures on $X$ and of all invariant111In
a topological dynamical system $(X,T)$, a measure $\mu$ on $X$ is _invariant_
if $\int f\,d\mu=\int f\circ T\,d\mu$, for every continuous
$f:X\to\mathbb{R}$. Borel probability measures on $X$, respectively. By saying
“measure” we will always mean a Borel probability, since no other measures
will be considered.

It is well known that $\mathcal{M}_{T}(X)$ endowed with the weak-star topology
is a nonempty metrizable simplex222A metrizable _simplex_ is a compact convex
metric space, such that every point is the barycenter of a unique probability
measure supported by the set of extreme points. and that the extreme points of
$\mathcal{M}_{T}(X)$ are the ergodic measures333A measure
$\mu\in\mathcal{M}_{T}(X)$ is _ergodic_ if all $T$-invariant Borelsets have
measure zero or one.. The set of ergodic measures will be denoted by
$\mathcal{M}^{\mathsf{e}}_{T}(X)$. When $\mathcal{M}_{T}(X)$ is a singleton
(equivalently, when $\mathcal{M}_{T}(X)=\mathcal{M}^{\mathsf{e}}_{T}(X)$), the
system $(X,T)$ is called _uniquely ergodic_. A system $(X,T)$ is called
_strictly ergodic_ if it is uniquely ergodic and minimal444A topological
dynamical system is _minimal_ if it contains no proper closed invariant
subsets.. It is elementary to see that if $(X,T)$ is uniquely ergodic then $X$
contains a unique minimal subset $M$ which equals the topological
support555Topological support of a measure is the smallest closed set of
measure 1. of the unique invariant measure. Then $(M,T)$666We write $T$, but
in fact, we mean here the restriction of $T$ to $M$. is a strictly ergodic
system and $\mathcal{M}_{T}(X)=\mathcal{M}_{T}(M)$.

By a _measure-preserving system_ we will understand a quadruple
$(X,\Sigma,\mu,T)$, where $(X,\Sigma,\mu)$ is a standard (Lebesgue)
probability space and $T:X\to X$ is a measure automorphism. When a topological
dynamical system $(X,T)$ is considered with a fixed invariant measure
$\mu\in\mathcal{M}_{T}(X)$, it becomes a measure-preserving system
$(X,\mu,T)$. The indication of the sigma-algebra is omitted intentionally, as
this role will always be played by the sigma-algebra of Borel sets of $X$
(technically, in order to have a standard probability space, we need to
consider the Borel sigma-algebra completed with respect to $\mu$).

Two measure-preserving systems, say $(X,\Sigma,\mu,T)$ and
$(\bar{X},\bar{\Sigma}_{\bar{\mu}},\bar{\mu},\bar{T})$, are said to be
_isomorphic_ if there exists an isomorphism $\phi:X\to\bar{X}$ of the measure
spaces $(X,\Sigma,\mu)$ and $(\bar{X},\bar{\Sigma},\bar{\mu})$, which
intertwines the actions of $T$ and $\bar{T}$, i.e. such that $\phi\circ
T=\bar{T}\circ\phi$. In such case, by a commonly accepted abuse of
terminology, we will briefly say that $\mu$ and $\bar{\mu}$ are isomorphic.

Two topological dynamical systems $(X,T)$ and $(\bar{X},\bar{T})$ are
_topologically conjugate_ (briefly _conjugate_) if there exists a
homeomorphism $\phi:X\to\bar{X}$ which intertwines the actions of $T$ and
$\bar{T}$.

A topological dynamical system $(\bar{X},\bar{T})$ is called a _model_ for a
measure-preserving system $(X,\Sigma,\mu,T)$, if there exists an invariant
measure $\bar{\mu}\in\mathcal{M}_{\bar{T}}(\bar{X})$ isomorphic to $\mu$. If
$(\bar{X},\bar{T})$ is also uniquely ergodic, we call it a _uniquely ergodic
model_ of $(X,\Sigma,\mu,T)$. If $(X,T)$ is strictly ergodic, we call it a
_strictly ergodic_ (or _Jewett–Krieger_) _model_ of $(X,\Sigma,\mu,T)$. If
$(\bar{X},\bar{T})$ is a uniquely ergodic model of $(X,\Sigma,\mu,T)$, then
$(\bar{M},\bar{T})$ is a strictly ergodic model of $(X,\Sigma,\mu,T)$, where
$\bar{M}$ is a unique minimal subset of $\bar{X}$.

##  3\. Upper and lower semicontinuous partitions and multifunctions

Let $X$ be a compact metric space. By $2^{X}$ we denote the space of all
compact subsets of $X$ topologized by the Hausdorff metric (then $2^{X}$ is
also a compact metric space).

######  Definition 3.1.

A partition $\mathcal{P}$ of $X$ with closed atoms is _upper_ (resp.  _lower_)
_semicontinuous_ if, whenever $(A_{n})_{n\geq 1}$ is a sequence of atoms of
$\mathcal{P}$, converging (in the Hausdorff metric) to a set $S$, then $S$ is
contained in (resp. contains) an atom of $\mathcal{P}$. If $\mathcal{P}$ is
both upper and lower semicontinuous, it is called _continuous_.

The following facts are true. The last statement, for which we give no
reference, is an easy exercise.

######  Proposition 3.2.

  1. (1)

[HY, Theorem 3-31] Let $f:X\to Y$ be any function between any spaces. Define
the _fiber partition_ as the partition of $X$ whose atoms are the sets
$f^{-1}(y)$, $y\in Y$. If $X$ and $Y$ are compact metric and $f$ is continuous
then the fiber partition has closed atoms and is upper semicontinuous.

  2. (2)

[HY, Theorem 3-34] Any partition $\mathcal{P}$ of a compact metric space $X$,
with closed atoms and upper semicontinuous, equals the fiber partition of some
continuous function $f:X\to Y$ onto a compact metric space $Y$.

  3. (3)

The common refinement of any collection of upper semicontinuous partitions
with closed atoms has closed atoms and is upper semicontinuous.

######  Definition 3.3.

A set-valued function $f:Y\to 2^{X}$, where $X$ and $Y$ are compact metric
spaces, is _upper_ (resp.  _lower_) _semicontinuous_ if, whenever
$(y_{n})_{n\geq 1}$ is a sequence in $Y$ converging to some $y\in Y$ and
$f(y_{n})\to S\in 2^{X}$, then $S$ is contained in (resp. contains) $f(y)$.

It is elementary to see, that a function $f:Y\to 2^{X}$ is continuous if and
only if it is both upper and lower semicontinuous.

Note the following obvious fact:

######  Proposition 3.4.

If $f:Y\to 2^{X}$, where $X$ and $Y$ are compact metric spaces, is upper
(resp. lower) semicontinuous, and the collection of values, $\\{f(y):y\in
Y\\}$, happens to be a partition of $X$, then this partition is upper (resp.
lower) semicontinuous.

We remark that the converse need not hold. Indeed, suppose that a set-valued
function $f:Y\to 2^{X}$ is such that $\\{f(y):y\in Y\\}$ is an upper (resp.
lower) semicontinuous partition of $X$. For any bijection $g:X\to X$, the
partition by images of the composition $f\circ g$ is the same as that for $f$,
while the composition may destroy any continuity properties of the function.

A very useful class of lower semicontinuous set-valued functions is provided
by the following fact, whose elementary proof we skip:

######  Proposition 3.5.

Let $X$ be a compact metric space and let $\mathcal{M}(X)$ denote the set of
all probability measures on $X$, equipped with the weak-star topology. Then
the function $\mathsf{supp}:\mathcal{M}(X)\to 2^{X}$, assigning to each
measure its topological support, is lower semicontinuous.

##  4\. Strict uniformity and related notions

Our goal is to introduce and compare several properties, of similar flavor, of
topological dynamical systems. Although none of these properties is new and
they appear in various contexts in the literature, to our knowledge they have
never been given short names (exception: “semi-simple”). For convenience of
our further discussion, we introduce such names below.

######  Definition 4.1.

A topological dynamical system $(X,T)$ will be called:

  1. (I)

_Pointwise ergodic_ , if every point $x\in X$ is generic for an ergodic
measure. In this case, we let $\Phi:X\to\mathcal{M}^{\mathsf{e}}_{T}(X)$
denote the map associating to each point $x\in X$ the ergodic measure for
which $x$ is generic. Also, we let $\mathcal{P}$ denote the fiber partition of
the function $\Phi$.

  2. (II)

_Semi-simple_ , if $X$ is a union of minimal sets.

  3. (III)

_Partitionable_ , if $X$ is a disjoint union of closed invariant uniquely
ergodic sets (we will say _uniquely ergodic subsystems_).

  4. (IV)

_Strictly partitionable_ , if $X$ is a union of strictly ergodic subsystems.

  5. (V)

_Continuously pointwise ergodic_ if it is pointwise ergodic and $\Phi$ is
continuous.

  6. (VI)

_Continuously strictly pointwise ergodic_ if it is continuously pointwise
ergodic and semi-simple.

We begin the analysis of the connections between the notions (I)–(VI) by some
fairly obvious observations:

######  Proposition 4.2.

  1. (1)

A system $(X,T)$ is partitionable if and only if it is pointwise ergodic and
$\mathcal{P}$ has closed atoms.

  2. (2)

A system $(X,T)$ is strictly partitionable if and only if it is pointwise
ergodic and semi-simple.

  3. (3)

A system $(X,T)$ is continuously pointwise ergodic if and only if it is
partitionable and $\mathcal{P}$ and is upper semicontinuous.

  4. (4)

A system $(X,T)$ is continuously strictly pointwise ergodic if and only if it
is strictly partitionable and $\mathcal{P}$ is upper semicontinuous.

  5. (5)

If $(X,T)$ is continuously pointwise ergodic then
$\mathcal{M}^{\mathsf{e}}_{T}(X)$ is compact.

###### Proof.

(1) A partitionable system splits into uniquely ergodic subsystems and each of
these subsystems is pointwise ergodic (see [O]). So the whole system is
pointwise ergodic as well. The atoms of $\mathcal{P}$ coincide with the
uniquely ergodic subsystems, so they are closed. The converse implication is
obvious: the atoms of $\mathcal{P}$, if closed, are uniquely ergodic
subsystems.

(2) A strictly partitionable system is pointwise ergodic by (1), and semi-
simple by definition. If a semi-simple system is pointwise ergodic then so is
its every minimal subsystem. A pointwise ergodic minimal system is uniquely
ergodic (see [KW, Proposition 3.2]), which implies strict partitionability.

(3) The fact that continuity of $\Phi$ implies closed atoms and upper
semicontinuity of $\mathcal{P}$ follows directly from Proposition 3.2 (1).
Suppose that $\mathcal{P}$ has closed atoms and is upper semicontinuous, and
consider a sequence $(x_{n})_{n\geq 1}$ of points in $X$ converging to a point
$x\in X$. Denote $\mu_{n}=\Phi(x_{n})$, $\mu=\Phi(x)$, and let $A_{n}$ and $A$
denote the atoms of $\mathcal{P}$ containing $x_{n}$ and $x$, respectively. By
passing to a subsequence, we may assume that $\mu_{n}$ converge to some
invariant measure $\mu^{\prime}\in\mathcal{M}_{T}(X)$ and that $A_{n}$
converge in $2^{X}$ to some compact set $A^{\prime}\subset X$. By upper
semicontinuity of $\mathcal{P}$, we have $A^{\prime}\subset A$. On the other
hand, by lower semicontinuity of the set-valued function $\mathsf{supp}$ on
$\mathcal{M}_{T}(X)$, we also have $\mathsf{supp}(\mu^{\prime})\subset
A^{\prime}$, which implies that $\mu^{\prime}$ is carried by the atom $A$.
Since $A$ carries only one invariant measure $\mu=\Phi(x)$, we obtain that
$\mu^{\prime}=\mu$, which concludes the proof of continuity of $\Phi$.

(4) is (3) applied to semi-simple systems.

(5) is obvious: a continuous image of a compact space is compact. ∎

In addition to the notation (I)–(VI), let us denote by (P) the condition that
$\mathcal{P}$ has closed atoms and is upper semicontinuous, (equivalently,
that $\Phi$ is continuous), by (B) the property that
$\mathcal{M}^{\mathsf{e}}_{T}(X)$ is compact (in other words, that
$\mathcal{M}_{T}(X)$ is a Bauer simplex777A _Bauer simplex_ is a simplex whose
set of extreme points is closed.), and by (S) the condition that the set-
valued function $\mathsf{supp}$ is continuous on
$\mathcal{M}^{\mathsf{e}}_{T}(X)$.

The property (P) has a special interpretation, which follows immediately from
Proposition 3.2 (2):

######  Proposition 4.3.

Let $(X,T)$ be a pointwise ergodic dynamical system. The partition
$\mathcal{P}$ has closed atoms and is upper semicontinuous if and only if
there exists a topological factor $\pi:X\to Y$ from $(X,T)$ to a system
$(Y,S)$ such that $S$ is the identity on $Y$, and each fiber $\pi^{-1}(y)$ is
uniquely ergodic. In this case, for each invariant measure
$\mu\in\mathcal{M}_{T}(X)$, $\pi$ coincides (up to measure) with the measure-
theoretic factor map from $(X,\mu,T)$ onto the sigma-algebra of invariant
sets.

We continue by establishing more equivalences and implications.

######  Proposition 4.4.

_(VI) $\iff$(IV)$\wedge$(B)$\wedge$(S)_. A system $(X,T)$ is continuously
strictly pointwise ergodic if and only if it is strictly partitionable,
$\mathcal{M}^{\mathsf{e}}_{T}(X)$ is compact, and the set-valued function
$\mathsf{supp}:\mathcal{M}^{\mathsf{e}}_{T}(X)\to 2^{X}$ is continuous. The
partition $\mathcal{P}$ is then continuous.

###### Proof.

Assume that $(X,T)$ is continuously strictly pointwise ergodic. We already
know that then $\mathcal{M}^{\mathsf{e}}_{T}(X)$ is compact. Let
$(\mu_{n})_{n\geq 1}$ be a sequence of ergodic measures converging to an
ergodic measure $\mu$. Let $A_{n}$ and $A$ denote the topological supports of
$\mu_{n}$ and $\mu$, respectively. Consider a sequence of points $x_{n}\in
A_{n}$ converging to a point $x\in X$. Since $\Phi$ is continuous, the
measures $\mu_{n}=\Phi(x_{n})$ converge to $\mu=\Phi(x)$, i.e. $x$ is generic
for $\mu$. Since $(X,T)$ is strictly partitionable, this implies that
$x\in\mathsf{supp}(\mu)=A$. We have shown that $\mathsf{supp}$ is upper
semicontinuous on $\mathcal{M}^{\mathsf{e}}_{T}(X)$. Since $\mathsf{supp}$ is
lower semicontinuous on $\mathcal{M}(X)$, we have proved continuity of
$\mathsf{supp}$ on $\mathcal{M}^{\mathsf{e}}_{T}(X)$.

Conversely, assume that $(X,T)$ is strictly partitionable,
$\mathcal{M}^{\mathsf{e}}_{T}(X)$ is compact, and $\mathsf{supp}$ is
continuous on $\mathcal{M}^{\mathsf{e}}_{T}(X)$. Then $\mathcal{P}$ equals the
partition by images of the continuous set-valued function $\mathsf{supp}$
defined on a compact space, so it is continuous, in particular upper
semicontinuous. Proposition 4.2 (4) now implies that $(X,T)$ is continuously
strictly pointwise ergodic. ∎

We are interested in two other properties of topological dynamical systems,
which have seemingly a much more “ergodic-theoretic” flavor than all the
“topological” properties discussed so far. Nonetheless, we will be able to
provide their topological charactarizations. Both notions have been introduced
in [H], but the respective properties seem to originate in a much earlier work
[O].

######  Definition 4.5.

  * •

A topological dynamical system $(X,T)$ is called _uniform_ if, for every
continuous function $f:X\to\mathbb{R}$, the Cesàro means

$A^{f}_{n}(x)=\frac{1}{n}\sum_{i=0}^{n-1}f(T^{i}x)$

converge uniformly on $X$.

  * •

A system $(X,T)$ which is both uniform and semi-simple is called _strictly
uniform_.

Clearly, in a uniform system, the limit function $\tilde{f}=\lim_{n}A^{f}_{n}$
is continuous. As we have already mentioned several times, uniquely ergodic
systems are uniform. We have the following nearly obvious implication:

######  Proposition 4.6.

Any uniform system is continuously pointwise ergodic. Any strictly uniform
system is continuously strictly pointwise ergodic.

###### Proof.

Clearly, a uniform system $(X,T)$ is pointwise ergodic. It is now crucial to
observe that the partition $\mathcal{P}$ equals the common refinement over all
continuous function $f:X\to\mathbb{R}$ of the fiber partitions of the limit
functions $\tilde{f}$. Now, by Proposition 3.2 (1) and (3), $\mathcal{P}$ is
upper semicontinuous, and by Proposition 4.2 (3), $(X,T)$ is continuously
pointwise ergodic. The statement about strictly uniform systems follows from
the above applied to semi-simple systems. ∎

The second part of Proposition 4.6 places strict uniformity at the top of the
hierarchy of properties discussed in this paper, as the most restrictive one.
A notable role of strict uniformity in the theory of dynamical systems is
sanctioned by two theorems, the first of which is the celebrated
Jewett–Krieger Theorem ([J] and [K]), the second, although vastly generalizes
the first one, is much less commonly known.

######  Theorem 4.7.

(Jewett–Krieger888Jewett showed this assuming weak mixing, Krieger relaxed
this assumption to ergodicicty., 1970) Every ergodic measure-preserving system
$(X,\Sigma,\mu,T)$ has a strictly ergodic topological model.

As a corollary, we obtain that

  * •

_every ergodic measure-preserving system has a strictly uniform topological
model_.

The second theorem is due to G. Hansel [H], and generalizes the above to non-
ergodic measure-preserving systems:

######  Theorem 4.8.

(Hansel, 1974) Every measure-preserving system $(X,\Sigma,\mu,T)$ has a
strictly uniform topological model.

Hansel’s theorem has the following remarkable interpretation: the class of
strictly uniform systems, although the smallest (among the classes discussed
in this paper), is still reach enough to capture “the entire ergodic theory”.

We will denote uniformity by (U) and strict uniformity by (sU). The following
theorem, which to our best knowledge has not been noted up to date,
constitutes the main result of this paper. Combined with the equivalence
(V)$\iff$(IV)$\wedge$(P), it characterizes the “ergodic” property of
uniformity by means of “topological organization” of the uniquely ergodic
subsystems.

######  Theorem 4.9.

_(U) $\iff$(V), (sU)$\iff$(VI)_. A system $(X,T)$ is uniform if and only if it
is continuously pointwise ergodic. It is strictly uniform if and only if it is
continuously strictly pointwise ergodic.

###### Proof.

In view of Proposition 4.6, we only need to prove that continuously pointwise
ergodic systems are uniform. So, assume that $(X,T)$ is continuously pointwise
ergodic. Fix any continuous function $f:X\to[0,1]$ and a point $x_{0}\in X$.
Let $\mu_{0}=\Phi(x_{0})$. Given $\varepsilon>0$, there exists an
$n_{0}=n(x_{0})$ such that

$\left|A_{n_{0}}^{f}(x_{0})-\int f\,d\mu_{0}\right|<\frac{\varepsilon}{2}.$

The $|A_{n_{0}}^{f}(x)-\int f\,d\mu|$ is a doubly continuous function of
$(x,\mu)$, so the inequality $|A_{n_{0}}^{f}(x)-\int
f\,d\mu|<\frac{\varepsilon}{2}$ holds for all measures $\mu$ in a sufficiently
small neighborhood $V_{0}$ of $\mu_{0}$ and all points $x$ is a sufficiently
small neighborhood $U_{0}$ of $x_{0}$. By continuity of $\Phi$, we can choose
$U_{0}$ so small that $\Phi(U_{0})\subset V_{0}$. Then we have

$\left|A_{n_{0}}^{f}(x)-\int f\,d\mu_{x}\right|<\frac{\varepsilon}{2},$

for all $x\in U_{0}$, where $\mu_{x}=\Phi(x)$. By compactness, $X$ is covered
by finitely many neighborhoods $U_{i}$ created analogously (with the same
$\varepsilon$) for some points $x_{i}\in X$ ($i=1,2,\dots,l$). This cover
defines finitely many numbers $n_{i}=n(x_{i})$. Let $n$ be any integer larger
than

$M=\frac{2\max\\{n_{1},\dots,n_{l}\\}}{\varepsilon}.$

Pick any $x\in X$. We can divide the forward orbit of $x$ of length $n$ into
“portions”, as follows: $x$ belongs to some $U_{i_{0}}$, so we choose
$n_{i_{0}}$ as the length of first portion. Next, $T^{n_{i_{0}}}x$ belongs to
some $U_{i_{1}}$, so we choose $n_{i_{1}}$ as the length of second portion.
Next, $T^{n_{i_{0}}+n_{i_{1}}}x$ belongs to some $U_{i_{2}}$, so we choose
$n_{i_{2}}$ as the length of third portion. And so on, until we are left with
a “tail” of some length of length $m<\max\\{n_{1},\dots,n_{l}\\}$, starting at
$T^{n_{i_{0}}+n_{i_{1}}+\cdots+n_{i_{q}}}x$ for some $q\in\mathbb{N}$. We have

$A_{n}^{f}(x)=\frac{n_{i_{0}}}{n}A_{n_{i_{0}}}x+\sum_{j=1}^{q}\frac{n_{i_{j}}}{n}A_{n_{i_{j}}}^{f}(T^{n_{i_{0}}+n_{i_{1}}+\cdots+n_{i_{j-1}}}x)+\frac{m}{n}A_{m}^{f}(T^{n_{i_{1}}+n_{i_{2}}+\cdots+n_{i_{q}}}x).$

Because all the points $T^{n_{i_{j}}}x$ are generic for the same measure
$\mu_{x}$, we have

$\left|A_{n_{i_{0}}}^{f}(x)-\int f\,d\mu_{x}\right|<\frac{\varepsilon}{2},$

and for each $j=1,2,\dots,q$,

$\left|A_{n_{i_{j}}}^{f}(T^{n_{i_{0}}+n_{i_{1}}+\cdots+n_{i_{j-1}}}x)-\int
f\,d\mu_{x}\right|<\frac{\varepsilon}{2},$

while the last term
$\frac{m}{n}A_{m}^{f}(T^{n_{i_{1}}+n_{i_{2}}+\cdots+n_{i_{q}}}x)$ has absolute
value smaller than $\frac{\varepsilon}{2}$. By averaging, we obtain that the
inequality

$\left|A_{n}^{f}(x)-\int f\,d\mu_{x}\right|<\varepsilon$

holds for any $n>M$ uniformly for all $x\in X$, which implies uniformity.

The equivalence between strict uniformity and continuous strict pointwise
ergodicity is the (just proved) equivalence between continuous pointwise
ergodicity and uniformity, applied to semi-simple systems. ∎

The following diagram shows all the implications between the conditions
discussed in this paper: (I)–(VI), (U), (sU), (S), (P) and (B):

$\begin{matrix}&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\text{(IV)}\wedge\text{(B)}\wedge\text{(S)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!&&\\!\\!\\!\\!\text{(I)}\wedge\text{(II)}\\!\\!\\!\\!\\!&&\\\
&&\Updownarrow&&\Updownarrow&&\\\
&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\text{(IV)}\wedge\text{(P)}\\!\\!\\!\\!\\!\\!&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\text{(III)}\wedge\text{(II)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!&&\\\
&&\Updownarrow&&\Updownarrow\scriptstyle{df}\\!\\!\\!\\!\\!&&\\\
\text{(V)}\wedge\text{(II)}\\!\\!\\!&\overset{df}{\iff}&\text{(VI=sU)}&\implies&\text{(IV)}&\implies&\text{(II)}\\\
&&\Downarrow&&\Downarrow&&\\\
\text{(III)}\wedge\text{(P)}\\!\\!\\!&\overset{df}{\iff}&\text{(V=U)}\,&\implies&\text{(III)}&\implies&\text{(I)}\\\
&&\Downarrow&&&&\\\ &&\text{(B)}&&&&\end{matrix}$

By analogy to the conditions equivalent to (VI), one might ask: is (V)
equivalent to (III)$\wedge$(B)$\wedge$(S)? This question has a negative
answer: as we will show in examples, neither (III)$\wedge$(B)$\wedge$(S)
implies (V) nor (V) implies (S).

Nonetheless, the conjunction (III)$\wedge$(B)$\wedge$(S) has a special
meaning. Recall that in a uniquely ergodic system $(X,T)$, the unique minimal
set $M$ produces a strictly ergodic (hence strictly uniform) subsystem $(M,T)$
carrying the same invariant measure as $(X,T)$. Notice that if $(X,T)$ is
partitionable then the union of all minimal sets (regardless of whether it is
closed or not) supports the same invariant measures as $(X,T)$. Thus, a
natural question arizes: under what conditions does the union of minimal
subsets of a partitionable system $(X,T)$ produce a strictly uniform
subsystem. Below we give a complete answer.

######  Theorem 4.10.

Let $(X,T)$ be partitionable. Let $M$ denote the union of all minimal subsets
of $X$. Then $M$ is closed and the system $(M,T)$ is strictly uniform if and
only if $\mathcal{M}^{\mathsf{e}}_{T}(X)$ is compact and the set-valued
function $\mathsf{supp}:\mathcal{M}^{\mathsf{e}}_{T}(X)\to 2^{X}$ is
continuous.

###### Proof.

First notice that partitionability of $(X,T)$ implies that the family of all
minimal subsets coincides with
$\mathsf{supp}(\mathcal{M}^{\mathsf{e}}_{T}(X))$, i.e. with the image of
$\mathcal{M}^{\mathsf{e}}_{T}(X)$ via $\mathsf{supp}$. Now assume that
$\mathcal{M}^{\mathsf{e}}_{T}(X)$ is compact and
$\mathsf{supp}:\mathcal{M}^{\mathsf{e}}_{T}(X)\to 2^{X}$ is continuous. It
follows that $\mathsf{supp}(\mathcal{M}^{\mathsf{e}}_{T}(X))$ is compact,
hence closed in $2^{X}$. This easily implies that the union $M$ of all minimal
sets is closed in $X$. The system $(M,T)$ satisfies partitionability (III) and
semi-simplicity (II), so it is strictly pointwise ergodic (IV). It also
satisfies (B) and (S). Since (IV)$\wedge$(B)$\wedge$(S)$\iff$(sU), $(M,T)$ is
strictly uniform.

Conversely, assume that $M$ is closed and that the system $(M,T)$ is strictly
uniform. Then $(M,T)$ satisfies (B) and (S), i.e.
$\mathcal{M}^{\mathsf{e}}_{T}(M)$ is compact and the function
$\mathsf{supp}:\mathcal{M}^{\mathsf{e}}_{T}(M)\to 2^{M}\subset 2^{X}$ is
continuous. By partitionability, we have
$\mathcal{M}^{\mathsf{e}}_{T}(M)=\mathcal{M}^{\mathsf{e}}_{T}(X)$, implying
that the conditions (B) and (S) hold also for $(X,T)$. ∎

We remark, that without partitionability of $(X,T)$ (even with pointwise
ergodicity assumed instead), Theorem 4.10 loses its valuable interpretation.
The conjunction (B)$\wedge$(S) still implies that $M$ is compact and whenever
$(M,T)$ is pointwise ergodic, it is strictly uniform. However, without
partitionability, there is no guarantee that
$\mathcal{M}_{T}(M)=\mathcal{M}_{T}(X)$ and $(M,T)$ cannot replace $(X,T)$ in
the role of a topological model of some measure-preserving systems. This
happens for instance in the example given by Katznelson and Weiss in [KW] (see
below for more details).

On the other hand, partitionability of $(X,T)$ does not follow from the
assumptions that $(X,T)$ is pointwise ergodic, $M$ is compact, even if $(M,T)$
is strictly uniform and carries the same invariant measures as $(X,T)$. The
simplest example here is $X$ being the two-point compactification of
$\mathbb{Z}$, $X=\mathbb{Z}\cup\\{-\infty,\infty\\}$, with $Tx=x+1$. In this
system $M=\\{-\infty,\infty\\}$ is closed and $(M,T)$ is trivially strictly
uniform. Nonetheless, $(X,T)$ is pointwise ergodic without being
partitionable; the set of points generic for $\delta_{\infty}$ equals
$\mathbb{Z}\cup\\{\infty\\}$ and is not closed.

##  5\. Examples

One of the first examples showing phenomena associated to the notions
discussed in the preceding paragraph is given in [KW]. The system $(X,T)$ is
pointwise ergodic but not partitionable. The set
$\mathcal{M}^{\mathsf{e}}_{T}(X)$ is homeomorphic to the unit interval by a
map $t\mapsto\mu_{t}$, in particular $\mathcal{M}^{\mathsf{e}}_{T}(X)$ is
compact and uncountable. The topological supports of the ergodic measure form
a nested family of sets $E_{t}$ intersecting at a fixpoint $x_{0}$ (which is a
unique minimal subset). It can be verified that also the set-valued map
$t\mapsto E_{t}$ is continuous (into $2^{X}$), implying that $\mathsf{supp}$
is continuous on $\mathcal{M}^{\mathsf{e}}_{T}(X)$. This example shows that
(I)$\wedge$(B)$\wedge$(S)$\,\ \not\\!\\!\\!\implies$(III).

We continue with six other examples. They are elementary, yet they resolve
possible questions about other implications. The space $X$ in the first five
of them is the same and consists of a sequence of circles $C_{k}$ ($k\geq 1$)
converging to a circle $C$ (in the last example we add to this space one more
circle $C^{\prime}$). The circles $C_{k}$ and $C$ can be imagined as all
having equal diameters and placed on parallel planes, or as concentric circles
with radii of $C_{k}$ converging to the radius of $C$. On each circle $C_{k}$
we pick one point $c_{k}$ so that these points converge to a point $c\in C$.
We will call the points $c_{k}$ and $c$ the _origins_ of the circles $C_{k}$
and $C$, respectively (they will be used in Examples 5.2, 5.3 and 5.6).

######  Example 5.1.

In this example, the transformation $T$ restricted to each circle $C_{k}$
(henceforth denoted by $T_{k}$) is an irrational rotation. The angles of the
rotations $T_{k}$ tend to zero with $k$, so that $T$ restricted to $C$ is the
identity map (see Figure 1).

Figure 1.

Clearly, this system is strictly partitionable. The strictly ergodic
subsystems are the circles $C_{k}$ and all single points on $C$. The set
$\mathcal{M}^{\mathsf{e}}_{T}(X)$ consists of the normalized Lebesgue measures
$\lambda_{k}$ on the circles $C_{k}$ and the Dirac measures $\delta_{x}$ at
all points $x\in C$. This set of measures is not closed in
$\mathcal{M}_{T}(X)$ (the measures $\lambda_{k}$ converge to the non-ergodic
Lebesgue measure on $C$). So, the system cannot be continuously pointwise
ergodic and, by Theorem 4.9, it is not uniform. It follows, that the partition
$\mathcal{P}$ connot be upper semicontinuous, and
$\Phi:X\to\mathcal{M}_{T}(X)$ cannot be continuous (all these negative
properties can be seen directly, but we want show how our diagram works). On
the other hand, the map $\mathsf{supp}$ is easily seen to be continuous, even
on the closure of $\mathcal{M}^{\mathsf{e}}_{T}(X)$, which consists of two
disjoint compact sets: one containing the Lebesgue measures on all the circles
(including $C$) and second, consisting of all Dirac measures at the points
$x\in C$. This example shows that without (B), (IV)$\wedge$(S)$\,\
\not\\!\\!\\!\implies$(VI) (not even (V)).

######  Example 5.2.

In the previous example we slightly change the transformations $T_{k}$. For
each $k$ we pick an arc $I_{k}$ containing the origin $c_{k}$. The diameters
of the arcs decrease to zero, so that the arcs $I_{k}$ converge to the
singleton $\\{c\\}$. On each $I_{k}$ we “slow down the motion”, meaning that
we locally bring the graph of $T_{k}$ closer to the diagonal (see Figure 2).
Since $T$ is not changed outside the arcs $I_{k}$, (and on each $I_{k}$ the
map is even closer to identity than before), the maps $T_{k}$ converge, as
before, to the identity map on $C$. We can easily arrange the modifications so
that each $T_{k}$ is conjugate to some irrational rotation (much slower than
before). We let $\mu_{k}$ denote the unique invariant (and ergodic) measure
supported by $C_{k}$. Each orbit in $C_{k}$ now spends much more time in
$I_{k}$ than before, so $\mu_{k}(I_{k})$ is larger than $\lambda_{k}(I_{k})$.
By slowing down strongly enough we can easily arrange that $\mu_{k}(I_{k})\to
1$, implying that the measures $\mu_{k}$ converge to $\delta_{c}$.

Figure 2.

The system is strictly partitionable with the partition $\mathcal{P}$ the same
as in the previous example. Since this partition is not upper semicontinuous,
the system is not continuously pointwise ergodic. However, this time, the set
of ergodic measures $\\{\mu_{k}:k\geq 1\\}\cup\\{\delta_{x}:x\in C\\}$ is
closed in $\mathcal{M}_{T}(X)$ (because $\mu_{k}\to\delta_{c}$), hence
compact. Since (IV)$\wedge$(B) holds and (VI) does not, it must be (S) that
fails (this can also be seen directly: $\mathsf{supp}(\mu_{k})=C_{k}\to
C\neq\mathsf{supp}(\delta_{c})$). This example shows that without (S),
(IV)$\wedge$(B)$\,\ \not\\!\\!\\!\implies$(VI) (not even (V)).

######  Example 5.3.

This is the declared example showing that (III)$\wedge$(S)$\wedge$(B)$\,\
\not\\!\\!\\!\implies$(V). Let the map $T_{k}$ on the circle $C_{k}$ fix the
origin $c_{k}$ and move all other points, say, clockwise, but with the speed
of the movement decaying as $k$ increases, so that on $C$ we have the identity
map (see figure 3).

Figure 3.

Clearly, the system is partitionable, with the same partition $\mathcal{P}$ as
in Examples 5.1 and 5.2. Since $\mathcal{P}$ is not upper semicontinuous, the
system is not continuously pointwise ergodic. The set of ergodic measures
equals $\\{\delta_{c_{k}}:k\geq 1\\}\cup\\{\delta_{x}:x\in C\\}$ and is
compact, and the set-valued function
$\mathsf{supp}:\mathcal{M}^{\mathsf{e}}_{T}(X)\to 2^{X}$ is obviously
continuous.

######  Example 5.4.

This is the other declared example showing that (V)$\,\
\not\\!\\!\\!\implies$(S) (and some other failing implications). The maps
$T_{k}$ on the circles $C_{k}$ are modifications of one fixed irrational
rotation. We can “slow down the motion” on each circle $C_{k}$ so that the
resulting maps $T_{k}$ are conjugate to irrational rotations, and converge to
a continuous map on $C$ which fixes just one point $c$ and moves all other
points, say, clockwise (see Figure 4). Then $\delta_{c}$ is the unique
invariant (and ergodic) measure supported by $C$. The unique invariant (and
ergodic) measures $\mu_{k}$ supported by $C_{k}$ must accumulate at invariant
measures supported by $C$, so they have no other choice than to converge to
$\delta_{c}$.

Figure 4.

The the system is partitionable but not semi-simple. The partition
$\mathcal{P}$ into closed uniquely ergodic sets equals $\\{C_{k}:k\geq
1\\}\cup\\{C\\}$ and is continuous, so $(X,T)$ is continuously pointwise
ergodic. In particular, $\mathcal{M}^{\mathsf{e}}_{T}(X)$ is compact (the last
condition can also be seen more directly, by noting that
$\mathcal{M}^{\mathsf{e}}_{T}(X)=\\{\mu_{k}:k\geq 1\\}\cup\\{\delta_{c}\\}$).
However, since the supports of $\mu_{k}$ are whole circles, while that of the
limit measure $\delta_{c}$ is a singleton, the function
$\mathsf{supp}:\mathcal{M}^{\mathsf{e}}_{T}(X)\to 2^{X}$ is not continuous.
This example shows that (V)$\,\ \not\\!\\!\\!\implies$(II) and (V)$\,\
\not\\!\\!\\!\implies$(S). This example also shows that although (VI) implies
continuity of $\mathcal{P}$, (V) combined with this continuity still does not
imply (VI).

######  Example 5.5.

Now we change the maps $T_{k}$ so that they all are the same as that on $C$
(see figure 5).

Figure 5.

Like in the previous example, the system is partitionable but not semi-simple,
with $\mathcal{P}$ continuous, hence $(X,T)$ is continuously pointwise
ergodic. This time the map $\mathsf{supp}:\mathcal{M}^{\mathsf{e}}_{T}(X)\to
2^{X}$ is continuous. The example shows that (V) combined with (S) (and
additionally with the continuity of $\mathcal{P}$) still does not suffice for
(VI).

######  Example 5.6.

In the above example, we can add to the space one more circle $C^{\prime}$
that is tangent to $C$ at $c$. The dynamics on $C^{\prime}$ is symmetric to
that on $C$, so that all points on $C^{\prime}$ are generic for $\delta_{c}$
and $C\cup C^{\prime}$ is an atom of $\mathcal{P}$ (see figure 6).

Figure 6.

So modified system has the same properties as in the preceding example, except
that the partition $\mathcal{P}$ is now upper semicontinuous not continuous.
This example shows that (V) (contrary to (VI)) does not imply continuity of
$\mathcal{P}$, even when combined with (S).

## Acknowledgements

The authors thank Mariusz Lemańczyk for raising a question that triggered this
research.

Tomasz Downarowicz is supported by National Science Center, Poland (Grant No.
2018/30/M/ST1/00061) and by the Wrocław University of Science and Technology
(Subsidy for 2020, budget no. 8201003902 MPK: 9130730000).

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