# Integrated Sensing and Communications for Unsourced Random Access:
Fundamental Limits

Mohammad Javad Ahmadi⋆ and Rafael F. Schaefer⋆,†
⋆ Chair of Information Theory and Machine Learning, Technische Universität
Dresden
⋆ Cluster of Excellence CeTI, † BMBF Research Hub 6G-life
† Barkhausen Institut, Dresden, Germany
email: {mohammad_javad.ahmadi<EMAIL_ADDRESS>H. Vincent Poor
This work of M. J. Ahmadi and R. F. Schaefer was supported by the German
Research Foundation (DFG) as part of Germany’s Excellence Strategy - EXC
2050/1 - Project ID 390696704 - Cluster of Excellence _“Centre for Tactile
Internet with Human-in-the-Loop” (CeTI)_ of Technische Universität Dresden.
This work of R. F. Schaefer was further supported in part by the German
Federal Ministry of Education and Research (BMBF) within the national
initiative on 6G Communication Systems through the research hub _6G-life_
under Grant 16KISK001K. This work of H. V. Poor was supported by the U.S.
National Science Foundation under Grant ECCS-2335876. Dept. of Electrical and
Computer Engineering
Princeton University
email<EMAIL_ADDRESS>
###### Abstract

This work considers the problem of integrated sensing and communication (ISAC)
with a massive number of unsourced and uncoordinated users. In the proposed
model, known as the unsourced ISAC system (UNISAC), all active communication
and sensing users share a short frame to transmit their signals, without
requiring scheduling with the base station (BS). Hence, the signal received
from each user is affected by significant interference from numerous
interfering users, making it challenging to extract the transmitted signals.
UNISAC aims to decode the transmitted message sequences from communication
users while simultaneously detect active sensing users, regardless of the
identity of the decoded and detected users. In this paper, we derive an
achievable performance limit for UNISAC and demonstrate its superiority over
conventional approaches such as ALOHA, time-division multiple access, treating
interference as noise, and multiple signal classification. Through numerical
simulations, we validate the UNISAC’s effectiveness in detecting and decoding
a large number of users.

###### Index Terms:

Integrated Sensing and Communications, Unsourced Random Access, Achievability
Bound, Massive Random Access.

##  I Introduction

In unsourced random access (URA), introduced by Polyanskiy in [1], a massive
number of unidentified users perform sporadic transmission within a short
period of time without engaging in any scheduling with the BS. Eliminating the
need for scheduling significantly reduces the signaling overhead and latency
of the URA system, thereby granting it the potential to support an unbounded
number of users. In addition, the BS focuses solely on the information
transmitted without considering the identity of the sender of the message.
This approach is particularly suitable for the next generation of
communication networks, where millions of inexpensive devices are connected to
the system, with only a fraction of them active during each transmission [2,
7, 8, 3, 9, 10, 6, 4, 11, 5].

Existing research on URA has predominantly concentrated on extracting
information transmitted by unsourced users (communication purpose). However,
with the advent of sixth-generation (6G) mobile networks, additional
functionalities such as computing, sensing, and security have become
imperative [12, 13]. As hardware architectures, channel characteristics, and
signal processing methods become increasingly similar across these different
tasks, it is necessary to adapt the URA system to jointly accommodate these
diverse purposes rather than focusing solely on information transmission [14].

In this paper, we consider a URA system that provides both sensing and
communication functions concurrently, referred to as unsourced integrated
sensing and communications (UNISAC). Specifically, we incorporate the object
detection task alongside the communication task to accommodate a wide range of
users. The UNISAC model serves two user categories: communication users and
sensing users. Each category has distinct objectives for the receiver:
decoding transmitted messages from communication users and detecting sensing
users while estimating their angle of arrival (AOA). Due to the aggregation of
signals from various sensing and communication sources at the receiver, the
URA faces the complex task of analyzing each user’s signal among the
interference from other users. In this paper, we investigate the achievable
performance of the UNISAC model, taking into account the misdetection error
and mean-squared error of AOA estimation (MSEAOA) for sensing users, in
addition to the decoding error of communication users. Numerical results
indicate that the proposed UNISAC model exhibits superior performance compared
to conventional multiple access systems such as treat interference as noise
(TIN), time division multiple access (TDMA), ALOHA, and multiple signal
classification (MUSIC).

The rest of the paper is organized as follows: In Section II, we outline the
system model. Then, the achievable bound of the UNISAC model is derived in
Section III. Finally, before concluding the paper in Section V, we present
numerical results in Section IV.

Notation: Lower-case and upper-case boldface letters are used for denoting a
vector and a matrix, respectively; $\mathrm{diag}(\mathbf{A})$ generates a
diagonal matrix by setting all off-diagonal elements of $\mathbf{A}$ to zero;
$\mathrm{det}(\mathbf{A})$ calculates the determinant of the matrix
$\mathbf{A}$; $\mathbf{I}_{N}$ is an $N\times N$ identity matrix;
$\mathrm{Re}(t)$ gives the real part of $t$; $\|\mathbf{A}\|^{2}$ calculates
the squared Frobenius norm of the matrix $\mathbf{A}$; the notation
$\mathbf{A}\preceq\mathbf{B}$ indicates that the matrix
$\mathbf{A}-\mathbf{B}$ is negative semidefinite; $\mathcal{CN}(0,b)$ denotes
the zero-mean circularly symmetric complex white Gaussian noise with variance
$b$; $\mathbf{0}_{n,m}$ denotes a matrix of size $n\times m$ where all
elements are zero; $F_{\chi^{2}}(t)$ is the cumulative distribution function
of the chi-squared distribution with $t$ degrees of freedom and
$F_{\chi^{2}}^{-1}(t)$ is its inverse; $\chi_{T}^{2}$ denotes a chi-squared
random variable with $T$ degrees of freedom; the function $\mathbf{t}^{R}$
reverses the order of elements in the vector $\mathbf{t}$.

##  II System Model

Consider an ISAC system where $K_{T_{s}}$ sensing users (objects) and
$K_{T_{c}}$ communication users are connected to a BS. As shown in Fig. 1, the
BS is equipped with an $M$-element uniform linear array (ULA) with element
spacing of $d=\lambda/2$, where $\lambda$ represents the wavelength of the
received signal. Due to sporadic behaviour, only a small fraction of sensing
and communication users are active at each transmission, and a significant
portion of users do not transmit any signal. All the active sensing and
communication users simultaneously share the same length-$n$ frame for
transmitting signals. To generate its transmitted signal, each communication
user maps its bit sequence $\mathbf{w}_{j}\in\mathbb{C}^{B_{c}}$ to the first
$2^{B_{c}}$ rows of the codebook
$\mathbf{A}\in\mathbb{C}^{2^{B_{c}+B_{s}}\times n}$, and each sensing user
generates its transmitted signal by randomly choosing a row among the last
$2^{B_{s}}$ rows of the codebook $\mathbf{A}$. Note that since there are no
message sequences to be transmitted by the sensing users, the value of $B_{s}$
lacks a predefined value and must be optimally chosen according to the system
configuration.

Considering the synchronous transmission of all the users, the received signal
at the BS is written as

$\displaystyle\mathbf{Y}=\sum_{\text{\tiny$j\in\\{\mathcal{A}_{c},\mathcal{A}_{s}\\}$}}\mathbf{b}_{\theta_{j}}\mathbf{a}_{j}+\mathbf{Z}\in\mathbb{C}^{M\times
n},$ (1)

with

$\displaystyle\mathbf{b}_{\theta_{j}}=\left[1,e^{-j\pi\theta_{j}},e^{-j2\pi\theta_{j}},\cdots,e^{-j\pi(M-1)\theta_{j}}\right]^{T},$
(2)

where $\mathbf{a}_{j}\in\mathbb{C}^{n}$ denotes the $j$th row of the matrix
$\mathbf{A}$, $\theta_{j}$ represents the AOA corresponding to the codeword
$\mathbf{a}_{j}$, ${\mathcal{A}_{c}}\subset\\{1,2,...,2^{B_{c}}\\}$ and
${\mathcal{A}_{s}}\subset\\{2^{B_{c}}+1,...,2^{B_{s}+B_{c}}\\}$ are the sets
of active row indices of the codebook $\mathbf{A}$ associated with
communication and sensing users, respectively (active row index refers to a
row that is selected and transmitted by an active user), and $\mathbf{Z}$
comprises symmetric complex Gaussian elements $\mathcal{CN}(0,\sigma_{z}^{2})$
which serves as an additive Gaussian noise matrix. The objectives of the
receiver are: 1) to detect the presence of active sensing users
($j\in\mathcal{A}_{s}$) by detecting their transmitted signals as well as
estimating their corresponding AOAs, and 2) to decode the transmitted bit-
sequences of active communication users,
$\mathbf{w}_{j},j\in{\mathcal{A}_{c}}$.
For measuring the detection and decoding performance of the UNISAC model, the
per-user probability of error (PUPE) is adopted as

$\displaystyle\epsilon$
$\displaystyle=\dfrac{\mathbb{E}\left\\{{|\mathcal{L}_{c,md}|+|\mathcal{L}_{s,md}|}+{|\mathcal{L}_{c,coll}|+|\mathcal{L}_{s,coll}|}\right\\}}{{|{\mathcal{A}_{c}}|+|{\mathcal{A}_{s}}|}}$
(3)

where $\mathcal{L}_{c,md}$ and $\mathcal{L}_{s,md}$ denote the list of active
communication and sensing user indices that are not detected, respectively,
and $\mathcal{L}_{c,coll}$ and $\mathcal{L}_{s,coll}$ are the list of
communication and sensing user indices that are in collision, respectively.
Moreover, concerning the AOA estimation for the sensing users, the performance
metric is the MSEAOA for the successfully detected sensing users, defined as
follows:

$\displaystyle\Delta$
$\displaystyle=\dfrac{1}{|\mathcal{L}_{s,d}|}\sum_{i\in\mathcal{L}_{s,d}}\mathbb{E}\left\\{|\theta_{i}-\hat{\theta}_{i}|^{2}\right\\},$
(4)

where $\mathcal{L}_{s,d}$ represents the list of active sensing user indices
that are successfully detected, and $\hat{\theta}_{i}$ is the estimate of
$\theta_{i}$.

Figure 1: Illustration of the UNISAC model. The system accommodates multiple
users for sensing and communication purposes, with only a limited subset
active at any given moment.

##  III Achievable Performance limits

In this section, we derive achievable bounds for the PUPE and MSEAOA of the
UNISAC model.

######  Proposition 1.

For the UNISAC model, there exists a transceiver configuration where the
communication and sensing users satisfy the power constraints $\bar{P}_{c}$
and $\bar{P}_{s}$, respectively, and the performance metrics in (3) and (4)
satisfy

$\displaystyle\epsilon$ $\displaystyle\leq P_{cons}+P_{coll}+P_{md},$ (5)
$\displaystyle\Delta$
$\displaystyle\leq\sum_{K_{s}=1}^{|\mathcal{A}_{s}|}\sum_{K_{c}=1}^{|\mathcal{A}_{c}|}P_{K_{s},K_{c}}\mathbb{E}_{o_{g}}\left\\{\left|\mathrm{Re}\left(\dfrac{\log(o_{g})}{j\pi(M-1)}\right)\right|^{2}\right\\},$
(6)

where $P_{cons}$ represents the probability that at least one
communication/sensing user surpasses the power constraint,
$P_{coll}=\dfrac{\mathbb{E}\left\\{{|\mathcal{L}_{c,coll}|+|\mathcal{L}_{s,coll}|}\right\\}}{{|{\mathcal{A}_{c}}|+|{\mathcal{A}_{s}}|}}$,
$P_{md}=\dfrac{\mathbb{E}\left\\{{|\mathcal{L}_{c,md}|+|\mathcal{L}_{s,md}|}\right\\}}{{|{\mathcal{A}_{c}}|+|{\mathcal{A}_{s}}|}}$,
and
$P_{K_{s},K_{c}}=\mathbb{P}\left(|\mathcal{L}_{c,md}|=K_{c},|\mathcal{L}_{s,md}|=K_{s}\right)$.
These values are obtained as:

$\displaystyle P_{cons}$
$\displaystyle=1-F_{\chi^{2}}\left({\dfrac{2n\bar{P}_{s}}{P_{s}^{\prime}}},2n\right)^{|{\mathcal{A}_{s}}|}F_{\chi^{2}}\left({\dfrac{2n\bar{P}_{c}}{P_{c}^{\prime}}},2n\right)^{|{\mathcal{A}_{c}}|},$
(7a) $\displaystyle P_{coll}$
$\displaystyle\leq\dfrac{\sum_{i=2}^{\infty}\dfrac{i\binom{|\mathcal{A}_{s}|}{i}}{2^{B_{s}(i-1)}}+\sum_{j=2}^{\infty}\dfrac{j\binom{|\mathcal{A}_{c}|}{j}}{2^{B_{c}(j-1)}}}{|\mathcal{A}_{c}|+|\mathcal{A}_{s}|},$
(7b) $\displaystyle P_{md}$
$\displaystyle=\sum_{K_{s}=1}^{|\mathcal{A}_{s}|}\sum_{K_{c}=1}^{|\mathcal{A}_{c}|}\dfrac{K_{c}+K_{s}}{|\mathcal{A}_{c}|+|\mathcal{A}_{s}|}P_{K_{s},K_{c}},$
(7c) $\displaystyle P_{K_{s},K_{c}}$
$\displaystyle\leq\mathbb{P}(x_{g}>\eta)+e^{{L_{s}+L_{c}+L_{0}}},$ (7d)
$\displaystyle L_{0}$
$\displaystyle=-\lambda_{*}g-nM\log\left(1+\lambda_{*}\sigma_{t}^{2}-\lambda_{*}^{2}\sigma_{z}^{2}\sigma_{t}^{2}\right),$
(7e) $\displaystyle g$ $\displaystyle=-M(\sigma_{z}^{2}+\sigma_{t}^{2})\eta,$
(7f) $\displaystyle\sigma_{t}^{2}$
$\displaystyle=K_{c}P_{c}^{\prime}+K_{s}P_{s}^{\prime},$ (7g) $\displaystyle
L_{c}$
$\displaystyle=\sum_{i=0}^{K_{c}-1}\log\left(\dfrac{|\mathcal{A}_{c}|-i}{K_{c}-i}\right)+\log\left(\dfrac{2^{B_{c}}-i}{K_{c}-i}\right),$
(7h) $\displaystyle L_{s}$
$\displaystyle=\sum_{i=0}^{K_{s}-1}\log\left(\dfrac{|\mathcal{A}_{s}|-i}{K_{s}-i}\right)+\log\left(\dfrac{2^{B_{s}}-i}{K_{s}-i}\right),$
(7i) $\displaystyle\lambda_{*}$
$\displaystyle=\frac{{-b_{1}+\sqrt{{b_{1}^{2}-4a_{1}c_{1}}}}}{{2a_{1}}},$ (7j)
$\displaystyle a_{1}$ $\displaystyle=\sigma_{t}^{2}\sigma_{z}^{2}g,\ \
b_{1}=-g\sigma_{t}^{2}+2nM\sigma_{z}^{2}\sigma_{t}^{2},$ (7k) $\displaystyle
c_{1}$ $\displaystyle=-g-nM\sigma_{t}^{2},$ (7l) $\displaystyle o_{g}$
$\displaystyle=1+(g_{2}+\mathbf{g}^{T}\mathbf{g}^{R})/M,$ (7m)

where $P_{c}^{\prime}$ and $P_{s}^{\prime}$ denote the average transmitted
powers of communication and sensing users, respectively, with
$P_{c}^{\prime}\leq\bar{P}_{c}$ and $P_{s}^{\prime}\leq\bar{P}_{s}$,
$x_{g}=\mathbf{a}\mathbf{A}_{d}^{H}(\mathbf{A}_{d}\mathbf{A}_{d}^{H})^{-1}\mathbf{A}_{d}\mathbf{a}^{H}$,
elements of
$\mathbf{A}_{d}\in\mathbb{C}^{(|\mathcal{A}_{c}|+|\mathcal{A}_{s}|)\times n}$
and $\mathbf{a}\in\mathbb{C}^{n}$ are randomly chosen from
$\mathcal{CN}(0,1)$, $\eta$ serves as a design parameter, each element of
$\mathbf{g}\in\mathbb{C}^{M}$ follows
$\mathcal{CN}\left(0,{(\sigma_{t}^{2}+\sigma_{z}^{2})}/{(fP_{s}^{\prime})}\right)$,
and
$g_{2}\sim\mathcal{CN}\left(0,4M(\sigma_{t}^{2}+\sigma_{z}^{2})/(fP_{s}^{\prime})\right)$,
with $f$ following the chi-squared distribution with $2n$ degrees of freedom.

###### Proof:

In our analysis, we consider a random codebook
$\mathbf{A}\in\mathbb{C}^{2^{B_{c}+B_{s}}\times n}$, where every element in
the first $2^{B_{c}}$ rows are drawn from $\mathcal{CN}(0,P_{c}^{\prime})$,
and the last $2^{B_{s}}$ rows consist of $\mathcal{CN}(0,P_{s}^{\prime})$.
Each communication user randomly selects its transmitted signal from the first
$2^{B_{c}}$ rows of the codebook $\mathbf{A}$, and each sensing user selects
from the last $2^{B_{s}}$ rows. Note that in order to fulfill the
achievability assumption, all the assumptions outlined in this section should
result in an increase in errors.

For obtaining $\epsilon$ as shown in (5), we consider three sources of errors
as $P_{cons}$, $P_{coll}$, and $P_{md}$. According to the definition, we have

$\displaystyle
P_{cons}=1-\prod_{j\in\\{\mathcal{A}_{c},\mathcal{A}_{s}\\}}\mathbb{P}\left(\|\mathbf{a}_{i}\|^{2}/n<\bar{P}_{l}\right).$
(8)

Considering $\frac{2}{P_{l}^{\prime}}\|\mathbf{a}_{i}\|^{2}\sim\chi_{2n}^{2}$
for $l\in\\{c,s\\}$, then we can derive $P_{cons}$ as in (7a). We calculate
$P_{coll}$ in (7b) by considering that the probability that $j$ users select
the same signal is upper bounded by
$\dfrac{\binom{|\mathcal{A}_{l}|}{j}}{2^{B_{l}(j-1)}}$, for $l\in\\{c,s\\}$.
In the rest of this part, we calculate an upper bound for $P_{md}$.
To find active $\mathbf{a}_{i}$’s available in the received signal in (1), the
decoder selects $|\mathcal{A}_{c}|$ signals from the the first $2^{B_{c}}$
rows of $\mathbf{A}$, and $|\mathcal{A}_{s}|$ signals from the last
$2^{B_{s}}$ rows of it. Then, appending these signals together, the matrix
$\mathbf{A}_{d}\in\mathbb{C}^{(|\mathcal{A}_{c}|+|\mathcal{A}_{s}|)\times n}$
is constructed. Note that there are
$\binom{2^{B_{s}}}{|\mathcal{A}_{s}|}\binom{2^{B_{c}}}{|\mathcal{A}_{c}|}$
possible choices for the matrix $\mathbf{A}_{d}$. The receiver declares the
matrix that satisfies the following strategy as the collection of detected
signals from active communication and sensing users.

$\displaystyle\hat{\mathbf{A}}_{d}=\operatorname*{arg\,min}_{\mathbf{A}_{d}}\|\mathbf{Y}f_{p}\left(\mathbf{A}_{d}\right)\|^{2},$
(9)

where

$\displaystyle
f_{p}\left(\mathbf{A}_{d}\right)=\mathbf{I}_{n}-\mathbf{A}_{d}^{H}(\mathbf{A}_{d}\mathbf{A}_{d}^{H})^{-1}\mathbf{A}_{d}.$
(10)

We can write the received signal in (1) in the following form

$\displaystyle\mathbf{Y}=\mathbf{B}_{a}\mathbf{A}_{a}+\mathbf{Z},$ (11)

where $\mathbf{b}_{\theta_{j}}$’s and $\mathbf{a}_{j}$’s for
$j\in\\{\mathcal{A}_{c},\mathcal{A}_{s}\\}$ construct the columns of
$\mathbf{B}_{a}\in\mathbb{C}^{M\times(|\mathcal{A}_{c}|+|\mathcal{A}_{s}|)}$
and the rows of
$\mathbf{A}_{a}\in\mathbb{C}^{(|\mathcal{A}_{c}|+|\mathcal{A}_{s}|)\times n}$,
respectively. Let $\mathbf{A}_{a}=\begin{bmatrix}\mathbf{A}_{a,1}\\\
\mathbf{A}_{a,2}\end{bmatrix}$ and
$\mathbf{B}_{a}=\begin{bmatrix}\mathbf{B}_{a,1},\mathbf{B}_{a,2}\end{bmatrix}$,
where $\mathbf{B}_{a,2}$ and $\mathbf{A}_{a,2}$ consist of $K_{s}+K_{c}$
columns and rows, respectively. We define
$\mathbf{A}_{e}=\begin{bmatrix}\mathbf{A}_{a,1}\\\
\mathbf{A}_{e,f}\end{bmatrix}$, where the rows of
$\mathbf{A}_{e,f}\in\mathbb{C}^{\left(K_{c}+K_{s}\right)\times n}$ are $K_{s}$
sensing signals and $K_{c}$ communication signals that are not sent (false
signals). Considering the definition of $P_{K_{s},K_{c}}$ in the statement of
the Proposition 1 and the strategy in (9), we write

$\displaystyle P_{K_{s},K_{c}}$
$\displaystyle\leq\mathbb{P}\left(\|\mathbf{Y}f_{p}\left(\mathbf{A}_{e}\right)\|^{2}<\|\mathbf{Y}f_{p}\left(\mathbf{A}_{a}\right)\|^{2}\right)$
(12) $\displaystyle\leq\mathbb{P}(x_{g}>\eta)+\tilde{P}_{K_{s},K_{c}},$ (13)

where
$\tilde{P}_{K_{s},K_{c}}=\mathbb{P}\left(\|\mathbf{Y}f_{p}\left(\mathbf{A}_{e}\right)\|^{2}<\|\mathbf{Y}f_{p}\left(\mathbf{A}_{a}\right)\|^{2}\
\big{|}x_{g}\leq\eta\right)$, $x_{g}$ and $\eta$ are defined in the statement
of the Proposition 1, and in (12), we use the property

$\displaystyle\mathbb{P}\left(\bigcap_{i\in\mathcal{S}}A_{i}\right)\leq\mathbb{P}\left(A_{j}\right),\
j\in\mathcal{S}.$ (14)

Since
$\mathbf{A}_{a}f_{p}\left(\mathbf{A}_{a}\right)=\mathbf{0}_{(|\mathcal{A}_{c}|+|\mathcal{A}_{s}|),n}$,
$f_{p}\left(\mathbf{A}_{a}\right)\preceq\mathbf{I}_{n}$, and
$f_{p}\left(\mathbf{A}_{a}\right)f_{p}\left(\mathbf{A}_{a}\right)^{H}=f_{p}\left(\mathbf{A}_{a}\right)$,
then

$\displaystyle\|\mathbf{Y}f_{p}\left(\mathbf{A}_{a}\right)\|^{2}\leq\|\mathbf{Z}\|^{2}.$
(15)

The received signal in (11) can be written as

$\displaystyle\mathbf{Y}=\begin{bmatrix}\mathbf{B}_{a,1},\mathbf{0}_{M,(K_{s}+K_{c})}\end{bmatrix}\mathbf{A}_{e}+\mathbf{Z}^{\prime}+\mathbf{Z},$
(16)

where $\mathbf{Z}^{\prime}=\mathbf{B}_{a,2}\mathbf{A}_{a,2}$. Assuming various
components of the steering vector are mutually independent, each element of
$\mathbf{Z}^{\prime}$ belongs to a $\mathcal{CN}\left(0,\sigma_{t}^{2}\right)$
distribution, where $\sigma_{t}^{2}=K_{c}P_{c}^{\prime}+K_{s}P_{s}^{\prime}$.
Using
$\mathbf{A}_{e}f_{p}\left(\mathbf{A}_{e}\right)=\mathbf{0}_{(|\mathcal{A}_{c}|+|\mathcal{A}_{s}|),n}$
and $x_{g}\leq\eta$, we get

$\displaystyle\|\mathbf{Y}f_{p}\left(\mathbf{A}_{e}\right)\|^{2}$
$\displaystyle\geq\|\mathbf{Z}+\mathbf{Z}^{\prime}\|^{2}+g,$ (17)

where $g=-M(\sigma_{z}^{2}+\sigma_{t}^{2})\eta$. Substituting (15) and (17)
into (13), and taking into account that the vectorized version of a matrix has
the same Frobenius norm as the matrix itself, we have

$\displaystyle\tilde{P}_{K_{s},K_{c}}$
$\displaystyle\leq\mathbb{P}\left(\|\mathbf{z}+\mathbf{z}^{\prime}\|^{2}+g<\|\mathbf{z}\|^{2}\right)$
(18a)
$\displaystyle\leq\mathbb{E}\left\\{e^{-\lambda_{1}\|\mathbf{z}+\mathbf{z}^{\prime}\|^{2}-\lambda_{1}g+\lambda_{1}\|\mathbf{z}\|^{2}}\right\\}$
(18b)
$\displaystyle=\dfrac{e^{-\lambda_{1}g+L_{s}+L_{c}}}{(1+\lambda_{1}\sigma_{t}^{2})^{nM}}\mathbb{E}\left\\{e^{\lambda_{1}\|\mathbf{z}\|^{2}}e^{\dfrac{-\lambda_{1}\|\mathbf{z}\|^{2}}{(1+\lambda_{1}\sigma_{t}^{2})}}\right\\}$
(18c)
$\displaystyle=\dfrac{e^{-\lambda_{1}g+L_{s}+L_{c}}}{\left(1+\lambda_{1}\sigma_{t}^{2}-\lambda_{1}^{2}\sigma_{z}^{2}\sigma_{t}^{2}\right)^{nM}}$
(18d)
$\displaystyle=e^{{L_{s}+L_{c}-\lambda_{*}g-nM\log\left(1+\lambda_{*}\sigma_{t}^{2}-\lambda_{*}^{2}\sigma_{z}^{2}\sigma_{t}^{2}\right)}},$
(18e)

where $\mathbf{z}$ and $\mathbf{z}^{\prime}$ are the vectorized versions of
$\mathbf{Z}$ and $\mathbf{Z}^{\prime}$, respectively, and $L_{s}$, $L_{c}$,
and $\lambda_{*}$ are defined in the statement of the Proposition 1. Note that
$\lambda_{*}$ represents the optimal value of $\lambda_{1}$ that minimizes the
expression in (18d). To establish (18b), we utilize the Chernoff bound, which
states that $\mathbb{P}(x>0)\leq\mathbb{E}(e^{\lambda_{1}x})$ for any
$\lambda_{1}>0$, and to obtain (18c) and (18d), we use the identity

$\displaystyle\mathbb{E}\left\\{e^{x\|\mathbf{a}+\mathbf{b}\|^{2}}\right\\}=\dfrac{1}{(1-x\sigma_{a}^{2})^{n}}e^{x\|\mathbf{b}\|^{2}/(1-x\sigma_{a}^{2})},$
(19)

for $x\sigma_{a}^{2}<1$,
$\mathbf{a}\sim\mathcal{CN}\left(\mathbf{0}_{n,1},\sigma_{a}^{2}\mathbf{I}_{n}\right)$,
and $\mathbf{b}\in\mathbb{C}^{n}$. In (18c), we consider that there are
$e^{L_{s}+L_{c}}=\binom{2^{B_{s}}}{K_{s}}\binom{|\mathcal{A}_{s}|}{K_{s}}\binom{2^{B_{c}}}{K_{c}}\binom{|\mathcal{A}_{c}|}{K_{c}}$
different realizations for the matrix $\mathbf{z}^{\prime}$. By plugging (18e)
into (13), an upper limit for $P_{K_{s},K_{c}}$ is obtained as in (7d), which
directly leads to the result in (7c).
After detecting the sensing and communication signals, we can obtain a soft
estimation of their corresponding steering vectors using least-squares (LS)
estimator. Applying LS on the received signal in (16), we have

$\displaystyle\hat{\mathbf{B}}_{a,1}$
$\displaystyle=\mathbf{Y}\mathbf{A}_{e}^{H}(\mathbf{A}_{e}\mathbf{A}_{e}^{H})^{-1}$
(20) $\displaystyle=\mathbf{B}_{a,1}+\tilde{\mathbf{Z}},$ (21)

where
$\tilde{\mathbf{Z}}=(\mathbf{Z}+\mathbf{Z}^{\prime})\mathbf{A}_{e}^{H}\left(\mathbf{A}_{e}\mathbf{A}_{e}^{H}\right)^{-1}$.
It is clear that every row of $\tilde{\mathbf{Z}}$ follows
$\mathcal{CN}\left(\mathbf{0}_{1,J},(\sigma_{z}^{2}+\sigma_{t}^{2})\left(\mathbf{A}_{a}\mathbf{A}_{a}^{H}\right)^{-1}\right)$,
where $J=|\mathcal{A}_{c}|+|\mathcal{A}_{s}|-K_{s}-K_{c}$. For every
$i\in\mathcal{A}_{s}$, the estimation of the steering vector is obtained by
selecting the corresponding column of $\hat{\mathbf{B}}$, which yields

$\displaystyle\hat{\mathbf{b}}_{\theta_{i}}=\mathbf{b}_{\theta_{i}}+\mathbf{z}_{i},$
(22)

where $\mathbf{z}_{i}$ is the column of $\tilde{\mathbf{Z}}$ corresponding to
the index $i$. Any positive definite matrix $\mathbf{T}$ satisfies the
condition $\mathbf{T}^{-1}\preceq\mathrm{diag}\left(\mathbf{T}\right)^{-1}$.
Therefore, assuming every element of $\mathbf{z}_{i}$ to follow
$\mathcal{CN}\left(0,\dfrac{\sigma_{z}^{2}+\sigma_{t}^{2}}{\|\mathbf{a}_{i}\|^{2}}\right)$
is a pessimistic assumption, leading to a higher MSEAOA due to increased noise
variance. The receiver estimates $\theta_{i}$ as

$\displaystyle\hat{\theta}_{i}=\mathrm{Re}\left(\dfrac{1}{-j\pi(M-1)}\log\left(\dfrac{1}{M}\hat{\mathbf{b}}_{\theta_{i}}^{T}\hat{\mathbf{b}}_{\theta_{i}}^{R}\right)\right),$
(23)

Plugging (22) into (23) and using the structure in (2), we get

$\displaystyle\hat{\theta}_{i}$
$\displaystyle=\mathrm{Re}\left(\dfrac{1}{-j\pi(M-1)}\log\left(e^{-j\pi(M-1)\theta_{i}}+z_{i}^{\prime}\right)\right)$
(24) $\displaystyle=\theta_{i}+z_{i}^{\prime\prime},$ (25)

where
$z_{i}^{\prime}=\dfrac{2}{M}\mathbf{b}_{\theta_{i}}^{T}\mathbf{z}_{i}^{R}+\dfrac{1}{M}\mathbf{z}_{i}^{T}\mathbf{z}_{i}^{R}$,
and
$z_{i}^{\prime\prime}=\mathrm{Re}\left(\dfrac{1}{-j\pi(M-1)}\log(1+e^{j\pi(M-1)\theta_{i}}z_{i}^{\prime})\right)$.
Considering that randomly shifting of $\mathbf{z}_{i}$ does not change its
distribution, the random variable $o_{g}$ in the statement of the Proposition
1 has the same probability distribution function as
$1+e^{j\pi(M-1)\theta_{i}}z_{i}^{\prime}$. Hence, the MSEAOA can be upper
bounded as in (6). ∎

######  Remark 1.

Finding $\mathbb{E}_{o_{g}}\left\\{.\right\\}$ and $\mathbb{P}(x_{g}>\eta)$ in
(6) and (7d) becomes challenging because of the complicated probability
distribution functions of $o_{g}$ and $x_{g}$. Hence, we employ the Monte
Carlo method to estimate them by generating multiple realizations.

##  IV Numerical Results

To assess the performance of the proposed UNISAC model, we compare its
achievability result derived in Proposition 1 with that of traditional
multiple-access models. We evaluate various configurations under the following
conditions: The bit-sequence length of communication users is $B_{c}=100$, the
antenna array consists of $M=5$ elements, and the number of channel-uses is
$n=5000$. Other parameters are selected to reach the target PUPE and the
target MSEAOA of $\epsilon_{o}=0.1$ and $\Delta_{o}=5\times 10^{-4}$,
respectively.

######  Definition 1.

(Approximate channel coding rate in the finite blocklength regime [15]): For
any $P_{er}>0$, there exists a $(2^{B},n,P_{er})$ code in an additive white
Gaussian noise channel, where

$\displaystyle
R\approx\log(1+S)-\sqrt{\dfrac{1}{n}\dfrac{S(S+2)\log_{2}^{2}e}{2(S+1)^{2}}}Q^{-1}(P_{er}),$
(26)

where $R$ and $S$ denote channel coding rate and the signal-to-noise ratio
(SNR), respectively, $P_{er}$ is the block error rate, $Q(.)$ denotes the
standard $Q$-function, and $e$ is the Euler’s number.

######  Definition 2.

(Theorem 4.1 [16]): For the channel model in (11) with $K$ active users, the
lower bound for the MSEAOA is obtained as

$\displaystyle{\Delta=\dfrac{\sigma_{z}^{2}}{2}\left(\sum_{t=1}^{n}\mathrm{Re}\left(\mathbf{X}_{t}\mathbf{D}^{H}f_{p}(\mathbf{B}_{a}^{H})\mathbf{D}\mathbf{X}_{t}^{H}\right)\right)^{-1},}$
(27)

where
$f_{p}\left(\mathbf{B}\right)=\mathbf{I}_{n}-\mathbf{B}^{H}(\mathbf{B}\mathbf{B}^{H})^{-1}\mathbf{B}$,
$\mathbf{X}_{t}\in\mathbb{C}^{K\times K}$ is a diagonal matrix with the $t$th
column of the matrix $\mathbf{A}_{a}$ on its diagonal, and
$\mathbf{D}\in\mathbb{C}^{M\times K}$ represents the element-wise partial
derivative of $\mathbf{B}_{a}$ with respect to $\theta$,
$\mathbf{D}=\partial\mathbf{B}_{a}/\partial\theta$.

######  Definition 3.

(Chapter 8 [17]): Let $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ be the null and
alternative hypotheses, respectively, with corresponding channel models
represented as follows: $\mathcal{H}_{0}:\mathbf{y}=\mathbf{z}$ and
$\mathcal{H}_{1}:\mathbf{y}=\mathbf{s}+\mathbf{z}$, where
$\mathbf{z}\sim\mathcal{CN}\left(\mathbf{0}_{T,1},\delta\mathbf{I}_{T}\right)$
represents the additive noise term and $\mathbf{s}\in\mathbb{C}^{T}$ is the
desired signal. The misdetection probability for detecting the signal
$\mathbf{s}$ can be computed as follows:

$\displaystyle
p_{miss}=Q\left(\sqrt{\dfrac{\|\mathbf{s}\|^{2}}{2\delta}}\right).$ (28)

The benchmark strategies are described below. The first model is termed the
“optimistic” model, where we adopt optimistic assumptions leading to lower
bounds on PUPE and MSEAOA. The lower bound for
$\mathbb{E}\left\\{{|\mathcal{L}_{s,coll}|+|\mathcal{L}_{c,coll}|}\right\\}$
is obtained as
$\sum_{l\in\\{c,s\\}}\dfrac{\binom{|\mathcal{A}_{l}|}{2}}{2^{B_{l}}}\left(\dfrac{2^{B_{l}}-1}{{2^{B_{l}}}}\right)^{|\mathcal{A}_{l}|-2}$.
To ensure that
$\mathbb{E}\left\\{|\mathcal{L}_{s,md}|+|\mathcal{L}_{c,md}|\right\\}/({|\mathcal{A}_{c}|+|\mathcal{A}_{s}|})\xrightarrow{}0$,
we employ Shannon limit as

$\displaystyle\dfrac{B_{T}}{n}\xrightarrow{}\mathbb{E}\left\\{\log_{2}\left(\mathrm{det}\left(\mathbf{I}_{M}+\dfrac{P_{cl}}{\sigma_{z}^{2}}\mathbf{B}_{a}\mathbf{B}_{a}^{H}\right)\right)\right\\},$
(29)

where $B_{T}=|\mathcal{A}_{c}|B_{c}+|\mathcal{A}_{s}|B_{s}$ and $P_{cl}$
denote the total number of transmitted bits and the transmitted power by each
sensing and communication user, respectively. To obtain the lower bound of the
MSEAOA, we use Definition 2 for a single user to obtain

$\displaystyle\Delta=\dfrac{0.5\sigma_{z}^{2}}{\pi^{2}n\bar{P}_{s}\sum_{i=1}^{M-1}i^{2}}.$
(30)

In the second model, which is called “TDMA” model, we divide the frame into
$|\mathcal{A}_{c}|+|\mathcal{A}_{s}|$ subframes, and every active sensing and
communication user transmits its signal through an individual subframe. In
this case, we can obtain an approximation of the
$\mathbb{E}\left\\{|\mathcal{L}_{c,md}|\right\\}/|\mathcal{A}_{c}|$ by
adopting Definition 1 with $S=M\bar{P}_{c}/\sigma_{z}^{2}$ and
$R=B_{c}/N_{\text{\tiny TDMA}}$, where $N_{\text{\tiny
TDMA}}=n/(|\mathcal{A}_{c}|+|\mathcal{A}_{s}|)$. Using (28), we obtain
$\mathbb{E}\left\\{|\mathcal{L}_{s,md}|\right\\}/|\mathcal{A}_{s}|=Q\left(\sqrt{\dfrac{\bar{P}_{s}N_{\text{\tiny
TDMA}}M}{2\sigma_{z}^{2}}}\right)$ . For MSEAOA, we employ the result in
Definition 2 for a single user, which gives
$\Delta=0.5\sigma_{z}^{2}/(\pi^{2}N_{\text{\tiny
TDMA}}\bar{P}_{s}\sum_{i=1}^{M-1}i^{2})$. The third model is known as “TIN”
model, where all users transmit their signals simultaneously, and the receiver
treats signals of all users except a desired user as noise. In this case,
$\mathbb{E}\left\\{|\mathcal{L}_{c,md}|\right\\}/|\mathcal{A}_{c}|$ is
approximated by inserting $S=M\bar{P}_{c}/\sigma_{n}^{2}$ and and $R=B_{c}/n$
into (26), and
$\mathbb{E}\left\\{|\mathcal{L}_{s,md}|\right\\}/|\mathcal{A}_{s}|\approx
Q\left(\sqrt{\dfrac{\bar{P}_{s}nM}{2\sigma_{n}^{2}}}\right)$, where
$\sigma_{n}^{2}=\bar{P}_{c}|\mathcal{A}_{c}|+\bar{P}_{s}|\mathcal{A}_{s}|+\sigma_{z}^{2}$.
The MSEAOA is calculated using Definition 2 by employing the single-user
scenario as
$\Delta=0.5\sigma_{n}^{2}/(\pi^{2}n\bar{P}_{s}\sum_{i=1}^{M-1}i^{2})$. The
“TDMA-MUSIC” model divides the length-$n$ frame into two subframes with
lengths $n/2$. In the first subframe, the TDMA strategy is employed for
decoding the communication users, and in the second subframe,
$|\mathcal{A}_{s}|$ sensing users transmit their signals simultaneously. The
AOAs of the sensing users in the second subframe are determined using MUSIC
estimator. For the communication part,
$\mathbb{E}\left\\{|\mathcal{L}_{c,md}|\right\\}/|\mathcal{A}_{c}|$ is
approximated by putting $S=M\bar{P}_{c}/\sigma_{z}^{2}$ and
$R=B_{c}/N_{\text{\tiny TDMA2}}$ into (26), where $N_{\text{\tiny
TDMA2}}=0.5n/|\mathcal{A}_{c}|$. The MSEAOA is obtained by (27), considering
$|\mathcal{A}_{s}|$ users’ transmission through $n/2$ channel uses. In the
“ALOHA” model, we divide the frame into $T$ subframes, and each user randomly
selects a subframe to transmit its signal. The receiver only identifies
signals from users that are not involved in collisions. The block error of the
sensing and communication users is approximated by
$\mathbb{E}\left\\{|\mathcal{L}_{l,md}|\right\\}/|\mathcal{A}_{l}|\approx
1-(1-P_{md,l})\left(1-f_{po}(1;\alpha_{po})/\sum_{i=1}^{\infty}f_{po}(i;\alpha_{po})\right)$,
where $\alpha_{po}=(|\mathcal{A}_{s}|+|\mathcal{A}_{c}|)/T$, $l\in\\{c,s\\}$,
$f_{po}(i;a)$ denotes the probability distribution function of the Poisson
distribution with the parameter $a$, $P_{md,c}$ is approximated by plugging
$S=M\bar{P}_{c}/\sigma_{z}^{2}$ and $R=B_{c}T/N_{\text{\tiny ALOHA}}$ in (26),
and $P_{md,s}$ is calculated as
$P_{md,s}=Q\left(\sqrt{\dfrac{\bar{P}_{s}N_{\text{\tiny
ALOHA}}M}{2\sigma_{z}^{2}}}\right)$, with $N_{\text{\tiny ALOHA}}=n/T$. In
these computations, we use the approximation that the number of users engaged
in an $i$-collision follows a Poisson distribution with a parameter of
$\alpha_{po}$ [18]. Also, the MSEAOA is obtained using Definition 2, which
gives $\Delta=0.5\sigma_{z}^{2}/(\pi^{2}N_{\text{\tiny
ALOHA}}\bar{P}_{s}\sum_{i=1}^{M-1}i^{2})$.
In Fig. 2, we compare the achievability performance of the proposed UNISAC
with the existing multiple access models by plotting the required
${E}/{N_{0}}$ to reach the target PUPE and MSEAOA, where ${E}/{N_{0}}$ is the
average per-user energy of the system which is defined as

$\displaystyle\dfrac{E}{N_{0}}=\dfrac{(\bar{P}_{c}|\mathcal{A}_{c}|n_{c}+\bar{P}_{s}|\mathcal{A}_{s}|n_{s})}{(|\mathcal{A}_{c}|+|\mathcal{A}_{s}|)\sigma_{z}^{2}},$
(31)

where $n_{c}$ and $n_{s}$ denote the signal lengths of each communication and
sensing user, respectively. The $x$-axis in this figure is
$|\mathcal{A}_{s}|+|\mathcal{A}_{c}|$ with
$|\mathcal{A}_{s}|=|\mathcal{A}_{c}|$. Observing Fig. 2, it is clear that the
UNISAC model demonstrates significantly better performance than other models
under conditions with a high number of active users. The TDMA-MUSIC model’s
underperformance stems from its inability to handle cases where the number of
sensing users surpasses the number of antenna elements. The poor performance
of ALOHA, TIN, and TDMA with a large number of users are attributed to dense
collisions, high interference levels, and short signal lengths, respectively.
Conversely, the UNISAC model has potential to achieve superior performance by
mitigating collisions, reducing interference, and assigning longer signal
lengths.

Figure 2: The required $E/N_{0}$ as a function of the number of active sensing
and communication users (with $|\mathcal{A}_{c}|=|\mathcal{A}_{s}|$),
comparing UNISAC’s achievable result against existing multiple access models
to achieve a target PUPE of $\epsilon_{0}=0.1$ and a target MSEAOA of
$\Delta_{0}=5\times 10^{-4}$.

In order to evaluate the effectiveness of the achievable result for the
MSEAOA, Fig. 3 compares the MSEAOA presented in (6) with the optimistic
outcome illustrated in (30), considering various values of $M$. To obtain this
result, we assume that $P_{K_{s},K_{c}}=1$ when $K_{s}=0$ and $K_{c}=0$, and
$P_{K_{s},K_{c}}=0$ otherwise. This figure illustrates the enhancement in the
AOA estimation achieved by the UNISAC model as the number of antenna elements
is increased. Additionally, the minimal difference between the achievability
and optimistic results indicates a tight capacity region for the AOA
estimation provided by the UNISAC model.

Figure 3: Achievable and optimistic results of MSEAOA for different numbers of
antenna elements $M$.

##  V Conclusions

In this paper, we have addressed the problem of ISAC in scenarios with a
massive number of unsourced and uncoordinated users. The proposed system
allows active communication and sensing users to share a short transmission
frame without coordination with the base station. We introduced an achievable
performance bound for the proposed ISAC system and compared its effectiveness
against traditional models like ALOHA, TDMA, TIN, and MUSIC. Through computer
simulations, we have verified the effective performance of the proposed model
in detecting and decoding a large number of users. This research contributes
to advancing the understanding and practical implementation of ISAC systems in
real-world scenarios with massive user counts.

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# Neutralized Local Entropy

Snir Ben Ovadia, Federico Rodriguez-Hertz

###### Abstract.

We introduce a notion of a point-wise entropy of measures (i.e local entropy)
called neutralized local entropy, and compare it with the Brin-Katok local
entropy and with the Ledrappier-Young local entropy on unstable leaves. We
show that the neutralized local entropy must coincide with the two other
notions of local entropies, and so all three quantities are equal almost
everywhere. Neutralized local entropy is computed by measuring open sets with
a relatively simple geometric description. Our proof uses a measure density
lemma for Bowen balls, and a version of a Besicovitch covering lemma for Bowen
balls. In particular this gives an elementary proof to the fact that the
unstable entropy carries all entropy.

##  1\. Introduction and main results

One of the most useful tools in studying dynamical systems are dynamical
covers and partitions of a measure. That is, given an invariant measure, a
cover or a partition modulo the measure, whose elements have a significant
structure w.r.t the dynamics. The pursuit for such objects creates a tension
between searching for elements with a simple geometric description, with which
it is easy to work, and searching for elements with significant dynamical
structure, which allows to control their orbits and measure. Two
charecteristic examples of this tension are geometric balls, and Bowen balls.

Geometric balls allow one to utilize strong geometric properties such as the
Lebesgue density theorem, or the Besicovitch covering lemma. Bowen balls on
the other hand, allow one to estimate their measure in terms of the entropy
(see [BK83]), and control their image under the dynamics for a fixed amount of
iterations; while having a possibly very complicated geometric shape.

Pesin theory allows one to linearize locally the action of the dynamics on
typical orbits. However, the size of the neighborhood where the linearization
is valid may deteriorate along the orbit, although in a sub-exponential rate.
Given an ergodic measure with positive metric entropy, the Ruelle inequality
says it must have some positive and negative Lyapunov exponents ([Rue78]).
When an orbit admits a central direction, i.e an invariant sub-space of the
tangent space corresponding to $0$ Lyapunov exponents, the central direction
may not be integrable into an invariant manifold. In addition, the
differential may contract or expand tangent vectors in the central direction,
in a sub-exponential way. These effects make it very hard, and generally not
attainable, to get a simple description of the set of points which remain
close to the orbit.

Furthermore, even in the absence of $0$ Lyapunov exponents, the decay of the
size of the Pesin chart (i.e neighborhood with a local linearization of the
dynamics) does not allow to control the set of points which remain close to
the orbit by a fixed distance (i.e Bowen balls). Hence, phasing-out sub-
exponential effects such as the central direction or the deterioration of the
size of Pesin charts becomes very useful. Removing these effect allows one to
treat the action of the dynamics along an orbit as if they were linear and
hyperbolic, and so simplifying greatly the geometric description of the set of
point which shadow the orbit.

Our goal in this paper is to address exactly this difficulty. Our proof relies
on a sub-exponential measure density lemma over Bowen balls (rather than
geometric balls as in the Lebesgue density lemma).

In our setup $M$ is a closed Riemannian manifold, $d=\mathrm{dim}M\geq 2$, and
$f\in\mathrm{Diff}^{1+\beta}(M)$, $\beta>0$. Let $\mu$ be an $f$-invariant
Borel probability measure. The purpose of this paper is to compute the
neutralized local entropy defined by,

$\mathcal{E}_{\mu}(x):=\lim_{r\to
0}\limsup_{n\to\infty}\frac{-1}{n}\log\mu(B(x,n,e^{-rn})),$ (1)

where $B(x,n,e^{-rn}):=\\{y\in M:d(f^{i}(y),f^{i}(y)(x))\leq e^{-rn},\forall
0\leq i\leq n\\}$. One can check the following properties for the neutralized
local entropy:

  1. (a)

$\mathcal{E}_{\mu}(x)\in[h_{\mu}^{\mathrm{BK}}(x),d\cdot\log M_{f}]\text{
}\mu\text{-a.e,}$ (2)

where $h_{\mu}^{\mathrm{BK}}(x)$ is the local entropy at $x$ given by the
Brin-Katok formula, [BK83].111$h_{\mu}^{\mathrm{BK}}(x)=\lim_{\epsilon\to
0}\limsup\frac{-1}{n}\log\mu(B(x,n,\epsilon))=\lim_{\epsilon\to
0}\liminf\frac{-1}{n}\log\mu(B(x,n,\epsilon))$, where the limits exist and the
inequality holds $\mu$-a.e, and $B(x,n,\epsilon)=\\{y\in
M:d(f^{i}(x),f^{i}(y))\leq\epsilon,\forall 0\leq i\leq n\\}$. The upper bound
is by the classical Lemma which states that $\lim_{s\to
0}\frac{\log\mu(B(x,s))}{\log s}\leq d$ for $\mu$-a.e $x$ (see, for example,
[LY85a, Lemma 4.1.4]), and by the fact that $B(x,n,e^{-rn})\supseteq
B(x,M_{f}^{-n})$ for all $n\geq 0$ and sufficiently small $r>0$.

  2. (b)

$\mathcal{E}_{\mu}\circ f=\mathcal{E}_{\mu}\text{ }\mu\text{-a.e.}$ (3)

This can be seen by the following two inequalities:

    1. (I)

$\mathcal{E}_{\mu}(f(x))\geq\mathcal{E}_{\mu}(x)$: since

$\mu(B(f^{-1}(x),n,e^{-rn}))=\mu(f[B(f^{-1}(x),n,e^{-rn})])\leq\mu(B(x,n,M_{f}e^{-rn})),$

    2. (II)

$\mathcal{E}_{\mu}(f(x))\leq\mathcal{E}_{\mu}(x)$: since

$\mu(B(f^{-1}(x),n,e^{-rn}))\geq\mu(f^{-1}[B(x,n+1,e^{-r\frac{n}{n+1}\cdot(n+1)})])\geq\mu(B(x,n+1,e^{-r(n+1)})).$

In particular, if $\mu$ is ergodic, then the neutralized local entropy of
$\mu$ is constant almost everywhere, and dominates the metric entropy of $\mu$
(and lies within a globally-bounded interval).

The significance of the neutralized entropy is that it estimates the
asymptotic measure of sets with a distinctive geometric shape. Unlike the
sequence $\\{B(x,n,r)\\}_{n\geq 0}$ which can develop a very complicated
geometric shape for large $n$\- due to a central direction, or even for a non-
uniformly hyperbolic trajectory- the sequence $\\{B(x,n,e^{-rn})\\}_{n\geq 0}$
can have a nice description a.e for any $r>0$, by neutralizing any sub-
exponential effects. Sets with a more explicit geometric description are very
useful for the construction of covers (or consequently even partitions), and
so controlling the measure of such sets is important.

In fact, one can guess that by the lack of diversity for dynamical intrinsic
invariants, the neutralized local entropy must coincide with other notions of
local entropy (or the metric entropy in the ergodic case). This paper is
dedicated to the proof of this statement for smooth systems. In [LY85a, LY85b]
the authors introduce a notion of local entropy through conditional measures
w.r.t a measurable partition sub-ordinated to stable leaves,
$h_{\mu}^{\mathrm{LY}}(x)$.222$h^{\mathrm{LY}}_{\mu}(x)=\lim_{\epsilon\to
0}\limsup\frac{-1}{n}\log\mu_{\xi(x)}(B^{u}(x,n,\epsilon))=\lim_{\epsilon\to
0}\liminf\frac{-1}{n}\log\mu_{\xi(x)}(B^{u}(x,n,\epsilon))$, where the limits
exist and the equality holds $\mu$-a.e, $B^{u}(x,n,\epsilon):=\\{y\in
W^{u}(x):\forall 0\leq i\leq n,d(f^{i}(x),f^{i}(y))\leq\epsilon\\}$,
$W^{u}(x)$ is global stable leaf of $x$, $\xi$ is a measurable partition
subordinated to the unstable foliation, and $\mu_{\xi(x)}$ is the conditional
of $\mu$ on the partition element which contains $x$, which is well-defined
$\mu$-a.e by the Rokhlin disintegration theorem. When $\mu$ is ergodic, both
$h_{\mu}^{\mathrm{BK}}$ and $h_{\mu}^{\mathrm{LY}}$ are constant a.e and are
equal to the metric entropy of $\mu$. The identity

$\displaystyle\mathcal{E}_{\mu}(x)$
$\displaystyle=h_{\mu}^{\mathrm{BK}}(x)\text{ }\mu\text{-a.e},$ (4)

gives an explicit dynamical meaning to the heuristic “the central direction
carries no entropy”, without an ambient partition into leaves or its
conditional measures.

In particular, it follows as a corollary from the proof that

$h_{\mu}^{\mathrm{LY}}(\cdot)=h_{\mu}^{\mathrm{BK}}(\cdot)\text{
}\mu\text{-a.e,}$ (5)

and so all three notions of local entropy coincide a.e; and that

$\mathcal{E}_{\mu}(x)=\lim_{r\to
0}\liminf_{n\to\infty}\frac{-1}{n}\log\mu(B(x,n,e^{-rn}))\text{,
}\mu\text{-a.e.}$ (6)

In addition, as a corollary one gets that the Ledrappier-Young entropy of $f$
coincides with Ledrappier-Young local entropy of $f^{-1}$ almost everywhere.

As pointed out to the authors by Ledrappier ([Led]), the unstable entropy (i.e
$h^{\mathrm{LY}}_{\mu}$) is always less or equal to the total entropy of the
system when integrated, as the entropy is defined as the supremum over
countable partitions (and the measurable unstable partition is countably
generated). Hence the inequality
$h^{\mathrm{BK}}_{\mu}\leq\mathcal{E}_{\mu}\leq h^{\mathrm{LY}}_{\mu}$ gives
an elementary proof to the fact that $\int
h^{\mathrm{LY}}_{\mu}(x)d\mu(x)=h_{\mu}(f)$.

Thieullen [Thi92a] studied a similar notion to the neutralized local entropy,
called $\alpha$-entropy, for certain systems on infinite-dimensional systems
(see also [Thi92b, Thi91]). Other generalizations of entropy have also been
studied in terms of ergodic theory for some systems by [TV02, TV03].

##  2\. Proof of the Main Results

######  Lemma 2.1.

Let $\mu=\sum_{i=1}^{N}a_{i}\mu^{i}$ where $\sum_{i=1}^{N}a_{i}=1$ and
$\mu^{i}$ is an $f$-invariant Borel probability measure for all $i\leq N$.
Assume in addition that $\\{\mu^{i}\\}_{i=1}^{N}$ are all mutually singular.
Then there exist mutually disjoint sets $\\{A_{i}\\}_{i=1}^{N}$ s.t $\forall
i\leq N$, $\mu^{i}(A_{i})=1$ and $\forall x\in A_{i}$,

$h_{\mu^{i}}^{\mathrm{LY}}(x)=h_{\mu}^{\mathrm{LY}}(x).$

###### Proof.

It is enough to prove in the case $N=2$. By the mutual singularity of
$\mu^{1}$ and $\mu^{2}$, there are measurable sets $S^{1}$ and $S^{2}$ s.t
$\mu^{i}(S^{i})=1$, while $\mu^{3-i}(S^{i})=0$. We may assume w.l.o.g that
$f[S_{i}]=S_{i}$. Assume that $\chi^{u}(\cdot)>0$ $\mu^{i}$-a.e on $S_{i}$,
and that $\mu^{i}$-a.e point in $S^{i}$ admits a local stable leaf, $i=1,2$.
Assume further that $S_{i}\subseteq\\{\text{generic points}\\}$ and that for
every $x\in S_{i}$, $\frac{1}{n}\sum_{k=0}^{n-1}\delta_{x}\circ
f^{-k}\xrightarrow[n\to\pm\infty]{\text{weak-*}}\nu_{x}$ where $\nu_{x}$ is an
ergodic $f$-invariant probability measure s.t $\nu_{x}(S_{i})=1$.333By the
ergodic decomposition theorem $\mu^{i}=\int_{S_{i}}\mu_{x}^{i}d\mu^{i}(x)$,
and so $\mu^{i}=\int_{\\{x\in
S_{i}:\mu_{x}^{i}(S_{i})=1\\}}\mu_{x}^{i}d\mu^{i}(x)$.

Let $\xi^{u}$ be a measurable partition of $\mu$ subordinated to the unstable
leaves, as in [LY85a, LY85b]. If $x\in S_{i}$, and $\mu_{\xi^{u}(x)}$ and
$\mu^{i}_{\xi^{u}(x)}$ are well-defined, then
$\mu_{\xi^{u}(x)}=\mu^{i}_{\xi^{u}(x)}$ by Birkhoff’s ergodic theorem and the
Riesz-Markov-Kakutani representation theorem (since if $\xi^{u}(x)$ contained
any point from $S_{3-i}$, then $\mu^{i}=\mu^{3-i}$). Therefore,
$h_{\mu}^{\mathrm{LY}}(x)=h_{\mu^{i}}^{\mathrm{LY}}(x)$, if both limits are
proper for $x$. Set $A_{i}\subseteq S_{i}$ the set of points which satisfy it.

Now, let $\mu=a_{0}\mu^{0}+a_{+}\mu^{+}$ where $\chi^{u}(\cdot)=0$
$\mu^{0}$-a.e and $\chi^{u}(\cdot)>0$ $\mu^{+}$-a.e, then $h_{\mu^{0}}(f)=0$.
For any $x\in A_{+}$ we have that $\mu_{\xi^{u}(x)}=\mu^{+}_{\xi^{u}(x)}$, and
so $h_{\mu}^{\mathrm{LY}}(x)=h_{\mu^{+}}^{\mathrm{LY}}(x)$. Finally, for a
point $x$ with $\chi^{u}(x)=0$,
$h_{\mu}^{\mathrm{LY}}(x)=0=h_{\mu^{0}}^{\mathrm{LY}}(x)$ by definition. ∎

Remark: The proof extends also to the case where $N=\infty$ (and $\mu$ is a
countable convex combination).

######  Definition 2.2.

Let $\mu$ be an $f$-invariant ergodic probability measure s.t $\chi(\mu)>0$.
Denote by
$(\chi_{1},m_{1},\ldots,\chi_{\ell_{\underline{\chi}}},m_{\ell_{\underline{\chi}}})=\underline{\chi}(\mu)$
the Lyapunov exponents and dimension of $\mu$ in a decreasing order.

  1. (1)

Let
$0<\tau\leq\tau_{\underline{\chi}}:=\frac{1}{100d}\min\\{\chi_{\ell_{\underline{\chi}}},\chi_{i+1}-\chi_{i}:i\leq\ell_{\underline{\chi}}-1\\}$,
and let $C_{\underline{\chi},\tau}(\cdot)$ be the Lyapunov change of
coordinates for points in $\mathrm{LR}_{\underline{\chi}}=\\{\text{Lyapunov
regular points with an index }\tau_{\underline{\chi}}\\}$ (see [KM95]).

  2. (2)

Let
$\mathrm{PR}_{\underline{\chi}}=\\{x\in\mathrm{LR}_{\underline{\chi}}:\limsup_{n\to\pm\infty}\frac{1}{n}\log\|C_{\underline{\chi},\tau}^{-1}(f^{n}(x))\|=0,\forall
0<\tau\leq\tau_{\underline{\chi}}\\}$, the set of $\chi$-Pesin regular points
which carries $\mu$.
$\mathrm{PR}:=\bigcup_{\underline{\chi}}\mathrm{PR}_{\underline{\chi}}$ is
called the set of Pesin regular points.

  3. (3)

Given $x\in\mathrm{PR}_{\underline{\chi}}$, let $E_{j}(x)$ be the Oseledec
subspace of $x$ corresponding to $\chi_{j}$.

  4. (4)

Let $\widehat{E}_{j}:=C^{-1}_{\underline{\chi},\tau}(x)[E_{j}(x)]$, and notice
that $j\neq
j^{\prime}\Rightarrow\widehat{E}_{j}\perp\widehat{E}_{j^{\prime}}$. Write
$\widehat{E}^{\mathrm{cs}}:=\oplus_{\chi_{j}\leq 0}\widehat{E}_{j}$.

  5. (5)

A Pesin block $\Lambda^{(\underline{\chi},\tau)}$ is a subset of
$\bigcup_{|\underline{\chi}^{\prime}-\underline{\chi}|_{\infty}<\tau}\mathrm{PR}_{\underline{\chi}^{\prime}}$
which is a level set $[q_{\tau}\geq\frac{1}{l}]$ of a measurable function
$q_{\tau}:\bigcup_{|\underline{\chi}^{\prime}-\underline{\chi}|_{\infty}<\tau}\mathrm{PR}_{\underline{\chi}^{\prime}}\to(0,1)$
s.t (a) $\frac{q_{\tau}\circ f}{q_{\tau}}=e^{\pm\tau}$, (b)
$q_{\tau}(\cdot)\leq\frac{1}{\|C^{-1}_{\underline{\chi},\tau}(\cdot)\|^{d}}$.

######  Lemma 2.3 (Besicovitch-Bowen covering lemma).

Let $\Lambda^{(\underline{\chi},\tau)}$ ($0<\tau\leq\tau_{\underline{\chi}}$)
be a Pesin block, and let $x_{0}\in\Lambda^{(\underline{\chi},\tau)}$. Let
$B(x_{0})$ be the Pesin chart of $x_{0}$ for
$\Lambda^{(\underline{\chi},\tau)}$. Let
$A\subseteq\Lambda^{(\underline{\chi},\tau)}\cap B(x_{0})$ be a measurable
subset. Then $A$ can be covered by a cover of exponential Bowen balls of
points in $A$ (i.e $B(\cdot,n,e^{-n\epsilon})$), with multiplicity bounded by
$e^{3d\tau n}$, where $n$ is sufficiently large w.r.t
$\Lambda^{(\underline{\chi},\tau)}$, and $\epsilon\geq 2\tau$.

The idea of the proof follows the principle steps of the proof of the
classical Besicovitch covering lemma (see [DiB02, § 18]), using a certain
volume doubling property for exponential Bowen balls.

###### Proof.

Let $n\in\mathbb{N}$ sufficiently large so $e^{-\epsilon n}$ is smaller than
the Pesin chart size for $\Lambda^{(\underline{\chi},\tau)}$. Set $A_{1}:=A$,
and choose $x_{1}\in A_{1}$. Given $k$, set
$A_{k+1}:=A\setminus\bigcup_{j=1}^{k}B(x_{j},n,e^{-n\epsilon})$ and choose
$x_{k+1}\in A_{k+1}$. Continue in this process as long as
$A\setminus\bigcup_{j=1}^{k}B(x_{j},n,e^{-n\epsilon})$ is not empty.

Claim 1: $\forall k<j$, $B(x_{j},n,\frac{1}{3}e^{-n\epsilon})\cap
B(x_{k},n,\frac{1}{3}e^{-n\epsilon})=\varnothing$.

Proof: Otherwise, $\forall i\leq n$,
$d(f^{i}(x_{k}),f^{i}(x_{j}))\leq\frac{2}{3}e^{-\epsilon n}$, whence $x_{j}\in
B(x_{k},n,e^{-n\epsilon})$, in contradiction to the choice of
$\\{x_{l}\\}_{l\geq 1}$.

Claim 2: $\exists N\in\mathbb{N}$ s.t $\bigcup^{N}B(x_{k},n,e^{-\epsilon
n})\supseteq A$.

Proof: For any $k\geq 1$, $\mathrm{Vol}(B(x_{k},n,\frac{1}{3}e^{-\epsilon
n}))=C^{\pm 1}\cdot e^{-\underline{\chi}^{u}n}\cdot
e^{-d_{\mathrm{cs}}\epsilon n}e^{\pm d\tau n}$, where
$e^{-\underline{\chi}^{u}n}=\prod_{i\leq\ell_{\underline{\chi}}:\chi_{i}>0}e^{-\chi_{i}n\cdot
d_{i}}$, $d_{i}$ is the multiplicity of $\chi_{i}$,
$d_{\mathrm{cs}}:=\sum_{i:\chi_{i}\leq 0}d_{i}$, and $C$ is a constant
depending on $M$, $d$, and $\Lambda^{(\underline{\chi},\epsilon)}$. Then
$N\leq\frac{\mathrm{Vol}(M)}{C^{-1}\cdot e^{-\underline{\chi}^{u}n}\cdot
e^{-d\tau n}\cdot e^{-d_{\mathrm{cs}}\epsilon n}}$.

Claim 3: $\bigcup_{k=1}^{N}A_{k}\supseteq A$.

Proof: The process continues unless $A$ is covered.

Claim 4: $\forall k\leq N$, $\\#\\{j\leq N:B(x_{j},n,e^{-n\epsilon})\cap
B(x_{k},n,e^{-n\epsilon})\neq\varnothing\\}\leq C_{d}e^{2dn\tau}$, where
$C_{d}$ is a constant depending on $M$ and on
$\Lambda^{(\underline{\chi},\tau)}$.

Proof: Let $k\leq N$, and let $k\neq j\leq N$ s.t
$B(x_{j},n,e^{-n\epsilon})\cap B(x_{k},n,e^{-n\epsilon})\neq\varnothing$. Then

$B(x_{j},n,\frac{1}{3}e^{-n\epsilon})\subseteq
B(x_{j},n,e^{-n\epsilon})\subseteq B(x_{k},n,3e^{-n\epsilon}).$

Then

$\\#\\{j\leq N:B(x_{j},n,e^{-n\epsilon})\cap
B(x_{k},n,e^{-n\epsilon})\neq\varnothing\\}\leq\frac{\max_{k\leq
N}\mathrm{Vol}(B(x_{k},n,3e^{-n\epsilon}))}{\min_{j\leq
N}\mathrm{Vol}(B(x_{j},n,\frac{1}{3}e^{-n\epsilon}))}\leq
C^{2}3^{2d_{\mathrm{cs}}}e^{2d\tau n}.$

Claim 5: We can divide $\\{x_{k}\\}_{k\leq N}$ into sub-collections
$\mathcal{C}_{i}$, $i=1,\ldots,\lceil C_{d}e^{2nd\tau}+1\rceil$, where for any
$i\leq\lceil C_{d}e^{2dn\tau}+1\rceil$, $\\{B(x,n,e^{-\epsilon
n}):x\in\mathcal{C}_{i}\\}$ is a mutually disjoint collection. In particular,
for $n$ sufficiently large, $\lceil C_{d}e^{2n\tau}+1\rceil\leq e^{3dn\tau}$.

Proof: Let $K_{n}:=\lceil C_{d}e^{2nd\tau}+1\rceil$. We associate $x_{k}$ with
$\mathcal{C}_{k}$ for all $k\leq K_{n}$. For each $k>K_{n}$, we allocate it
into one of the pre-existing collections in the following way.

Given $B(x_{K_{n}+1},n,e^{-\epsilon n})$, by claim 4, there exists at least
one $i_{K_{n}+1}\leq K_{n}$ s.t $B(x_{K_{n}+1},n,e^{-\epsilon n})\cap
B(x_{i_{K_{n}+1}},n,e^{-\epsilon n})=\varnothing$; so allocate $x_{K_{n}+1}$
to $\mathcal{C}_{i_{K_{n}+1}}$.

Next, consider $B(x_{K_{n}+2},n,e^{-\epsilon n})$, at least two of the
$K_{n}+1$-first balls do not intersect it. If any of the balls
$2,\ldots,K_{n}$ do not intersect $B(x_{K_{n}+2},n,e^{-\epsilon n})$, allocate
it to the first associated collection as such. If all of the balls of
$1,\ldots,i_{K_{n}+1}-1,i_{K_{n}+1}+1,\ldots,K_{n}$ intersect it, then
allocate it to $\mathcal{C}_{i_{K_{n}+1}}$ which now contains three disjoint
balls.

We continue by induction. Assume that the balls $\\{B(x_{j},n,e^{-\epsilon
n})\\}{j\leq K_{n}+l-1}$ have all been allocated into one of the $K_{n}$-many
disjoint sub-collections. Consider $B(x_{K_{n}+l},n,e^{-\epsilon n})$, which
is disjoint from at least $l$ balls in $\\{B(x_{j},n,e^{-\epsilon n})\\}{j\leq
K_{n}+l-1}$. This implies that least one of the collections $\mathcal{C}_{i}$,
$i\leq K_{n}$, is disjoint from $B(x_{K_{n}+l},n,e^{-\epsilon n})$, to which
we may allocate it. ∎

The following lemma is paramount to our proof. Its significance is by
providing a density statement over sets which have unbounded eccentricity, for
a general measure.

######  Lemma 2.4 (Bowen-Lebesgue density lemma).

Let $\mu$ be an $f$-invariant probability measure s.t $\chi^{s}(\mu)>0$. Let
$A$ be a measurable set s.t $\mu(A)>0$. Then for $\mu$-a.e $x\in A$,

$\lim_{r\to 0}\limsup_{n\to\infty}\frac{-1}{n}\log\frac{\mu(B(x,n,e^{-nr})\cap
A)}{\mu(B(x,n,e^{-nr}))}=0.$

###### Proof.

First, assume that $\exists\lambda>0$ s.t (w.l.o.g) for $\mu$-a.e $x\in A$,

$\lim_{r\to 0}\limsup\frac{-1}{n}\log\frac{\mu(B(x,n,e^{-rn})\cap
A)}{\mu(B(x,n,e^{-rn}))}\geq\lambda.$

Let $0<r<\frac{1}{3d}\frac{\lambda}{4}$ s.t $\mu(A^{(0)})>0$ where

$A^{(0)}:=\\{x\in A:\limsup\frac{-1}{n}\log\frac{\mu(B(x,n,e^{-rn})\cap
A)}{\mu(B(x,n,e^{-rn}))}\geq\frac{7\lambda}{8}\\}.$

Then it follows that $\mu(A^{(1)})>0$ where
$A^{(1)}:=A^{(0)}\cap\Lambda^{(\underline{\chi},\tau)}$ for some index
$\underline{\chi}$ and $\tau\leq\min\\{\frac{r}{2},\tau_{\underline{\chi}}\\}$
(this can be achieved by first dividing the parameter space of
$\underline{\chi}$ by the Oseledec dimensions, then by boxes around each
$\underline{\chi}^{\prime}$ of size
$\frac{1}{2}\tau_{\underline{\chi}^{\prime}}$, and then further by boxes of
size
$\tau:=\min\\{\frac{r}{2},\frac{1}{2}\tau_{\underline{\chi}^{\prime}}\\}$;
whence
$\frac{1}{2}\tau_{\underline{\chi}^{\prime}}\leq\tau_{\underline{\chi}}$).

Let $N\geq 1$ large and $x\in A^{(1)}$, and set

$\displaystyle n_{x}^{N}:=\min\\{n\geq
N:\frac{-1}{n}\log\frac{\mu(B(x,n,e^{-rn})\cap
A)}{\mu(B(x,n,e^{-rn}))}\geq\frac{6\lambda}{8}\\}.$

Set for all $n\geq N$,

$A_{n}^{(1)}:=\\{x\in A^{(1)}:n_{x}^{N}=n\\}.$

By Lemma 2.3, we can choose a finite subset $\mathcal{C}_{n}\subseteq
A^{(1)}_{n}$ s.t $\bigcup_{x\in\mathcal{C}_{n}}B(x,n,e^{-rn})\supseteq
A^{(1)}_{n}$ while $\\{B(x,n,e^{-rn})\\}_{x\in\mathcal{C}_{n}}$ has an overlap
bound smaller than $e^{3d\tau n}\leq e^{3d\frac{r}{2}n}\leq
e^{\frac{\lambda}{8}n}$ (for all $n$ large enough depending on $M$ and
$\Lambda^{(\underline{\chi},\tau)}$). Then,

$\displaystyle\mu(A^{(1)}_{n})=$
$\displaystyle\mu\left((\bigcup_{x\in\mathcal{C}_{n}}B(x,n,e^{-rn}))\cap
A^{(1)}_{n}\right)\leq\sum_{x\in\mathcal{C}_{n}}\mu(B(x,n,e^{-rn})\cap
A^{(1)}_{n})\leq\sum_{x\in\mathcal{C}_{n}}\mu(B(x,n,e^{-rn})\cap A)$
$\displaystyle\leq$
$\displaystyle\sum_{x\in\mathcal{C}_{n}}\mu(B(x,n,e^{-rn}))e^{-\frac{6\lambda}{8}n}\leq
e^{-\frac{6\lambda}{8}n}e^{\frac{\lambda}{8}n}\mu(\bigcup_{x\in\mathcal{C}_{n}}B(x,n,e^{-rn}))\leq
e^{-\frac{5\lambda}{8}n}.$

Then,

$\mu(A^{(1)})=\sum_{n\geq N}\mu(A^{(1)}_{n})\leq\sum_{n\geq
N}e^{-\frac{5\lambda}{8}n}\leq(\sum_{n\geq 0}e^{-\frac{5\lambda}{8}n})\cdot
e^{-\frac{5\lambda}{8}N}\xrightarrow[]{N\to\infty}0,\text{ a contradiction!}$

∎

Remark: A possible heuristic way to interpret Lemma 2.4 is the following: we
think of belonging to a measurable set as satisfying some property. Then, for
almost any point which satisfies a certain property, more and more points
which spend a long portion of their orbit close to this point inherit this
property as well. That is, a bigger and bigger exponential portion of the
exponential Bowen ball lies in the measurable set as well.

######  Theorem 2.5.

Let $f\in\mathrm{Diff}^{1+\beta}(M)$, $\beta>0$, where $M$ is a closed
Riemannian manifold with $\mathrm{dim}M=d\geq 2$. Let $\mu$ be an
$f$-invariant Borel probability on $M$. Then

$\mathcal{E}_{\mu}(x)\leq h_{\mu}^{\mathrm{LY}}(x)\text{ }\mu\text{-a.e.}$

The idea of the key computation in the proof (see (7)) is the following: think
of an exponential Bowen ball as a narrow rectangle, foliated by strong leaves
corresponding to the direction of the narrow side of the rectangle. Some of
these leaves admit good estimates for the unstable entropy, for some points in
them. The Bowen-Lebesgue density lemma (Lemma 2.4) allows us to choose the
center of the rectangle so even when we take a twice-smaller rectangle in the
same center, still those “nice” leaves and points constitute a substantial
portion of its measure, thus allowing us to bound from below the measure of
the original rectangle.

###### Proof.

First, assume that for $\mu$-a.e $x$, $\chi^{u}(x)\geq\chi_{\mathrm{min}}>0$,
and assume that almost all the ergodic components of $\mu$ have the same
Oseledec dimensions. Let $0<\delta<1$, and let
$0<\tau=\tau(\delta)\ll\delta\cdot\max\\{1,\chi_{\mathrm{min}}\\}$ (e.g
$\tau=\frac{\max\\{1,\chi_{\mathrm{min}}\\}}{100d}\delta^{3}$). Assume further
that $\exists\underline{i}\in\mathbb{N}^{d+1}$ s.t for $\mu$-a.e ergodic
component $\mu_{x}$,

  1. (1)

$h_{\mu_{x}}(f)\in[i_{1}\delta,(i_{1}+1)\delta]$,

  2. (2)

$\chi_{j}(\mu_{x})\in[i_{j}\delta,(i_{j}+1)\delta]$, $j=2,\ldots,d+1$.

Notice that in particular, for $\mu$-a.e $x$,
$h^{\mathrm{LY}}_{\mu}(x)\in[i_{1}\delta,(i_{1}+1)\delta]$. Write
$h_{\mu}:=\delta\cdot\frac{2i_{1}+1}{2}$, and let $\underline{\chi}_{\mu}$ be
similarly the associated index by centers of intervals.

Let $\Lambda^{(\underline{\chi},\tau)}$ be a Pesin block s.t
$\mu(\Lambda^{(\underline{\chi},\tau)})\geq e^{-\tau}$. Let
$x_{0}\in\Lambda^{(\underline{\chi},\tau)}$, and let $B(x_{0},\frac{1}{l})$ be
its Pesin chart and set $Q:=\Lambda^{(\underline{\chi},\tau)}\cap
B(x_{0},\frac{1}{l})$.

For $x\in Q$, let $V^{s}(x)$ be its local stable leaf in
$B(x_{0},\frac{1}{l})$. The lamination $V^{s}(\cdot)$ of $Q$ is Lipschitz, and
extends to a Lipschitz foliation of $B(x_{0},\frac{1}{l})$, $\mathcal{F}^{u}$
([BPS99]). Denote its Lipschitz constant by $L$.

Let $H^{\mathrm{cs}}:=$connected component of
$\psi_{x_{0}}[\widehat{E}^{\mathrm{cs}}]$ in $B(x_{0},\frac{1}{l})$ which
contains $x_{0}$. Let $\pi_{u}:B(x_{0},\frac{1}{\ell})\to H^{\mathrm{cs}}$ be
the projection along the foliation $\mathcal{F}^{u}$.

Let $\Omega_{\tau}:=\\{y\in Q:\forall n\geq
n_{\tau},\mu_{\xi^{u}(y)}(B^{u}(y,n,\tau))\geq e^{-h_{\mu}-\tau n-\delta
n}\\}$, where $n_{\tau}$ is sufficiently large so $\mu(\Omega_{\tau})\geq
e^{-\tau}\mu(Q)$.

First we write $\mu|_{\Omega_{\tau}}$ as the leaves measures integrated by a
base measure:

$\mu|_{\Omega_{\tau}}=\int\mu_{\xi^{u}(y)}|_{\Omega_{\tau}}d\mu(y)=\int\mu_{\xi^{u}(\pi_{u}(y))}|_{\Omega_{\tau}}d\mu(y)=\int\mu_{\xi^{u}(\pi_{u}(z))}|_{\Omega_{\tau}}d\nu(y),$

where $\nu:=\mu\circ\pi_{u}^{-1}$ is a measure on $H^{\mathrm{cs}}$. For
$n^{\prime}:=\lceil n(1+\frac{r}{\chi_{\min}})\rceil+\lceil\frac{\log
3}{\chi_{\min}}\rceil$, we have that

$\mu(B(x,n^{\prime},e^{-n^{\prime}r})\cap\Omega_{\tau})\geq
e^{-n^{\prime}\tau}\mu(B(x,n^{\prime},e^{-n^{\prime}r}))$

for almost any $x$, for all $n$ large enough. Moreover, the choice of
$n^{\prime}$ guarantees that $\bigcup_{y\in
B(x,n^{\prime},e^{-n^{\prime}r})\cap\Omega_{\tau}}B^{u}(y,n^{\prime},\tau)\subseteq
B(x,n,e^{-nr})$. Assume also w.l.o.g that for almost every $y\in
B(x,n,e^{-nr})$, $B^{u}(y,n^{\prime},\tau)\subseteq\xi^{u}(y)$. Then,

$\displaystyle\mu(B(x,n,e^{-nr}))\geq$
$\displaystyle\mu(B(x,n,e^{-nr})\cap\Omega_{\tau})$ (7) $\displaystyle\geq$
$\displaystyle\int\mu_{\xi^{u}(z)}(B(x,n,e^{-nr})\cap\Omega_{\tau})d\nu(z)$
$\displaystyle\geq$ $\displaystyle
e^{-n^{\prime}\tau}\int_{B_{H^{\mathrm{cs}}}(\pi_{u}(x),e^{-(2r+\tau)n^{\prime}-\log
L})}e^{-(h_{\mu}+\tau)n^{\prime}}d\nu(y)$ $\displaystyle\geq$ $\displaystyle
e^{-(h_{\mu}+2\tau)n^{\prime}}\cdot\nu(B_{H^{\mathrm{cs}}}(\pi_{u}(x),e^{-(2r+\tau)n^{\prime}-\log
L}).$

For $\nu$-a.e $t\in H^{\mathrm{cs}}$,
$\limsup\frac{\log\nu(B_{H^{\mathrm{cs}}}(t,s))}{\log s}\leq d$, then choose
$x$ s.t

$\limsup\frac{-1}{n}\log\nu(B_{H^{\mathrm{cs}}}(\pi_{u}(x),e^{-(2r+\tau)n^{\prime}}))\leq(2r+\tau)d(1+\frac{r}{\chi_{\min}}),$

then

$\limsup\frac{-1}{n}\log\mu(B(x,n,e^{-nr}))\leq\left(h_{\mu}+2\tau+d(2r+\tau)+\delta\right)(1+\frac{r}{\chi_{\min}}).$

Finally, since $\tau>0$ was arbitrary, $\mathcal{E}_{\mu}(x)\leq
h_{\mu}+\delta$ $\mu$-a.e. Moreover, if $\delta>0$ is smaller than $\epsilon
h_{\mu}$ (the $0$-entropy case is similar), then

$\mathcal{E}_{\mu}(x)\leq e^{\epsilon}h_{\mu}(x)\text{ for }\mu\text{-a.e }x.$
(8)

This almost completes the case where $\chi^{s}(\mu)>0$.

Now for the case where $\chi^{u}=0$. Write $\mu=a_{0}\mu^{0}+a_{+}\mu^{+}$
where $\mu^{0}$ is all the ergodic components of $\mu$ with $\chi^{u}=0$
(whence it has entropy $0$). Similarly to the first case,
$B(x,n,e^{-rn})\supseteq B(x,e^{-2rn})$ a.e for large enough $n$. By the
classical lemma, $\limsup\frac{-1}{n}\log\mu(B(x,e^{-2rn}))\leq 2dr$ a.e.
Then, $\mathcal{E}_{\mu}=0\leq h_{\mu}^{\mathrm{LY}}$ $\mu^{0}$-a.e by
definition. If we knew that $\mathcal{E}_{\mu}\leq\mathcal{E}_{\mu^{+}}$
$\mu^{+}$-a.e, we would be done by the first case and by Lemma 2.1. Indeed,

$\displaystyle\mathcal{E}_{\mu}(x)=$ $\displaystyle\lim_{r\to
0}\limsup_{n\to\infty}\frac{-1}{n}\log\mu(B(x,n,e^{-nr}))$ $\displaystyle=$
$\displaystyle\lim_{r\to
0}\limsup_{n\to\infty}\frac{-1}{n}\log\left(a_{0}\mu^{0}(B(x,n,e^{-nr}))+a_{+}\mu^{+}(B(x,n,e^{-nr}))\right)$
$\displaystyle\leq$ $\displaystyle\lim_{r\to
0}\limsup_{n\to\infty}\frac{-1}{n}\log\mu^{+}(B(x,n,e^{-nr}))=\mathcal{E}_{\mu^{+}}(x).$

Now to treat the general case and to complete the proof. Let $\mu$, and write

$\mu=a_{0}\cdot\mu_{0}+\sum_{j\geq 1}a_{j}\mu_{j}^{\delta},\text{
}\\{\mu_{j}^{\delta}\\}_{j\geq 1}\text{ mutually singular},$

where $\mu_{0}$ constitutes of all the ergodic components of $\mu$ with
$\chi^{u}=0$, and for every $j\geq 1$ the indices are constant
$\mu_{j}^{\epsilon}$-a.e and for $\mu_{j}^{\epsilon}$-a.e ergodic component
$\mu_{j,x}^{\epsilon}$,

$\chi_{\ell}(\mu_{j,x}^{\epsilon})=e^{\pm\epsilon}\int\chi_{\ell}d\mu_{j}^{\epsilon}\text{,
}\ell=1,\ldots,d.$

Take a sequence
$\mathbb{Q}^{+}\ni\epsilon_{j}\leq\epsilon\cdot\chi^{0}_{\mathrm{min}}(\mu_{j}^{\epsilon})\to
0$, and let $A_{j}$ be the set of points which satisfy (8) for
$\mu_{j}^{\epsilon}$ with $\epsilon_{j}$, and which satisfy Lemma 2.1 (see the
following remark). Let
$A:=\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits_{j\geq
0}A_{j}$. Then $\mu(A)=1-a_{0}$, and for all $x\in A$, let $j(x)$ s.t $x\in
A_{j(x)}$, and so

$\displaystyle\limsup\frac{-1}{n}\log\mu(B(x,n,e^{-rn}))\leq$
$\displaystyle\limsup\frac{-1}{n}\log\mu^{\epsilon}_{j(x)}(B(x,n,e^{-rn}))$
$\displaystyle\leq$ $\displaystyle
h_{\mu^{\epsilon}_{j(x)}}^{\mathrm{LY}}(x)e^{\epsilon_{j}}+2rd=h_{\mu}^{\mathrm{LY}}(x)e^{\epsilon_{j}}+2rd.$

By monotonicity it is enough to compute the limit for a countable sequence
$r_{l}\to 0$. So for $(\mu-a_{0}\mu_{0})$-a.e $x$,

$\mathcal{E}_{\mu}(x)\leq h_{\mu}^{\mathrm{LY}}(x)e^{\epsilon}.$

Since $\epsilon>0$ is arbitrarily small, $\mathcal{E}_{\mu}(x)\leq
h^{\mathrm{LY}}\mu(x)$ for $(\mu-a_{0}\mu_{0})$-a.e $x$, and this completes
the proof. ∎

######  Corollary 2.6.

Under the assumptions of Theorem 2.5,
$h_{\mu}^{\mathrm{BK}}=\mathcal{E}_{\mu}=h_{\mu}^{\mathrm{LY}}$ a.e.

###### Proof.

By Theorem 2.5 and (2), $h_{\mu}^{\mathrm{LY}}-h_{\mu}^{\mathrm{BK}}\geq 0$
$\mu$-a.e. Then,

$0=h_{\mu}(f)-h_{\mu}(f)\geq\int h_{\mu}^{\mathrm{LY}}(x)d\mu(x)-\int
h_{\mu}^{\mathrm{BK}}(x)d\mu(x)=\int(h_{\mu}^{\mathrm{LY}}(x)-h_{\mu}^{\mathrm{BK}}(x))d\mu(x).$

The integrand on the r.h.s is non-negative, and so it equals $0$ $\mu$-a.e.
Therefore $h_{\mu}^{\mathrm{BK}}\leq\mathcal{E}_{\mu}\leq
h_{\mu}^{\mathrm{LY}}\leq h_{\mu}^{\mathrm{BK}}$ $\mu$-a.e and we are done.

∎

Remark: Corollary 2.6 implies that $\mathcal{E}_{\mu}(x)=\lim_{r\to
0}\liminf_{n\to\infty}\frac{-1}{n}\log\mu(B(x,n,e^{-rn}))$ for $\mu$-a.e $x$:

$h_{\mu}^{\mathrm{BK}}(x)=\lim_{r\to
0}\liminf_{n\to\infty}\frac{-1}{n}\log\mu(B(x,n,r))\leq\lim_{r\to
0}\liminf_{n\to\infty}\frac{-1}{n}\log\mu(B(x,n,e^{-rn}))\leq\mathcal{E}_{\mu}(x)=h_{\mu}^{\mathrm{BK}}(x).$

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