# Dimension approximation in smooth dynamical systems

Yongluo Cao  Departament of Mathematics, Shanghai Key Laboratory of PMMP, East
China Normal University, Shanghai 200062, P.R. China  Departament of
Mathematics, Soochow University, Suzhou 215006, Jiangsu, P.R. China  Center
for Dynamical Systems and Differential Equations, Soochow University, Suzhou
215006, Jiangsu, P.R. China<EMAIL_ADDRESS>,  Juan Wang  School of
Mathematics, Physics and Statistics, Shanghai University of Engineering
Science, Shanghai 201620, P.R. China<EMAIL_ADDRESS>and  Yun Zhao
School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu,
P.R. China  Center for Dynamical Systems and Differential Equations, Soochow
University, Suzhou 215006, Jiangsu, P.R. China<EMAIL_ADDRESS>
###### Abstract.

For a non-conformal repeller $\Lambda$ of a $C^{1+\alpha}$ map $f$ preserving
an ergodic measure $\mu$ of positive entropy, this paper shows that the
Lyapunov dimension of $\mu$ can be approximated gradually by the Carathéodory
singular dimension of a sequence of horseshoes. For a $C^{1+\alpha}$
diffeomorphism $f$ preserving a hyperbolic ergodic measure $\mu$ of positive
entropy, if $(f,\mu)$ has only two Lyapunov exponents
$\lambda_{u}(\mu)>0>\lambda_{s}(\mu)$, then the Hausdorff or lower box or
upper box dimension of $\mu$ can be approximated by the corresponding
dimension of the horseshoes $\\{\Lambda_{n}\\}$. The same statement holds true
if $f$ is a $C^{1}$ diffeomorphism with a dominated Oseledet’s splitting with
respect to $\mu$.

###### Key words and phrases:

Dimension, hyperbolic measure, horseshoe, repeller.

00footnotetext: 2010 Mathematics Subject classification: 37C45, 37D25, 37D20

##  1\. Introduction

In smooth dynamical systems, a fundamental approximation result asserts that a
$C^{1+\alpha}$ diffeomorphism $f$ which preserves a hyperbolic ergodic measure
$\mu$ of positive entropy can be approximated gradually by compact invariant
locally maximal hyperbolic sets–horseshoes $\\{\Lambda_{n}\\}$, in the sense
that dynamical quantities on the horseshoes such as the topological entropy
and pressure, Lyapunov exponents and averages of continuous functions are
approaching to the ones of the measure $\mu$.

This type of results are widely referred to the landmark work by Katok [9] or
Katok and Mendoza (see [10]). Misiurewicz and Szlenk [29] earlier proved a
related result for continuous and for piecewise monotone maps of the interval.
Przytycki and Urbański [34] obtained corresponding properties for holomorphic
maps in the case of a measure with only positive Lyapunov exponent. A related
setting of dyadic diophantine approximations is established by Persson and
Schmeling in [32]. For a general $C^{1+\alpha}$ diffeomorphism $f$ preserving
a hyperbolic ergodic measure $\mu$ with positive entropy, assume that $\mu$
has $\ell$ different Lyapunov exponent $\\{\lambda_{j}\\}_{j=1}^{\ell}$, on
each approaching horseshoe $\Lambda_{n}$, Avila, Crovisier and Wilkinson [2]
obtained a continuous splitting

$T_{\Lambda_{n}}M=E_{1}\oplus E_{2}\oplus\cdots\oplus E_{\ell}$

and showed that the exponential growth of $D_{x}f^{n}|_{E_{i}}$ is roughly
$\lambda_{i}$ for each $i=1,2,\cdots,\ell$. A corresponding statement for
$C^{1+\alpha}$ non-conformal transformations (i.e., non-invertible maps) was
shown in [13]. See Chung [14], Gelfert [18, 19] and Yang [44] for other
results related to Katok’s approximation construction of $C^{1+\alpha}$ maps.

A natural question is how large that part of the dynamics described by these
horseshoes is. So, it is interesting to estimate the Hausdorff dimension of
the stable and/or unstable Cantor sets of a horseshoe. If $\mu$ is a SRB
measure (i.e. a measure with a particular absolute continuity property on
unstable manifolds; see [6] for precise definitions), it was showed in [36]
that $\mu$ can be approximated by ergodic measures supported on horseshoes
with arbitrarily large unstable dimensions, which generalized Mendoza’s result
in [26] for diffeomorphisms in higher dimensional manifold. The approach in
[36] was based on Markov towers that can be described by horseshoes with
infinitely many branches and variable return times. However, there is an
essential mistake in the proof of the key Proposition 5.1 in [36]. The authors
in [43] proved the same result by a different method. They used the u-Gibbs
property of the conditional measure of the equilibrium measure and the
properties of the uniformly hyperbolic dynamical systems. Furthermore, in [43]
the authors proved that the Hausdorff dimension of $\mu$ can be approximated
gradually by the Hausdorff dimension of the horseshoes $\\{\Lambda_{n}\\}$
provided that the stable direction is one dimension. See also [24, 25, 27, 28,
37, 38] that represent works close to this topic.

In this work, our main task is to compare the dimension of the horseshoes
$\\{\Lambda_{n}\\}$ and the given hyperbolic ergodic measure $\mu$ of a
$C^{r}$ ($r\geq 1$) diffeomorphism in a more general setting that $\mu$ may be
not a SRB measure. For a non-conformal repeller $\Lambda$ of a $C^{1+\alpha}$
map, utilizing the approximation result in [13], we show that the Lyapunov
dimension (see (3.1) for the definition) of an $f$-invariant ergodic measure
$\mu$ supported on $\Lambda$ can be approximated gradually by the Carathéodory
singular dimension (see (3.6) for the definition) of the horseshoes
$\\{\Lambda_{n}\\}$. For a $C^{1+\alpha}$ diffeomorphism $f$ preserving a
hyperbolic ergodic measure $\mu$ of positive entropy, if $(f,\mu)$ has only
two Lyapunov exponents $\lambda_{u}(\mu)>0>\lambda_{s}(\mu)$, then the
Hausdorff or lower box or upper box dimension of $\mu$ can be approximated by
the corresponding dimension of the horseshoes $\\{\Lambda_{n}\\}$. The same
statement holds true if $f$ is a $C^{1}$ diffeomorphism with a dominated
Oseledec’s splitting w.r.t. $\mu$.

We arrange the paper as follows. In Section 2, we give some basic notions and
properties about topological and measure theoretic pressures and dimensions of
sets and measures. Statements of our main results will be given in Section 3.
In Section 4, we will give the detailed proofs of the main results.

##  2\. Definitions and preliminaries

In this section, we recall the definitions of topological pressure and various
dimensions of subsets and/or of invariant measures.

###  2.1. Topological and measure theoretic pressures

Let $f:X\to X$ be a continuous transformation on a compact metric space $X$
equipped with metric $d$. A subset $F\subset X$ is called an
$(n,\epsilon)-$separated set with respect to $f$, if for any two different
points $x,y\in F$, we have $d_{n}(x,y):=\max_{0\leq k\leq
n-1}d(f^{k}(x),f^{k}(y))>\epsilon.$ A sequence of continuous functions
$\Phi=\\{\phi_{n}\\}_{n\geq 1}$ is called _sub-additive_ , if

$\phi_{m+n}\leq\phi_{n}+\phi_{m}\circ f^{n},~{}~{}\forall n,m\in\mathbb{N}.$

Furthermore, a sequence of continuous functions $\Psi=\\{\psi_{n}\\}_{n\geq
1}$ is called _super-additive_ if $-\Psi=\\{-\psi_{n}\\}_{n\geq 1}$ is sub-
additive.

####  2.1.1. Topological pressure defined via separated sets

Given a sub-additive potential $\Phi=\\{\phi_{n}\\}_{n\geq 1}$ on $X$, put

$P_{n}(f,\Phi,\epsilon)=\sup\Big{\\{}\sum_{x\in F}e^{\phi_{n}(x)}|F\ \mbox{is
an}\ (n,\epsilon)-\mbox{separated subset of }X\Big{\\}}.$

######  Definition 2.1.

We call the following quantity

$\displaystyle P_{\mathrm{top}}(f,\Phi)=\lim_{\epsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log P_{n}(f,\Phi,\epsilon)$ (2.1)

the _sub-additive topological pressure_ of $(f,\Phi)$.

######  Remark 2.1.

If $\varPhi=\\{\varphi_{n}\\}_{n\geq 1}$ is _additive_ in the sense that
$\varphi_{n}(x)=\varphi(x)+\varphi(fx)+\cdots+\varphi(f^{n-1}x)\triangleq
S_{n}\varphi(x)$ for some continuous function $\varphi:X\to\mathbb{R}$, we
simply denote the topological pressure $P_{\mathrm{top}}(f,\varPhi)$ as
$P_{\mathrm{top}}(f,\varphi)$.

Let $\mathcal{M}_{f}(X)$ denote the space of all $f-$invariant measures on
$X$. For $\mu\in\mathcal{M}_{f}(X)$, let $h_{\mu}(f)$ denote the metric
entropy of $f$ with respect to $\mu$ (see Walters’ book [39] for details of
metric entropy), and let

$\mathcal{L}_{*}(\Phi,\mu)=\lim_{n\to\infty}\frac{1}{n}\int\phi_{n}d\mu.$

The existence of the above limit follows from a sub-additive argument. In
[11], the authors proved the following variational principle.

######  Theorem 2.1.

Let $f:X\to X$ be a continuous transformation on a compact metric space $X$,
and $\Phi=\\{\phi_{n}\\}_{n\geq 1}$ a sub-additive potential on $X$, then we
have

$P_{\mathrm{top}}(f,\Phi)=\sup\Big{\\{}h_{\mu}(f)+\mathcal{L}_{*}(\Phi,\mu):\mu\in\mathcal{M}_{f}(X),~{}\mathcal{L}_{*}(\Phi,\mu)\neq-\infty\Big{\\}}.$

Though it is unknown whether the variational principle holds for super-
additive topological pressure, Cao, Pesin and Zhao gave an alternative
definition via variational principle in [13]. Given a sequence of super-
additive continuous potentials $\Psi=\\{\psi_{n}\\}_{n\geq 1}$ on a compact
dynamical system $(X,f)$, the super-additive topological pressure of $\Psi$ is
defined as

$P_{\mathrm{var}}(f,\Psi):=\sup\Big{\\{}h_{\mu}(f)+\mathcal{L}_{*}(\Psi,\mu):\mu\in\mathcal{M}_{f}(X)\Big{\\}}.$

where

$\mathcal{L}_{*}(\Psi,\mu)=\lim_{n\to\infty}\frac{1}{n}\int\psi_{n}d\mu=\sup_{n\geq
1}\frac{1}{n}\int\psi_{n}d\mu.$

The second equality is due to the standard sub-additive argument.

####  2.1.2. Measure theoretic pressure

We first follow the approach in [33] to give the definitions of topological
pressures on arbitrary subsets. Given a sub-additive potential
$\Phi=\\{\phi_{n}\\}_{n\geq 1}$ on $X$, a subset $Z\subset X$ and
$\alpha\in\mathbb{R}$, let

$\displaystyle M(Z,\Phi,\alpha,N,\epsilon)=\inf\Big{\\{}$
$\displaystyle\sum_{i}\exp\bigr{(}-\alpha n_{i}+\sup_{y\in
B_{n_{i}}(x_{i},\epsilon)}\phi_{n_{i}}(y)\bigr{)}:$
$\displaystyle\bigcup_{i}B_{n_{i}}(x_{i},\epsilon)\supset Z,\,x_{i}\in X\text{
and }n_{i}\geq N\text{ for all }i\Big{\\}}.$

Since $M(Z,\Phi,\alpha,N,\epsilon)$ is monotonically increasing with $N$, let

$\displaystyle
m(Z,\Phi,\alpha,\epsilon):=\lim_{N\rightarrow\infty}M(Z,\Phi,\alpha,N,\epsilon).$
(2.2)

We denote the jump-up point of $m(Z,\Phi,\alpha,\epsilon)$ by

$P_{Z}(f,\Phi,\epsilon)=\inf\\{\alpha:m(Z,\Phi,\alpha,\epsilon)=0\\}=\sup\\{\alpha:m(Z,\Phi,\alpha,\epsilon)=+\infty\\}.$

######  Definition 2.2.

We call the quantity

$\displaystyle P_{Z}(f,\Phi)=\liminf_{\epsilon\to 0}P_{Z}(f,\Phi,\epsilon)$

the _topological pressure_ of $(f,\Phi)$ on the set $Z$ (see [17] for the
weighted version of this quantity).

Similarly, for $\alpha\in\mathbb{R}$ and $Z\subset X$, define

$R(Z,\Phi,\alpha,N,\epsilon)=\inf\Big{\\{}\sum_{i}\exp\bigr{(}-\alpha
N+\sup_{y\in B_{N}(x_{i},\epsilon)}\phi_{N}(y)\bigr{)}:\\\
\bigcup_{i}B_{N}(x_{i},\epsilon)\supset Z,\,x_{i}\in X\Big{\\}}.$

We set

$\displaystyle\underline{r}(Z,\Phi,\alpha,\epsilon)$
$\displaystyle=\liminf_{N\rightarrow\infty}R(Z,\Phi,\alpha,N,\epsilon),$
$\displaystyle\overline{r}(Z,\Phi,\alpha,\epsilon)$
$\displaystyle=\limsup_{N\rightarrow\infty}R(Z,\Phi,\alpha,N,\epsilon)$

and define the jump-up points of $\underline{r}(Z,\Phi,\alpha,\epsilon)$ and
$\overline{r}(Z,\Phi,\alpha,\epsilon)$ as

$\displaystyle\underline{CP}_{Z}(f,\Phi,\epsilon)$
$\displaystyle=\inf\\{\alpha:\underline{r}(Z,\Phi,\alpha,\epsilon)=0\\}=\sup\\{\alpha:\underline{r}(Z,\Phi,\alpha,\epsilon)=+\infty\\},$
$\displaystyle\overline{CP}_{Z}(f,\Phi,\epsilon)$
$\displaystyle=\inf\\{\alpha:\overline{r}(Z,\Phi,\alpha,\epsilon)=0\\}=\sup\\{\alpha:\overline{r}(Z,\Phi,\alpha,\epsilon)=+\infty\\}$

respectively.

######  Definition 2.3.

We call the quantities

$\underline{CP}_{Z}(f,\Phi)=\liminf_{\epsilon\to
0}\underline{CP}_{Z}(f,\Phi,\epsilon)\,\,\text{and}\,\,\overline{CP}_{Z}(f,\Phi)=\liminf_{\epsilon\to
0}\overline{CP}_{Z}(f,\Phi,\epsilon)$

the _lower_ and _upper topological pressures_ of $(f,\Phi)$ on the set $Z$
respectively.

Given an $f$-invariant measure $\mu$, let

$\displaystyle
P_{\mu}(f,\Phi,\epsilon)=\inf\\{P_{Z}(f,\Phi,\epsilon)\colon\mu(Z)=1\\}$

and then we call the following quantity

$P_{\mu}(f,\Phi):=\liminf_{\epsilon\to 0}P_{\mu}(f,\Phi,\epsilon)$

the _measure theoretic pressure_ of $(f,\Phi)$ with respect to $\mu$. Let
further

$\displaystyle\underline{CP}_{\mu}(f,\Phi,\epsilon)=\lim_{\delta\to
0}\inf\\{\underline{CP}_{Z}(f,\Phi,\epsilon)\colon\mu(Z)\geq 1-\delta\\},$
$\displaystyle\overline{CP}_{\mu}(f,\Phi,\epsilon)=\lim_{\delta\to
0}\inf\\{\overline{CP}_{Z}(f,\Phi,\epsilon)\colon\mu(Z)\geq 1-\delta\\}.$

We call the following quantities

$\displaystyle\underline{CP}_{\mu}(f,\Phi)=\liminf_{\epsilon\to
0}\underline{CP}_{\mu}(f,\Phi,\epsilon),\ \ \
\overline{CP}_{\mu}(f,\Phi)=\liminf_{\epsilon\to
0}\overline{CP}_{\mu}(f,\Phi,\epsilon)$

the _lower and upper measure theoretic pressures_ of $(f,\Phi)$ with respect
to $\mu$ respectively. It is proved in [12, Theorem A] that

$\displaystyle
P_{\mu}(f,\Phi)=\underline{CP}_{\mu}(f,\Phi)=\overline{CP}_{\mu}(f,\Phi)=h_{\mu}(f)+\mathcal{L}_{*}(\Phi,\mu)$
(2.3)

for any $f$-invariant ergodic measure $\mu$ with
$\mathcal{L}_{*}(\Phi,\mu)\neq-\infty$.

######  Remark 2.2.

In fact, one can show that

$\mathcal{P}_{\mu}(f,\Phi)=\inf\\{\mathcal{P}_{Z}(f,\Phi):\mu(Z)=1\\}$

here $\mathcal{P}$ denotes either $P$ or $\underline{CP}$ or $\overline{CP}$,
see [46] for a proof.

###  2.2. Dimensions of sets and measures

Now we recall the definitions of Hausdorff and box dimensions of subsets and
measures. Given a subset $Z\subset X$, For any $s\geq 0$, let

$\mathcal{H}_{\delta}^{s}(Z)=\inf\Big{\\{}\sum\limits_{i=1}^{\infty}(\mbox{diam}U_{i})^{s}:\\{U_{i}\bigr{\\}}_{i\geq
1}\ \mbox{is\ a cover\ of }Z\ \mbox{with }\mbox{diam}U_{i}\leq\delta,\forall
i\geq 1\Big{\\}}$

and

$\displaystyle\mathcal{H}^{s}(Z)=\lim\limits_{\delta\rightarrow
0}\mathcal{H}_{\delta}^{s}(Z).$

The above limit exists, though the limit may be infinity. We call
$\mathcal{H}^{s}(Z)$ the $s-$dimensional Hausdorff measure of $Z$.

######  Definition 2.4.

The following jump-up value of $\mathcal{H}^{s}(Z)$

$\dim_{H}Z=\inf\\{s:\mathcal{H}^{s}(Z)=0\\}=\sup\\{s:\mathcal{H}^{s}(Z)=\infty\\}$

is called the _Hausdorff dimension_ of $Z$. The _lower and upper box
dimension_ of $Z$ are defined respectively by

$\underline{\dim}_{B}Z=\liminf\limits_{\delta\to 0}\frac{\log
N(Z,\delta)}{-\log\delta}\ \text{and}\
\overline{\dim}_{B}Z=\limsup\limits_{\delta\to 0}\frac{\log
N(Z,\delta)}{-\log\delta},$

where $N(Z,\delta)$ denotes the least number of balls of radius $\delta$ that
are needed to cover the set $Z$. If
$\underline{\dim}_{B}Z=\overline{\dim}_{B}Z$, we will denote the common value
by $\dim_{B}Z$ and call it the _box dimension_ of $Z$.

The following two results are well-known in the field of fractal geometry,
e.g., see Falconer’s book [15] for proofs.

######  Lemma 2.1.

Let $X$ and $Y$ be metric spaces. For any $r\in(0,1)$, $\Phi:X\to Y$ is an
onto, $(C,r)$-H$\ddot{o}$lder continuous map for some $C>0$. Then

$\dim_{H}Y\leq r^{-1}\dim_{H}X,\quad\underline{\dim}_{B}Y\leq
r^{-1}\underline{\dim}_{B}X\quad\text{and}\quad\overline{\dim}_{B}Y\leq
r^{-1}\overline{\dim}_{B}X.$

######  Corollary 2.1.

Let $X$ and $Y$ be metric spaces, and let $\Phi:X\to Y$ be an onto, Lipschitz
continuous map. Then

$\dim_{H}Y\leq\dim_{H}X,\quad\underline{\dim}_{B}Y\leq\underline{\dim}_{B}X\quad\text{and}\quad\overline{\dim}_{B}Y\leq\overline{\dim}_{B}X.$

Given a Borel probability measure $\mu$ on $X$, the following quantity

$\displaystyle\dim_{H}\mu$ $\displaystyle=\inf\\{\dim_{H}Z:Z\subset
X~{}~{}\text{and}~{}~{}\mu(Z)=1\\}$ $\displaystyle=\lim_{\delta\to
0}\inf\\{\dim_{H}Z:Z\subset X~{}~{}\text{and}~{}~{}\mu(Z)\geq 1-\delta\\}$

is called the _Hausdorff dimension of the measure_ $\mu$. Similarly, we call
the following two quantities

$\underline{\dim}_{B}\mu=\lim_{\delta\to
0}\inf\\{\underline{\dim}_{B}Z:Z\subset X~{}~{}\text{and}~{}~{}\mu(Z)\geq
1-\delta\\}$

and

$\overline{\dim}_{B}\mu=\lim_{\delta\to 0}\inf\\{\overline{\dim}_{B}Z:Z\subset
X~{}~{}\text{and}~{}~{}\mu(Z)\geq 1-\delta\\}$

the _lower box dimension_ and _upper box dimension_ of $\mu$, respectively.

If $\mu$ is a finite measure on $X$ and there exists $d\geq 0$ such that

$\lim_{r\to 0}\frac{\log\mu(B(x,r))}{\log r}=d$

for $\mu$-almost every $x\in X$, then

$\dim_{H}\mu=\underline{\dim}_{B}\mu=\overline{\dim}_{B}\mu=d.$

This criterion was established by Young in [45].

##  3\. Statements of main results

In this section, we will give the statements of the main results in this
paper, and the proof will be postponed to the next section.

###  3.1. Dimension approximation for uniformly expanding systems

Let $f:M\to M$ be a smooth map of a $m_{0}$-dimensional compact smooth
Riemannian manifold $M$, and $\Lambda$ a compact $f$-invariant subset of $M$.
Let $\mathcal{M}_{f}(\Lambda)$ and $\mathcal{E}_{f}(\Lambda)$ denote
respectively the set of all $f$-invariant measures and ergodic measures on
$\Lambda$.

####  3.1.1. Definitions of repeller and Lyapunov dimension

We call $\Lambda$ a _repeller_ for $f$ or $f$ is _expanding_ on $\Lambda$ if

  1. (1)

there exists an open neighborhood $U$ of $\Lambda$ such that $\Lambda=\\{x\in
U:f^{n}(x)\in U\,\,\text{for all}\,\,n\geq 0\\}$;

  2. (2)

there is $\kappa>1$ such that

$\|D_{x}f(v)\|\geq\kappa\|v\|,\,\text{ for all }x\in\Lambda,\,\text{ and }v\in
T_{x}M,$

where $\|\cdot\|$ is the norm induced by the Riemannian metric on $M$, and
$D_{x}f:T_{x}M\rightarrow T_{f(x)}M$ is the differential operator.

Given an $f$-invariant ergodic measure $\mu$ supported on the repeller
$\Lambda$. Let
$\lambda_{1}(\mu)\geq\lambda_{2}(\mu)\geq\cdots\geq\lambda_{m_{0}}(\mu)$ and
$h_{\mu}(f)$ denote the Lyapunov exponents and the measure theoretic entropy
of $(f,\mu)$ respectively, we refer the reader to [6] and [39] for detailed
description of Lyapunov exponents and the the measure theoretic entropy. We
further define the _Lyapunov dimension_ of $\mu$ as follows:

$\dim_{\mathrm{L}}\mu:=\left\\{\begin{array}[]{ll}\ell+\frac{h_{\mu}(f)-\lambda_{m_{0}}(\mu)-\cdots-\lambda_{m_{0}-\ell+1}(\mu)}{\lambda_{m_{0}-\ell}(\mu)},&h_{\mu}(f)\geq\lambda_{m_{0}}(\mu)\\\
\frac{h_{\mu}(f)}{\lambda_{m_{0}}(\mu)},&0\leq
h_{\mu}(f)<\lambda_{m_{0}}(\mu)\end{array}\right.$ (3.1)

where $\ell=\max\\{i:\lambda_{m_{0}}(\mu)+\cdots+\lambda_{m_{0}-i+1}(\mu)\leq
h_{\mu}(f)\\}$.

The original definition of Lyapunov dimension in [1, 21, 22] is defined only
for hyperbolic systems as follows: assume that $\nu$ is an ergodic measure of
a smooth diffeomorphism $f$ with Lyapunov exponents
$\lambda_{1}\geq\cdots\geq\lambda_{u}>0\geq\lambda_{u+1}\geq\cdots\geq\lambda_{m_{0}}$,
then the the Lyapunov dimension is

$\mathrm{Lya}\dim\,\nu=\ell+\frac{\lambda_{1}+\cdots+\lambda_{u}+\cdots+\lambda_{\ell}}{|\lambda_{\ell+1}|}$

where $\ell=\max\\{i:\lambda_{1}+\cdots+\lambda_{i}\geq 0\\}$. Assume further
that $\nu$ is a SRB measure, then $h_{\nu}(f)=\lambda_{1}+\cdots+\lambda_{u}$.
In consequence,

$\mathrm{Lya}\dim\,\nu=\ell+\frac{h_{\nu}(f)+\lambda_{u+1}+\cdots+\lambda_{\ell}}{|\lambda_{\ell+1}|}$

and $\ell=\max\\{i:-\lambda_{u+1}-\cdots-\lambda_{i}\geq h_{\nu}(f)\\}$.
Hence, the definition in (3.1) is a reasonable substitute. For a $C^{1}$
expanding map $f$, Feng and Simon [16] defined the Lyapunov dimension of an
ergodic measure as the zero of the measure theoretic pressure
$P_{\mu}(f,\Phi_{f}(t))=0$ (see (3.4)). In this paper, we will prove the
unique solution of the equation $P_{\mu}(f,\Phi_{f}(t))=0$ is indeed our
definition of Lyapunov dimension (see Theorem A). Furthermore, this paper
shows that the Lyapunov dimension of an ergodic measure defined in (3.1)
equals to its Carathéodory singular dimension (see Proposition 3.1), so the
Carathéodory singular dimension (see Section 3.1.3 for the detailed
definition) can be regarded as a geometric explanation of the Lyapunov
dimension.

####  3.1.2. Singular valued potentials

Let $\Lambda$ be a repeller of a smooth map $f:M\to M$. Given $x\in\Lambda$
and $n\geq 1$, consider the differentiable operator $D_{x}f^{n}:T_{x}M\to
T_{f^{n}(x)}M$ and denote the singular values of $D_{x}f^{n}$ (square roots of
the eigenvalues of $(D_{x}f^{n})^{*}D_{x}f^{n}$) in the decreasing order by

$\alpha_{1}(x,f^{n})\geq\alpha_{2}(x,f^{n})\geq\dots\geq\alpha_{m_{0}}(x,f^{n}).$
(3.2)

For $t\in[0,m_{0}]$, set

$\varphi^{t}(x,f^{n}):=\sum_{i=m_{0}-[t]+1}^{m_{0}}\log\alpha_{i}(x,f^{n})+(t-[t])\log\alpha_{m_{0}-[t]}(x,f^{n}).$
(3.3)

Since $f$ is smooth, the functions $x\mapsto\alpha_{i}(x,f^{n})$,
$x\mapsto\varphi^{t}(x,f^{n})$ are continuous for any $n\geq 1$. It is easy to
see that for all $n,\ell\in\mathbb{N}$

$\varphi^{t}(x,f^{n+\ell})\geq\varphi^{t}(x,f^{n})+\varphi^{t}(f^{n}(x),f^{\ell}).$

It follows that the sequence of functions

$\Phi_{f}(t):=\\{-\varphi^{t}(\cdot,f^{n})\\}_{n\geq 1}$ (3.4)

is sub-additive, which is called the _sub-additive singular valued
potentials_.

####  3.1.3. Carathéodory singular dimension

We recall the definition of Carathéodory singular dimension of a repeller
which is introduced in [13].

Let $\Phi_{f}(t)=\\{-\varphi^{t}(\cdot,f^{n})\\}_{n\geq 1}$. Given a subset
$Z\subseteq\Lambda$, for each small number $r>0$, let

$m(Z,t,r):=\lim_{N\to\infty}\inf\Big{\\{}\sum_{i}\exp\bigr{(}\sup_{y\in
B_{n_{i}}(x_{i},r)}-\varphi^{t}(y,f^{n_{i}})\bigr{)}\Big{\\}},$

where the infimum is taken over all collections $\\{B_{n_{i}}(x_{i},r)\\}$ of
Bowen’s balls with $x_{i}\in\Lambda$, $n_{i}\geq N$ that cover $Z$. It is easy
to see that there is a jump-up value

$\dim_{C,r}Z:=\inf\\{t:m(Z,t,r)=0\\}=\sup\\{t:m(Z,t,r)=+\infty\\}.$ (3.5)

The following quantity

$\displaystyle{\dim_{C}Z:=\liminf_{r\to 0}\dim_{C,r}Z}$ (3.6)

is called the _Carathéodory singular dimension of $Z$_. Particularly, the
Carathéodory singular dimension of the repeller $\Lambda$ is independent of
the parameter $r$ for small values of $r>0$ (see [13, Theorem 4.1]).

For each $f$-invariant measure $\mu$ supported on $\Lambda$, let

$\dim_{C,r}\mu:=\inf\\{\dim_{C,r}Z:\mu(Z)=1\\},$

and the following quantity

$\dim_{C}\mu:=\liminf_{r\to 0}\dim_{C,r}\mu$

is called the _Carathéodory singular dimension_ of the measure $\mu$.

####  3.1.4. Approximation of Carathéodory singular dimension of repellers

Given a repeller $\Lambda$ of a $C^{1+\alpha}$ map $f$, the following result
shows that the zero of the measure theoretic pressure function is exactly the
Lyapunov dimension of an ergodic measure $\mu\in\mathcal{E}_{f}(\Lambda)$, and
the Lyapunov dimension of an ergodic measure of positive entropy can be
approximated by the Carathéodory singular dimension of a sequence of invariant
sets. Recall that $\Phi_{f}(t):=\\{-\varphi^{t}(\cdot,f^{n})\\}_{n\geq 1}$ is
the sub-additive singular valued potentials with respect to $f$ (See the
definition in (3.4).).

######  Theorem A.

Let $f:M\to M$ be a $C^{1+\alpha}$ map of an $m_{0}$-dimensional compact
smooth Riemannian manifold $M$, and $\Lambda$ a repeller of $f$. Then the
following statements hold:

  1. (1)

for every $f$-invariant ergodic measure $\mu$ supported on $\Lambda$, we have
that

$\dim_{\mathrm{L}}\mu=s_{\mu}$

where $s_{\mu}$ is the unique root of the equation $P_{\mu}(f,\Phi_{f}(t))=0$;

  2. (2)

let $\mu$ be an $f$-invariant ergodic measure on $\Lambda$ with
$h_{\mu}(f)>0$, for any $\varepsilon>0$ there exists an $f$-invariant compact
subset $\Lambda_{\varepsilon}\subset\Lambda$ such that
$\dim_{C}\Lambda_{\varepsilon}\rightarrow\dim_{\mathrm{L}}\mu$ as
$\varepsilon$ approaching zero.

Some comments on the previous theorem are in order. First, we would like to
point out that it is enough to require $f$ to be a $C^{1}$ map in the first
statement, however, the map of higher smoothness $C^{1+\alpha}$ is crucial in
the last statement as it allows us to utilize some powerful results of Pesin
theory. Second, if $f$ is a local diffeomorphism preserving an ergodic
expanding measure $\mu$ of positive entropy, i.e., $(f,\mu)$ has only positive
Lyapunov exponent, in this case one can also obtain a approximation result as
in [13] so that we can obtain the second statement in the previous theorem in
this setting. In [37], for a $C^{2}$ interval map $f$ with finitely many non-
degenerate critical points, the author proved that the Hausdorff dimension of
an expanding measure $\mu$ can be approximated gradually by the Hausdorff
dimension of a sequence of repellers.

For each $f$-invariant ergodic measure $\mu$ supported on $\Lambda$, the
following result shows that the Carathéodory singular dimension of $\mu$ is
exactly its Lyapunov dimension.

######  Proposition 3.1.

Let $f:M\to M$ be a $C^{1}$ map of an $m_{0}$-dimensional compact smooth
Riemannian manifold $M$, and $\Lambda$ a repeller for $f$. Then the following
statements hold:

  1. (1)

for each subset $Z\subset\Lambda$, we have that

$\dim_{C}Z=t_{Z}$

where $t_{Z}$ is the unique root of the equation $P_{Z}(f,\Phi_{f}(t))=0$;

  2. (2)

for each $f$-invariant ergodic measure $\mu$ supported on $\Lambda$, we have
that

$\dim_{C}\mu=\dim_{\mathrm{L}}\mu.$

###  3.2. Dimension approximation in non-uniformly hyperbolic systems

In this section, we first recall an approximation result in non-uniformly
hyperbolic systems that are proved by Avila et al [2], then we give the
statement of our dimension approximation result in non-uniformly hyperbolic
systems.

####  3.2.1. Lyapunov exponents and holonomy maps

Let $f:M\to M$ be a diffeomorphism on an $m_{0}$-dimensional compact smooth
Riemannian manifold $M$. By the Oseledec’s multiplicative ergodic theorem (see
[30]), there exists a total measure set $\mathcal{O}\subset M$ such that, for
each $x\in\mathcal{O}$ and each invariant measure $\mu$ there exist positive
integers $d_{1}(x),d_{2}(x),\cdots,d_{p(x)}(x)$, numbers
$\lambda_{1}(x)>\lambda_{2}(x)>\cdots>\lambda_{p(x)}(x)$ and a splitting

$T_{x}M=E_{1}(x)\oplus E_{2}(x)\oplus\cdots\oplus E_{p(x)}(x)$

satisfy that

  1. (1)

$D_{x}fE_{i}(x)=E_{i}(f(x))$ for each $i$ and
$\sum_{i=1}^{p(x)}d_{i}(x)=m_{0}$;

  2. (2)

for each $0\neq v\in E_{i}(x)$ we have that

$\lambda_{i}(x)=\lim_{n\to\infty}\frac{1}{n}\log\|D_{x}f^{n}(v)\|.$

Here we call the numbers $\\{\lambda_{i}(x)\\}_{i=1}^{p(x)}$ the Lyapunov
exponents of $(f,\mu)$. In the case that $\mu$ is an $f$-invariant ergodic
measure, the numbers $p(x)$, $\\{d_{i}(x)\\}$ and $\\{\lambda_{i}(x)\\}$ are
constants almost everywhere. We denote them simply by $p$,
$\\{d_{i}\\}_{i=1}^{p}$ and $\\{\lambda_{i}\\}_{i=1}^{p}$.

A compact invariant subset $\Lambda\subset M$ is called a _hyperbolic set_ ,
if there exists a continuous splitting of the tangent bundle
$T_{\Lambda}M=E^{s}\oplus E^{u}$, and constants $C>0,\ 0<\lambda<1$ such that
for every $x\in\Lambda$

  1. (1)

$D_{x}f(E^{s}(x))=E^{s}(f(x)),\ D_{x}f(E^{u}(x))=E^{u}(f(x))$;

  2. (2)

for all $n\geq 0,\ \|D_{x}f^{n}(v)\|\leq C\lambda^{n}\|v\|$ if $v\in
E^{s}(x)$, and $\|D_{x}f^{-n}(v)\|\leq C\lambda^{n}\|v\|$ if $v\in E^{u}(x)$.

Given a point $x\in\Lambda$, for each small $\beta>0$, the _local stable and
unstable manifolds_ are defined as follows:

$\displaystyle W_{\beta}^{s}(f,x)=\Big{\\{}y\in
M:d(f^{n}(x),f^{n}(y))\leq\beta,\ \forall n\geq 0\Big{\\}},$ $\displaystyle
W_{\beta}^{u}(f,x)=\Big{\\{}y\in M:d(f^{-n}(x),f^{-n}(y))\leq\beta,\ \forall
n\geq 0\Big{\\}}.$

The global stable and unstable sets of $x\in\Lambda$ are given as follows:

$W^{s}(f,x)=\bigcup_{n\geq
0}f^{-n}(W^{s}_{\beta}(f,f^{n}(x))),\,W^{u}(f,x)=\bigcup_{n\geq
0}f^{n}(W^{u}_{\beta}(f,f^{-n}(x))).$

A hyperbolic set is called _locally maximal_ , if there exists a neighbourhood
$U$ of $\Lambda$ such that $\Lambda=\bigcap_{n\in\mathbb{Z}}f^{n}(U)$. Recall
that a _horseshoe_ for a diffeomorphism $f$ is a transitive, locally maximal
hyperbolic set that is totally disconnected and not finite.

Let $W^{u}$ and $W^{s}$ be the unstable and stable foliations of hyperbolic
dynamical system $(f,\Lambda)$. For $x,y\in\Lambda$ with $x$ close to $y$, let
$W_{\beta}^{u}(f,x)$ and $W_{\beta}^{s}(f,x)$ be the local stable foliations
of $x$ and $y$. Define the map $h:W_{\beta}^{s}(f,x)\to W_{\beta}^{s}(f,y)$
sending $z$ to $h(z)$ by sliding along the leaves of $W^{u}$. The map $h$ is
called the holonomy map of $W^{u}$. The map $h$ is Lipschitz continuous if

$d_{y}(h(z_{1}),h(z_{2}))\leq Ld_{x}(z_{1},z_{2}),$

where $z_{1},z_{2}\in W_{\beta}^{s}(f,x)$ and $d_{x},d_{y}$ are natural path
metrics on $W_{\beta}^{s}(f,x)$, $W_{\beta}^{u}(f,y)$ with respect to a fixed
Riemannian structure on $M$. The constant $L$ is the Lipschitz constant, and
it is independent of the choice of $W^{s}$. The map $h$ is
$\alpha$-H$\ddot{o}$lder continuous if

$d_{y}(h(z_{1}),h(z_{2}))\leq Hd_{x}(z_{1},z_{2})^{\alpha},$

where $H$ is the Hölder constant. Similarly we can define the holonomy map of
$W^{s}$.

####  3.2.2. Approximation of Lyapunov exponents and entropy

For a $C^{1+\alpha}$ diffeomorphism $f:M\rightarrow M$, Katok [9] showed that
an $f$-invariant ergodic hyperbolic measure (a measure has no zero Lyapunov
exponents) with positive metric entropy can be approximated by horseshoes.
However, Katok’s result does not explicitly mention a control of the Oseledets
splitting over the horseshoes. Recently, Avila et al [2] showed that there is
a dominated splitting over the horseshoes, with approximately the same
Lyapunov exponents on each sub-bundle of the splitting.

Recall that $Df$-invariant splitting on a compact $f$-invariant subset
$\Lambda$

$T_{\Lambda}M=E_{1}\oplus E_{2}\oplus\cdots\oplus E_{\ell},~{}~{}(\ell\geq 2)$

is a _dominated splitting_ , if there exists $N\geq 1$ such that for every
$x\in\Lambda$, any unit vectors $v,w\in T_{x}M$:

$v\in E_{i}(x),w\in
E_{j}(x)~{}~{}\text{with}~{}~{}i<j\Longrightarrow\|D_{x}f^{N}(v)\|\geq
2\|D_{x}f^{N}(w)\|.$

We write $E_{1}\succeq E_{2}\succeq\cdots\succeq E_{\ell}$. Furthermore, if
there are numbers $\lambda_{1}>\lambda_{2}>\cdots>\lambda_{\ell}$, constants
$C>0$ and $\displaystyle{0<\varepsilon<\min_{1\leq
i<\ell}\frac{\lambda_{i}-\lambda_{i+1}}{100}}$ such that for every
$x\in\Lambda$, $n\in\mathbb{N}$, $1\leq j\leq\ell$ and each unit vector $u\in
E_{j}(x)$, it holds that

$C^{-1}e^{n(\lambda_{j}-\varepsilon)}\leq\|D_{x}f^{n}(u)\|\leq
Ce^{n(\lambda_{j}+\varepsilon)},$

then we say that

$T_{\Lambda}M=E_{1}\oplus E_{2}\oplus\cdots\oplus E_{\ell},~{}~{}(\ell\geq 2)$

is a $\\{\lambda_{j}\\}_{1\leq j\leq\ell}-$_dominated splitting_.

For the reader’s convenience, we recall Avila, Crovisier and Wilkinson’s
approximation results in the following:

######  Theorem 3.1.

Let $f:M\to M$ be a $C^{1+\alpha}$ diffeomorphism, and $\mu$ an $f$-invariant
ergodic hyperbolic measure with $h_{\mu}(f)>0$. For each $\varepsilon>0$ and a
weak-$*$ neighborhood $\mathcal{V}$ of $\mu$ in the space of $f$-invariant
probability measures on $M$. Then there exists a compact set
$\Lambda_{\varepsilon}^{*}\subset M$ and a positive integer $N$ such that the
following properties hold:

  1. (1)

$\Lambda_{\varepsilon}^{*}$ is a locally maximal hyperbolic set and
topologically mixing with respect to $f^{N}$;

  2. (2)

$h_{\mu}(f)-\varepsilon<h_{top}(f,\Lambda_{\varepsilon})<h_{\mu}(f)+\varepsilon$
where $\Lambda_{\varepsilon}=\Lambda_{\varepsilon}^{*}\cup
f(\Lambda_{\varepsilon}^{*})\cup\cdots f^{N-1}(\Lambda_{\varepsilon}^{*})$;

  3. (3)

$\Lambda_{\varepsilon}$ is $\varepsilon$-close to the support of $\mu$ in the
Hausdorff distance;

  4. (4)

each invariant probability measure supported on the horseshoe
$\Lambda_{\varepsilon}$ lies in $\mathcal{V}$;

  5. (5)

if $\lambda_{1}>\lambda_{2}>\cdots>\lambda_{\ell}$ are the distinct Lyapunov
exponents of $(f,\mu)$, with multiplicities $d_{1},d_{2},\cdots,d_{\ell}$,
then there exists a $\\{\lambda_{j}\\}_{1\leq j<\ell}-$dominated splitting
$T_{\Lambda_{\varepsilon}}M=E_{1}\oplus E_{2}\oplus\cdots\oplus E_{\ell}$ with
$\dim E_{i}=d_{i}$ for each $i$, and for each $x\in\Lambda_{\varepsilon}$,
$k\geq 1$ and each vector $v\in E_{i}(x)$

$e^{(\lambda_{i}-\varepsilon)kN}\leq\|D_{x}f^{kN}(v)\|\leq
e^{(\lambda_{i}+\varepsilon)kN},~{}~{}~{}\forall i=1,2,\cdots,\ell.$

######  Remark 3.1.

In the second statement, the original result does not give the inequality of
the right hand side. However, only a slightly modification can give the upper
bound of the topological entropy of $f$ on the horseshoe.

####  3.2.3. Statements of results

Let $f:M\to M$ be a $C^{1+\alpha}$ diffeomorphism of a compact Riemannian
manifold $M$, and let $\mu$ be a hyperbolic ergodic $f$-invariant probability
measure with positive entropy. Suppose that $(f,\mu)$ has only two Lyapunov
exponents $\lambda_{u}(\mu)>0>\lambda_{s}(\mu)$. Ledrappier, Young [23] and
Barreira, Pesin, Schmeling [7] proved that

$\mathrm{Dim}\mu=\frac{h_{\mu}(f)}{\lambda_{u}(\mu)}-\frac{h_{\mu}(f)}{\lambda_{s}(\mu)}$
(3.7)

where $\mathrm{Dim}$ denotes either $\dim_{H}$ or $\underline{\dim}_{B}$ or
$\overline{\dim}_{B}$. Our strategy used to prove the dimension approximation
in this setting is as follows. It follows from Theorem 3.1 that $\mu$ can be
approximated by a sequence of horseshoes
$\\{\Lambda_{\varepsilon}\\}_{\varepsilon>0}$. Using well-established
properties of dimension theory in uniform hyperbolic systems, one can show
that

$\mathrm{Dim}(\Lambda_{\varepsilon}\cap
W^{i}_{\beta}(f,x))\approx\frac{h_{\mu}(f)}{|\lambda_{i}(\mu)|}$

for $i=u,s$ and every $x\in\Lambda$. Burns and Wilkinson [8] proved the
holonomy maps of the stable and unstable foliations for
$(f,\Lambda_{\varepsilon})$ are Lipschitz continuous. Consequently, one can
show that

$\displaystyle\begin{aligned} &\dim_{H}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))+\dim_{H}(\Lambda_{\varepsilon}\cap W^{s}_{\beta}(f,x))\\\
\leq&\mathrm{Dim}\Lambda_{\varepsilon}\\\
\leq&\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))+\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{s}_{\beta}(f,x))\end{aligned}$

for every $x\in\Lambda_{\varepsilon}$. Hence, $\mathrm{Dim}\mu$ is
approximately equal to $\mathrm{Dim}\Lambda_{\varepsilon}$. The detailed
proofs will be given in the next section.

######  Theorem B.

Let $f:M\to M$ be a $C^{1+\alpha}$ diffeomorphism, and $\mu$ be an
$f$-invariant ergodic hyperbolic measure with $h_{\mu}(f)>0$. Assume that
$(f,\mu)$ has only two Lyapunov exponents
$\lambda_{u}(\mu)>0>\lambda_{s}(\mu)$. For each $\varepsilon>0$, there exists
a horseshoe $\Lambda_{\varepsilon}$ such that

$|\mathrm{Dim}\Lambda_{\varepsilon}-\mathrm{Dim}\mu|<\varepsilon$

where $\mathrm{Dim}$ denotes either $\dim_{H}$ or $\underline{\dim}_{B}$ or
$\overline{\dim}_{B}$.

In [42], the authors relaxed the smoothness of Theorem 3.1 to $C^{1}$ under
the additional condition that Oseledec’s splitting $E^{u}\oplus E^{s}$ of
$(f,\mu)$ is dominated. In this setting, one do not have Lipschitz continuity
of the holonomy map in general. However, using Palis and Viana’s method [31]
one can show that: for every $\gamma\in(0,1)$, there is some $D_{\gamma}>0$
such that the holonomy maps of the stable and unstable foliations for the
hyperbolic dynamical system $(f,\Lambda_{\varepsilon})$ (See Lemma 4.2) are
$(D_{\gamma},\gamma)$-Hölder continuous. Since $\gamma$ is arbitrary, using
the ideas in [41] one can prove the following theorem:

######  Theorem C.

Let $f:M\to M$ be a $C^{1}$ diffeomorphism, and let $\mu$ be an $f$-invariant
ergodic hyperbolic measure with $h_{\mu}(f)>0$. Assume that $(f,\mu)$ has only
two Lyapunov exponents $\lambda_{u}(\mu)>0>\lambda_{s}(\mu)$ and the
corresponding Oseledec’s splitting $E^{u}\oplus E^{s}$ is dominated. For each
$\varepsilon>0$, there exists a horseshoe $\Lambda_{\varepsilon}$ such that

$|\mathrm{Dim}\Lambda_{\varepsilon}-\mathrm{Dim}\mu|<\varepsilon$

where $\mathrm{Dim}$ denotes either $\dim_{H}$ or $\underline{\dim}_{B}$ or
$\overline{\dim}_{B}$.

##  4\. Proofs

In this section, we provide the proof of the main results presented in the
previous section.

###  4.1. Proof of Theorem A

Given an $f$-invariant ergodic measure $\mu$, let
$P(t):=P_{\mu}(f|_{\Lambda},\Phi_{f}(t))$, it is easy to see that the function
$t\mapsto P(t)$ is continuous and strictly decreasing on the interval
$[0,m_{0}]$. It follows from (2.3) that $P(0)=h_{\mu}(f)\geq 0$, and
$P(m_{0})\leq 0$ by Margulis-Ruelle’s inequality. Consequently, there exists a
unique root $s_{\mu}$ of the equation $P_{\mu}(f|_{\Lambda},\Phi_{f}(t))=0$.

If $h_{\mu}(f)=0$, it is easy to see that $h_{\mu}(f)=s_{\mu}=0$. Hence,
$\dim_{L}\mu=s_{\mu}$.

If $0<h_{\mu}(f)<\lambda_{m_{0}}(\mu)$, then $P(0)>0$ and $P(1)<0$. This
implies that $s_{\mu}\in(0,1)$ and
$0=P(s_{\mu})=h_{\mu}(f)-s_{\mu}\lambda_{m_{0}}(\mu)$. In consequence, we have
that

$s_{\mu}=\dim_{L}\mu=\frac{h_{\mu}(f)}{\lambda_{m_{0}}(\mu)}.$

If $h_{\mu}(f)\geq\lambda_{m_{0}}(\mu)$, note that

$\displaystyle 0$
$\displaystyle=h_{\mu}(f)+\mathcal{L}_{*}(\Phi_{f}(s_{\mu}),\mu)$
$\displaystyle=h_{\mu}(f)-\sum_{i=m_{0}-[s_{\mu}]+1}^{m_{0}}\lambda_{i}(\mu)-(s_{\mu}-[s_{\mu}])\lambda_{m_{0}-[s_{\mu}]}(\mu).$

Hence,

$s_{\mu}=[s_{\mu}]+\frac{h_{\mu}(f)-\sum_{i=m_{0}-[s_{\mu}]+1}^{m_{0}}\lambda_{i}(\mu)}{\lambda_{m_{0}-[s_{\mu}]}(\mu)}.$

On the other hand, since $t\mapsto P(t)$ is strictly decreasing in $t$, we
have that

$[s_{\mu}]=\max\\{i:\lambda_{m_{0}}(\mu)+\cdots+\lambda_{m_{0}-i+1}(\mu)\leq
h_{\mu}(f)\\}.$

This yields that

$s_{\mu}=\dim_{\mathrm{L}}\mu.$

To prove the second statement, by Theorem 5.1 in [13], for each $f$-invariant
ergodic measure $\mu$ with positive entropy, and for each $\varepsilon>0$
there exists an $f$-invariant compact subset
$\Lambda_{\varepsilon}\subset\Lambda$ such that the following statements hold:

  1. (i)

$h_{\text{top}}(f|_{\Lambda_{\varepsilon}})\geq h_{\mu}(f)-\varepsilon$;

  2. (ii)

there is a continuous invariant splitting $T_{x}M=E_{1}(x)\oplus
E_{2}(x)\oplus\cdots\oplus E_{\ell}(x)$ over $\Lambda_{\varepsilon}$ and a
constant $C>0$ so that

$C^{-1}\exp(n(\lambda_{j}(\mu)-\varepsilon))\leq\|D_{x}f^{n}(u)\|\leq
C\exp(n(\lambda_{j}(\mu)+\varepsilon))$

for any unit vector $u\in E_{j}(x)$, where
$\lambda_{1}(\mu)<\dots<\lambda_{\ell}(\mu)$ are distinct Lyapunov exponents
of $f$ with respect to the measure $\mu$.

By modifying the arguments in [13, Theorem 5.1], one may improve the estimate
in (i) as follows:

  1. (i)′

$h_{\mu}(f)+\varepsilon\geq h_{\text{top}}(f|_{\Lambda_{\varepsilon}})\geq
h_{\mu}(f)-\varepsilon$.

Since $\Lambda_{\varepsilon}$ is a repeller of $f$, one can choose an
$f$-invariant ergodic measure $\mu_{\varepsilon}$ on $\Lambda_{\varepsilon}$
so that $h_{\mu_{\varepsilon}}(f)=h_{\text{top}}(f|_{\Lambda_{\varepsilon}})$,
it yields that

$\displaystyle P_{\mathrm{top}}(f|_{\Lambda_{\varepsilon}},\Phi_{f}(t))$
$\displaystyle\geq
h_{\mu_{\varepsilon}}(f)+\mathcal{L}_{*}(\Phi_{f}(t),\mu_{\varepsilon})$
$\displaystyle\geq
h_{\mu}(f)+\mathcal{L}_{*}(\Phi_{f}(t),\mu)-(t+1)\varepsilon\qquad(~{}\text{by
(ii)}~{})$ $\displaystyle\geq
P_{\mu}(f|_{\Lambda},\Phi_{f}(t))-(m_{0}+1)\varepsilon.$

On the other hand, since $f$ is expanding, by the variational principle there
exists an $f$-invariant ergodic measure $\widetilde{\mu}_{\varepsilon}$ on
$\Lambda_{\varepsilon}$ so that

$\displaystyle P_{\mathrm{top}}(f|_{\Lambda_{\varepsilon}},\Phi_{f}(t))$
$\displaystyle=h_{\widetilde{\mu}_{\varepsilon}}(f)+\mathcal{L}_{*}(\Phi_{f}(t),\widetilde{\mu}_{\varepsilon})$
$\displaystyle\leq
h_{\mu}(f)+\mathcal{L}_{*}(\Phi_{f}(t),\mu)+(t+1)\varepsilon$
$\displaystyle\leq P_{\mu}(f|_{\Lambda},\Phi_{f}(t))+(m_{0}+1)\varepsilon.$

Hence,

$\Big{|}P_{\mathrm{top}}(f|_{\Lambda_{\varepsilon}},\Phi_{f}(t))-P_{\mu}(f|_{\Lambda},\Phi_{f}(t))\Big{|}\leq(m_{0}+1)\varepsilon.$

By Theorem 4.1 in [13], the Carathéodory singular dimension
$\dim_{C}\Lambda_{\varepsilon}$ of $\Lambda_{\varepsilon}$ is given by the
unique root of the following equation

$P_{\mathrm{top}}(f|_{\Lambda_{\varepsilon}},\Phi_{f}(t))=0.$

This together with the first statement yield that

$\displaystyle K|\dim_{C}\Lambda_{\varepsilon}-\dim_{\mathrm{L}}\mu|$
$\displaystyle\leq\Big{|}P_{\mu}(f|_{\Lambda},\Phi_{f}(\dim_{C}\Lambda_{\varepsilon}))-P_{\mu}(f|_{\Lambda},\Phi_{f}(\dim_{\mathrm{L}}\mu))\Big{|}$
$\displaystyle=\Big{|}P_{\mu}(f|_{\Lambda},\Phi_{f}(\dim_{C}\Lambda_{\varepsilon}))-P_{\mathrm{top}}(f|_{\Lambda_{\varepsilon}},\Phi_{f}(\dim_{C}\Lambda_{\varepsilon}))\Big{|}$
$\displaystyle\leq(m_{0}+1)\varepsilon$

where $K=\min_{x\in\Lambda}m(D_{x}f)$ and $m(\cdot)$ denotes the minimum norm
of an operator. Consequently, we have that
$\dim_{C}\Lambda_{\varepsilon}\rightarrow\dim_{\mathrm{L}}\mu$ as
$\varepsilon$ approaching zero.

###  4.2. Proof of Proposition 3.1

Given a subset $Z\subset\Lambda$, since $P_{Z}(f|_{\Lambda},\Phi_{f}(t))$ is
continuous and strictly decreasing in $t$, let $t_{Z}$ denote the unique root
of the equation $P_{Z}(f|_{\Lambda},\Phi_{f}(t))=0$. For every $t<t_{Z}$, we
have that $P_{Z}(f|_{\Lambda},\Phi_{f}(t))>0$. Fix such a number $t$, and take
$\beta>0$ so that $P_{Z}(f|_{\Lambda},\Phi_{f}(t))-\beta>0$. Since

$\displaystyle{P_{Z}(f|_{\Lambda},\Phi_{f}(t))=\liminf_{r\to
0}P_{Z}(f|_{\Lambda},\Phi_{f}(t),r)}$

there exists $r_{0}>0$ such that for each $0<r<r_{0}$ one has

$P_{Z}(f|_{\Lambda},\Phi_{f}(t),r)>P_{Z}(f|_{\Lambda},\Phi_{f}(t))-\beta.$

Fix such a small $r>0$. By the definition of topological pressure on arbitrary
subsets, one has

$m(Z,\Phi_{f}(t),P_{Z}(f|_{\Lambda},\Phi_{f}(t))-\beta,r)=+\infty.$

Hence, for each $\xi>0$, there exists $L\in\mathbb{N}$ so that for any $N>L$
we have that

$\displaystyle\exp(-N(P_{Z}(f|_{\Lambda},\Phi_{f}(t))-\beta))\inf\Big{\\{}\sum_{i}\exp(\sup_{y\in
B_{n_{i}}(x_{i},r)}-\varphi^{t}(y,f^{n_{i}}))\Big{\\}}$
$\displaystyle\geq\inf\Big{\\{}\sum_{i}\exp(-(P_{Z}(f|_{\Lambda},\Phi_{f}(t))-\beta)n_{i}+\sup_{y\in
B_{n_{i}}(x_{i},r)}-\varphi^{t}(y,f^{n_{i}}))\Big{\\}}>\xi$

where the infimum is taken over all collections $\\{B_{n_{i}}(x_{i},r)\\}$ of
Bowen’s balls with $n_{i}\geq N$, which covers $Z$. This yields that

$\inf\Big{\\{}\sum_{i}\exp(\sup_{y\in
B_{n_{i}}(x_{i},r)}-\varphi^{t}(y,f^{n_{i}}))\Big{\\}}>\xi\exp(N(P_{Z}(f|_{\Lambda},\Phi_{f}(t))-\beta)).$

Letting $N\to\infty$, we have that

$m(Z,t,r)=+\infty.$

Hence,

$\displaystyle\dim_{C,r}Z\geq t$

for all $0<r<r_{0}$. Consequently, since $t<t_{Z}$ is arbitrary, we have that

$\displaystyle\dim_{C}Z\geq t_{Z}.$ (4.1)

On the other hand, for each $t>t_{Z}$ one has that
$P_{Z}(f|_{\Lambda},\Phi_{f}(t))<0$. Fix such a number $t$, and take
$\widetilde{\beta}>0$ so that
$P_{Z}(f|_{\Lambda},\Phi_{f}(t))+\widetilde{\beta}<0$. By the definition of
topological pressure on arbitrary subsets, for any $R>0$, there exists $0<r<R$
such that

$P_{Z}(f|_{\Lambda},\Phi_{f}(t),r)<P_{Z}(f|_{\Lambda},\Phi_{f}(t))+\widetilde{\beta}.$

For such a small $r>0$ one has

$m(Z,\Phi_{f}(t),P_{Z}(f|_{\Lambda},\Phi_{f}(t))+\widetilde{\beta},r)=0$

Hence, for each small $\widetilde{\xi}>0$ there exists
$\widetilde{L}\in\mathbb{N}$ so that for any $N>\widetilde{L}$ we have that

$\displaystyle\exp(-N(P_{Z}(f|_{\Lambda},\Phi_{f}(t))+\widetilde{\beta}))\inf\Big{\\{}\sum_{i}\exp(\sup_{y\in
B_{n_{i}}(x_{i},r)}-\varphi^{t}(y,f^{n_{i}}))\Big{\\}}$
$\displaystyle\leq\inf\Big{\\{}\sum_{i}\exp(-(P_{Z}(f|_{\Lambda},\Phi_{f}(t))+\widetilde{\beta})n_{i}+\sup_{y\in
B_{n_{i}}(x_{i},r)}-\varphi^{t}(y,f^{n_{i}}))\Big{\\}}\leq\widetilde{\xi}$

where the infimum is taken over all collections $\\{B_{n_{i}}(x_{i},r)\\}$ of
Bowen’s balls with $n_{i}\geq N$, which covers $Z$. This yields that

$\inf\Big{\\{}\sum_{i}\exp(\sup_{y\in
B_{n_{i}}(x_{i},r)}-\varphi^{t}(y,f^{n_{i}}))\Big{\\}}\leq\widetilde{\xi}\exp(N(P_{Z}(f|_{\Lambda},\Phi_{f}(t))+\widetilde{\beta}))$

Letting $N\to\infty$, one has

$m(Z,t,r)=0.$

Consequently, for such $r>0$ one has

$\dim_{C,r}Z\leq t.$

Hence, we have that

$\displaystyle\dim_{C}Z=\liminf_{r\to 0}\dim_{C,r}Z\leq t_{Z}.$ (4.2)

It follows from (4.1) and (4.2) that

$\dim_{C}Z=t_{Z}.$

To show the second statement, for a given $f$-invariant ergodic measure $\mu$
supported on $\Lambda$, and a subset $Z\subset\Lambda$ with $\mu(Z)=1$, we
have that

$P_{Z}(f|_{\Lambda},\Phi_{f}(t))\geq P_{\mu}(f|_{\Lambda},\Phi_{f}(t)).$

By (1) of Theorem A and the first statement, one has

$\dim_{C}Z\geq\dim_{L}\mu.$

By the definition of Carathéodory singular dimension of arbitrary subsets, one
has

$\dim_{C,r}Z\geq\dim_{L}\mu$

for all sufficiently small $r>0$. Consequently, we have that

$\dim_{C}\mu=\liminf_{r\to 0}\dim_{C,r}\mu=\liminf_{r\to
0}\inf\\{\dim_{C,r}Z:\mu(Z)=1\\}\geq\dim_{L}\mu.$

To prove that $\dim_{C}\mu=\dim_{L}\mu$, we assume that
$\dim_{C}\mu>\widetilde{t}>\dim_{L}\mu$. By the first statement in Theorem A,
we have that

$P_{\mu}(f|_{\Lambda},\Phi(\widetilde{t}))<0.$

By the definition of measure theoretic pressure, for each $n\in\mathbb{N}$,
there exists $0<r_{n}<\frac{1}{n}$ so that

$\inf\\{P_{Z}(f|_{\Lambda},\Phi(\widetilde{t}),r_{n}):\mu(Z)=1\\}<0.$

Hence, there exists a subset $Z_{n}\subset\Lambda$ with $\mu(Z_{n})=1$ so that

$P_{Z_{n}}(f|_{\Lambda},\Phi(\widetilde{t}),r_{n})<0.$

Put $\widetilde{Z}:=\bigcap_{n\geq 1}Z_{n}$, then $\mu(\widetilde{Z})=1$ and

$\displaystyle P_{\widetilde{Z}}(f|_{\Lambda},\Phi(\widetilde{t}))$
$\displaystyle=\liminf_{r\to
0}P_{\widetilde{Z}}(f|_{\Lambda},\Phi(\widetilde{t}),r)$
$\displaystyle\leq\liminf_{n\to\infty}P_{Z_{n}}(f|_{\Lambda},\Phi(\widetilde{t}),r_{n})\leq
0$

It follows from the first statement and the definition of Carathéodory
singular dimension of $\mu$ that

$\dim_{C}\mu=\liminf_{r\to 0}\dim_{C,r}\mu\leq\liminf_{r\to
0}\dim_{C,r}\widetilde{Z}=\dim_{C}\widetilde{Z}\leq\widetilde{t},$

which yields a contraction. Hence, we have that $\dim_{C}\mu=\dim_{L}\mu$.

###  4.3. Proof of Theorem B

Ledrappier, Young [23] and Barreira, Pesin, Schmeling [7] proved that

$\mathrm{Dim}\mu=\frac{h_{\mu}(f)}{\lambda_{u}(\mu)}-\frac{h_{\mu}(f)}{\lambda_{s}(\mu)}$

where $\mathrm{Dim}$ denotes either $\dim_{H}$ or $\underline{\dim}_{B}$ or
$\overline{\dim}_{B}$. Fix a small number $\varepsilon>0$. By Theorem 3.1,
there exists a horseshoe $\Lambda_{\varepsilon}$ such that

  1. (i)

$|h_{\mathrm{top}}(f,\Lambda_{\varepsilon})-h_{\mu}(f)|<\varepsilon$;

  2. (ii)

there exists a dominated splitting $T_{\Lambda_{\varepsilon}}M=E^{u}\oplus
E^{s}$ with $\dim E^{i}=d_{i}\,(i=u,s)$, and for each
$x\in\Lambda_{\varepsilon}$, every $n\geq 1$ and each vector $v\in E^{i}(x)$
$(i=s,u)$

$e^{(\lambda_{i}(\mu)-\varepsilon)n}<\|D_{x}f^{n}(v)\|<e^{(\lambda_{i}(\mu)+\varepsilon)n}.$

Fixed any $k\in\mathbb{N}$, denote $F=f^{2^{k}}$. Since
$\Lambda_{\varepsilon}$ is a locally maximal hyperbolic set for $f$,
$\Lambda_{\varepsilon}$ is also a locally maximal hyperbolic set for $F$.
Notice that

$W^{u}_{\beta}(F,x)\cap\Lambda_{\varepsilon}=W^{u}_{\beta}(f,x)\cap\Lambda_{\varepsilon}\quad\text{and}\quad
W^{s}_{\beta}(F,x)\cap\Lambda_{\varepsilon}=W^{s}_{\beta}(f,x)\cap\Lambda_{\varepsilon}.$

Let $\|\cdot\|$ and $m(\cdot)$ denote the maximal and minimal norm of an
operator. For every $x\in\Lambda_{\varepsilon}$, Barreira [4] proved that

$\displaystyle\underline{t}_{u}^{k}\leq\dim_{H}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\leq\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\leq\overline{t}_{u}^{k}$

where $\underline{t}_{u}^{k}$, $\overline{t}_{u}^{k}$ are the unique solutions
of

$P_{\mathrm{top}}(F,-t\log\|D_{x}F|_{E^{u}(x)}\|)=0\quad\text{and}\quad
P_{\mathrm{top}}(F,-t\log m(D_{x}F|_{E^{u}(x)}))=0$

respectively. Using the same arguments as in the proof of Theorem $6.2$ and
Theorem $6.3$ in [3], one can prove that the sequences
$\\{\underline{t}_{u}^{k}\\}$ and $\\{\overline{t}_{u}^{k}\\}$ are monotone.
Furthermore, set

$\underline{t}_{u}:=\lim_{k\to\infty}\underline{t}_{u}^{k}\quad\text{and}\quad\overline{t}_{u}:=\lim_{k\to\infty}\overline{t}_{u}^{k},$

one can show that $\underline{t}_{u}$, $\overline{t}_{u}$ are the unique
solutions of the following equations

$P_{\mathrm{var}}(f,-t\\{\log\|D_{x}f^{n}|_{E^{u}}\|\\})=0,\quad
P_{\mathrm{top}}(f,-t\\{\log m(D_{x}f^{n}|_{E^{u}})\\})=0$

respectively.

Consequently, we have that

$\underline{t}_{u}\leq\dim_{H}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\leq\underline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\leq\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\leq\overline{t}_{u}$

and

$\displaystyle\begin{aligned}
\underline{t}_{u}&=\sup\Big{\\{}\frac{h_{\nu}(f)}{\displaystyle{\lim_{n\to\infty}}\frac{1}{n}\int\log\|D_{x}f^{n}|_{E^{u}}\|d\nu}:\nu\in\mathcal{M}_{f}(\Lambda_{\varepsilon})\Big{\\}},\\\
\overline{t}_{u}&=\sup\Big{\\{}\frac{h_{\nu}(f)}{\displaystyle{\lim_{n\to\infty}}\frac{1}{n}\int\log
m(D_{x}f^{n}|_{E^{u}})d\nu}:\nu\in\mathcal{M}_{f}(\Lambda_{\varepsilon})\Big{\\}}.\end{aligned}$

Combining with (i) and (ii), one has

$\displaystyle\frac{h_{\mu}(f)-\varepsilon}{\lambda_{u}(\mu)+\varepsilon}\leq\mathrm{Dim}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\leq\frac{h_{\mu}(f)+\varepsilon}{\lambda_{u}(\mu)-\varepsilon}$
(4.3)

for every $x\in\Lambda_{\varepsilon}$, where $\mathrm{Dim}$ denotes either
$\dim_{H}$ or $\underline{\dim}_{B}$ or $\overline{\dim}_{B}$. One can show in
a similar fashion that

$\displaystyle\begin{aligned}
-\frac{h_{\mu}(f)-\varepsilon}{\lambda_{s}(\mu)-\varepsilon}\leq\mathrm{Dim}(\Lambda_{\varepsilon}\cap
W^{s}_{\beta}(f,x))\leq-\frac{h_{\mu}(f)+\varepsilon}{\lambda_{s}(\mu)+\varepsilon}\end{aligned}$
(4.4)

for every $x\in\Lambda_{\varepsilon}$.

######  Lemma 4.1.

The holonomy maps of the stable and unstable foliations for
$(f,\Lambda_{\varepsilon})$ are Lipschitz continuous.

###### Proof.

Fix a positive integer $N$, put $F:=f^{N}$ and
$\Lambda=\Lambda_{\varepsilon}$. Since $\Lambda$ is a locally maximal
hyperbolic set for $f$, so is $\Lambda$ for $F$. Notice that

$W^{u}_{\beta}(F,x)\cap\Lambda=W^{u}_{\beta}(f,x)\cap\Lambda\quad\text{and}\quad
W^{s}_{\beta}(F,x)\cap\Lambda=W^{s}_{\beta}(f,x)\cap\Lambda.$ (4.5)

Let

$a_{F}=\|DF^{-1}|_{E^{u}}\|,\ b_{F}=\|DF|_{E^{s}}\|,c_{F}=\|DF|_{E^{u}}\|,\
d_{F}=\|DF^{-1}|_{E^{s}}\|.$

It follows from (ii) that

$1<\frac{\|D_{x}F|_{E^{i}(x)}\|}{m(D_{x}F|_{E^{i}(x)})}<e^{2\varepsilon
N},\text{ for every }x\in\Lambda~{}~{}\text{and}~{}~{}i\in\\{s,u\\}.$

Hence,

$a_{F}b_{F}c_{F}=\frac{\|DF|_{E^{s}}\|\cdot\|DF|_{E^{u}}\|}{m(DF|_{E^{u}})}<e^{(\lambda_{s}(\mu)+3\varepsilon)N}<1$

provided that $\varepsilon>0$ is sufficiently small such that
$\lambda_{s}(\mu)+3\varepsilon<0$. By Theorem $0.2$ in [8], we have that the
holonomy map of the stable foliation for $(F,\Lambda)$ is $C^{1}$. Similarly,
note that

$a_{F}b_{F}d_{F}=\frac{\|DF|_{E^{s}}\|}{m(DF||_{E^{s}})m(DF||_{E^{u}})}<e^{(-\lambda_{u}(\mu)+3\varepsilon)N}<1$

provided that $\varepsilon>0$ is sufficiently small such that
$\lambda_{u}(\mu)-3\varepsilon>0$. It follows from [8, Theorem 0.2] that the
holonomy map of the unstable foliation for $(F,\Lambda)$ is $C^{1}$. Combing
(4.5) one has the holonomy maps of the stable and unstable foliations for
$(f,\Lambda)$ are Lipschitz continuous. ∎

By Lemma 4.1 and the fact $f$ is topologically mixing on
$\Lambda_{\varepsilon}$, one has $\dim_{H}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))$, $\underline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))$ and $\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))$ are independent of $\beta$ and $x$ (see the proof of
Theorem $4.3.2$ in [5] for more details). Let

$A_{\varepsilon,x}=(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\times(\Lambda_{\varepsilon}\cap W^{s}_{\beta}(f,x)).$

By the properties of dimension (e.g. see [15, 33]), one has

$\displaystyle\begin{aligned} &\dim_{H}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))+\dim_{H}(\Lambda_{\varepsilon}\cap W^{s}_{\beta}(f,x))\\\
\leq\ &\dim_{H}A_{\varepsilon,x}\\\ \leq\
&\underline{\dim}_{B}A_{\varepsilon,x}\\\ \leq\
&\overline{\dim}_{B}A_{\varepsilon,x}\\\ \leq\
&\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))+\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{s}_{\beta}(f,x)).\end{aligned}$ (4.6)

Let $\Phi:A_{\varepsilon,x}\to\Lambda_{\varepsilon}$ be given by

$\Phi(y,z)=W^{s}_{\beta}(f,y)\cap W^{u}_{\beta}(f,z).$

It is easy to see $\Phi$ is a homeomorphism onto a neighborhood $V_{x}$ of $x$
in $\Lambda_{\varepsilon}$. It follows from Lemma 4.1 that $\Phi$ and
$\Phi^{-1}$ are Lipschitz continuous (see Theorem $4.3.2$ in [5] for detailed
proofs). It follows from Corollary 2.1 that

$\mathrm{Dim}V_{x}=\mathrm{Dim}A_{\varepsilon,x},$

where $\mathrm{Dim}$ denotes either $\dim_{H}$ or $\underline{\dim}_{B}$ or
$\overline{\dim}_{B}$. Since $\\{V_{x}:x\in\Lambda_{\varepsilon}\\}$ is an
open cover of $\Lambda_{\varepsilon}$, one can choose a finite open cover
$\\{V_{x_{1}},V_{x_{2}},\cdots,V_{x_{k}}\\}$ of $\Lambda_{\varepsilon}$. It
follows from (4.6) that

$\displaystyle\begin{aligned} &\dim_{H}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))+\dim_{H}(\Lambda_{\varepsilon}\cap W^{s}_{\beta}(f,x))\\\
\leq\ &\dim_{H}\Lambda_{\varepsilon}=\max_{1\leq i\leq k}\dim_{H}V_{x_{i}}\\\
\leq\ &\overline{\dim}_{B}\Lambda_{\varepsilon}=\max_{1\leq i\leq
k}\overline{\dim}_{B}V_{x_{i}}\\\ \leq\
&\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))+\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{s}_{\beta}(f,x)),\end{aligned}$

for every $x\in\Lambda_{\varepsilon}$. Combining (4.3) and (4.4) we obtain

$\lim_{\varepsilon\to
0}\mathrm{Dim}\Lambda_{\varepsilon}=\frac{h_{\mu}(f)}{\lambda_{u}(\mu)}-\frac{h_{\mu}(f)}{\lambda_{s}(\mu)}=\mathrm{Dim}\mu,$

where $\mathrm{Dim}$ denotes either $\dim_{H}$ or $\underline{\dim}_{B}$ or
$\overline{\dim}_{B}$. This completes the proof of Theorem B.

###  4.4. Proof of Theorem C

For every pair $(f,\mu)$ satisfying the assumptions, Wang and Cao [40,
Corollary 1] proved that

$\dim_{H}\mu=\frac{h_{\mu}(f)}{\lambda_{u}(\mu)}-\frac{h_{\mu}(f)}{\lambda_{s}(\mu)}.$

Fix a small number $\varepsilon>0$. Wang, Cao and Zou [42, Theorem 1.1] proved
that there exists a horseshoe $\Lambda_{\varepsilon}$ such that

  1. (i)

$|h_{\mathrm{top}}(f,\Lambda_{\varepsilon})-h_{\mu}(f)|<\varepsilon$;

  2. (ii)

there exists a dominated splitting $T_{\Lambda_{\varepsilon}}M=E^{u}\oplus
E^{s}$ with $\dim E^{i}=d_{i}\,(i=u,s)$, and for each
$x\in\Lambda_{\varepsilon}$, every $n\geq 1$ and each vector $v\in E^{i}(x)$
$(i=s,u)$,

$e^{(\lambda_{i}(\mu)-\varepsilon)n}<\|D_{x}f^{n}(v)\|<e^{(\lambda_{i}(\mu)+\varepsilon)n}.$

Fix a positive integer $k\in\mathbb{N}$, denote $F=f^{2^{k}}$. Since
$\Lambda_{\varepsilon}$ is a locally maximal hyperbolic set for $f$, so is
$\Lambda_{\varepsilon}$ for $F$. Notice that

$W^{u}_{\beta}(F,x)\cap\Lambda_{\varepsilon}=W^{u}_{\beta}(f,x)\cap\Lambda_{\varepsilon}\quad\text{and}\quad
W^{s}_{\beta}(F,x)\cap\Lambda_{\varepsilon}=W^{s}_{\beta}(f,x)\cap\Lambda_{\varepsilon}.$

For every $x\in\Lambda_{\varepsilon}$, it follows from [41, Lemmas 3.5 and
3.6] that

$\displaystyle\underline{t}_{u}^{k}\leq\dim_{H}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\leq\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\leq\overline{t}_{u}^{k},$

where $\underline{t}_{u}^{k}$, $\overline{t}_{u}^{k}$ are the unique roots of

$P_{\mathrm{top}}(F,-t\log\|D_{x}F|_{E^{u}(x)}\|)=0,\quad
P_{\mathrm{top}}(F,-t\log m(D_{x}F|_{E^{u}(x)}))=0$

respectively. Using the same arguments as in the proof of Theorem $6.2$ and
Theorem $6.3$ in [3], one can prove that the sequences
$\\{\underline{t}_{u}^{k}\\}$ and $\\{\overline{t}_{u}^{k}\\}$ are monotone.
Set

$\underline{t}_{u}:=\lim_{k\to\infty}\underline{t}_{u}^{k}\quad\text{and}\quad\overline{t}_{u}:=\lim_{k\to\infty}\overline{t}_{u}^{k},$

where $\underline{t}_{u}$, $\overline{t}_{u}$ are the unique solutions of the
following equations

$P_{\mathrm{var}}(f,-t\\{\log\|D_{x}f^{n}|_{E^{u}(x)}\|\\})=0,\quad
P_{\mathrm{top}}(f,-t\\{\log m(D_{x}f^{n}|_{E^{u}(x)})\\})=0$

respectively. Hence, we have that

$\underline{t}_{u}\leq\dim_{H}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\leq\underline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\leq\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\leq\overline{t}_{u}.$

Since

$\displaystyle\underline{t}_{u}$
$\displaystyle=\sup\Big{\\{}\frac{h_{\nu}(f)}{\displaystyle{\lim_{n\to\infty}}\frac{1}{n}\int\log\|D_{x}f^{n}|_{E^{u}(x)}\|d\nu}:\nu\in\mathcal{M}_{f}(\Lambda_{\varepsilon})\Big{\\}}$

and

$\displaystyle\overline{t}_{u}$
$\displaystyle=\sup\Big{\\{}\frac{h_{\nu}(f)}{\displaystyle{\lim_{n\to\infty}}\frac{1}{n}\int\log
m(D_{x}f^{n}|_{E^{u}(x)})d\nu}:\nu\in\mathcal{M}_{f}(\Lambda_{\varepsilon})\Big{\\}}$

using (i) and (ii) one can show that

$\displaystyle\frac{h_{\mu}(f)-\varepsilon}{\lambda_{u}(\mu)+\varepsilon}\leq\mathrm{Dim}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\leq\frac{h_{\mu}(f)+\varepsilon}{\lambda_{u}(\mu)-\varepsilon}$
(4.7)

for every $x\in\Lambda_{\varepsilon}$, where $\mathrm{Dim}$ denotes either
$\dim_{H}$ or $\underline{\dim}_{B}$ or $\overline{\dim}_{B}$. Similarly, we
obtain that

$\displaystyle-\frac{h_{\mu}(f)-\varepsilon}{\lambda_{s}(\mu)-\varepsilon}\leq\mathrm{Dim}(\Lambda_{\varepsilon}\cap
W^{s}_{\beta}(f,x))\leq-\frac{h_{\mu}(f)+\varepsilon}{\lambda_{s}(\mu)+\varepsilon}$
(4.8)

for every $x\in\Lambda_{\varepsilon}$.

######  Lemma 4.2.

Let $\Lambda$ be a locally maximal hyperbolic set of a $C^{1}$ diffeomorphism
such that $f$ is topologically mixing on $\Lambda$. Assume that the
diffeomorphism $f|_{\Lambda}$ possesses a
$\\{\lambda_{u}(\mu),\lambda_{s}(\mu)\\}$-dominated splitting
$T_{\Lambda}M=E^{u}\oplus E^{s}$ with $E^{u}\succeq E^{s}$ and
$\lambda_{u}(\mu)>0>\lambda_{s}(\mu)$. Then for every $\gamma\in(0,1)$ there
exists $D_{\gamma}>0$ such that the holonomy maps of the stable and unstable
foliations for $f$ are $(D_{\gamma},\gamma)$-H$\ddot{o}$lder continuous.

###### Proof.

For every $x\in\Lambda$, $n\in\mathbb{N}$ and each unit vector $v\in E^{i}(x)$
($i=u,s$),

$e^{n(\lambda_{i}(\mu)-\varepsilon)}\leq\|D_{x}f^{n}(v)\|\leq
e^{n(\lambda_{i}(\mu)+\varepsilon)}.$

Fix a positive integer $N$, put $F:=f^{N}$. This implies that for every
$x\in\Lambda$,

$1\leq\frac{\|D_{x}F|_{E^{u}}\|}{m(D_{x}F|_{E^{u}})}\leq
e^{2N\varepsilon},\quad
1\leq\frac{\|D_{x}F|_{E^{s}}\|}{m(D_{x}F|_{E^{s}})}\leq e^{2N\varepsilon}.$

Notice that $\Lambda$ is also a locally maximal hyperbolic set for $F$ and

$W^{u}_{\beta}(F,x)\cap\Lambda=W^{u}_{\beta}(f,x)\cap\Lambda\quad\text{and}\quad
W^{s}_{\beta}(F,x)\cap\Lambda=W^{s}_{\beta}(f,x)\cap\Lambda$

for every $x\in\Lambda$.

Let $\pi^{s}$ and $\pi^{u}$ be the holonomy maps of stable and unstable
foliations for $f$, i.e. for any $x\in\Lambda$, $x^{\prime}\in
W^{s}_{\beta}(f,x)$ and $x^{\prime\prime}\in W^{u}_{\beta}(f,x)$ close to $x$,

$\pi^{s}:\ W^{u}_{\beta}(f,x)\cap\Lambda\to
W^{u}_{\beta}(f,x^{\prime})\cap\Lambda\ \text{with}\
\pi^{s}(y)=W^{s}_{\beta}(f,y)\cap W^{u}_{\beta}(f,x^{\prime})$

and

$\pi^{u}:\ W^{s}_{\beta}(f,x)\cap\Lambda\to
W^{s}_{\beta}(f,x^{\prime\prime})\cap\Lambda\ \text{with}\
\pi^{u}(z)=W^{u}_{\beta}(f,z)\cap W^{s}_{\beta}(f,x^{\prime\prime}).$

Therefore, $\pi^{s}$ is also a map from $W^{u}_{\beta}(F,x)\cap\Lambda$ to
$W^{u}_{\beta}(F,x^{\prime})\cap\Lambda$ and $\pi^{u}$ is also a map from
$W^{s}_{\beta}(F,x)\cap\Lambda$ to
$W^{s}_{\beta}(F,x^{\prime\prime})\cap\Lambda$.

Let $U\subset M$ be an open subset such that
$\Lambda=\bigcap_{n\in\mathbb{Z}}f^{n}(U)$, and
$\mathcal{U}\subset\mbox{Diff}^{1}(M)$ be a neighbourhood of $f$ such that,
for each $g\in\mathcal{U}$, $\Lambda_{g}=\bigcap_{n\in\mathbb{Z}}g^{n}(U)$ is
a locally maximal hyperbolic set for $g$ and there is a homeomorphism
$h_{g}:\Lambda\rightarrow\Lambda_{g}$ satisfies that $g\circ h_{g}=h_{g}\circ
f$, with $h_{g}$ $C^{0}-$close to identity if $g$ is $C^{1}-$close to $f$. For
$g\in\mathcal{U}$, let $T_{\Lambda_{g}}M=E^{u}_{g}\oplus E^{s}_{g}$ denote the
hyperbolic splitting over $\Lambda_{g}$. For $i\in\\{u,s\\}$,
$\\{W^{i}_{\beta}(g,z):\ z\in\Lambda_{g}\\}$ is continuous on $g$ in the
following sense: there is $\\{\theta_{g,x}^{i}:\ x\in\Lambda\\}$ where
$\theta_{g,x}^{i}:\ W^{i}_{\beta}(f,x)\to W^{i}_{\beta}(g,h_{g}(x))$ is a
$C^{1}$ diffeomorphism with $\theta_{g,x}^{i}(x)=h_{g}(x)$, such that if $g$
is $C^{1}-$close to $f$ then, for all $x\in\Lambda$, $\theta_{g,x}^{i}$ is
uniformly $C^{1}-$close to the inclusion of $W^{i}_{\beta}(f,x)$ in $M$.

For any $\gamma\in(0,1)$, let $\mathcal{U}_{\gamma}^{F}$ be a small $C^{1}$
neighborhood of $F$ (recall $F=f^{N}$). Taking
$G\in\mathcal{U}_{\gamma}^{F}\cap\mathrm{Diff}^{2}(M)$ such that for every
$x\in\Lambda_{G}$ (here $\Lambda_{G}$ is a locally maximal hyperbolic set for
$G$), $n\in\mathbb{N}$ and $i=u,s$,

$e^{nN(\lambda_{i}(\mu)-2\varepsilon)}\leq\|D_{x}G^{n}|_{E^{i}(x)}\|\leq
e^{nN(\lambda_{i}(\mu)+2\varepsilon)}.$ (4.9)

######  Claim 4.1.

The following properties hold:

  * (a)

$h_{G}|_{W^{u}_{\beta}(F,x)\cap\Lambda}$ and
$\big{(}h_{G}|_{W^{u}_{\beta}(F,x)\cap\Lambda}\big{)}^{-1}$ are
$(C_{\gamma},\gamma)$-Hölder continuous for some $C_{\gamma}>0$.

  * (b)

the stable and unstable foliations

$\\{W^{s}(G,z):z\in\Lambda_{G}\\},\quad\\{W^{u}(G,z):z\in\Lambda_{G}\\}$

are $C^{1}$ and invariant for $G$. Thus the holonomy maps

$\displaystyle\begin{aligned} \pi^{s}_{G}:\
&W^{u}_{\beta}(G,h_{G}(x))\cap\Lambda_{G}\to
W^{u}_{\beta}(G,h_{G}(x^{\prime}))\cap\Lambda_{G}\ \text{with}\\\
&\pi^{s}_{G}(y)=W^{s}_{\beta}(G,y)\cap
W^{u}_{\beta}(G,h_{G}(x^{\prime})),\end{aligned}$

and

$\displaystyle\begin{aligned} \pi^{u}_{G}:\
&W^{s}_{\beta}(G,h_{G}(x))\cap\Lambda_{G}\to
W^{s}_{\beta}(G,h_{G}(x^{\prime\prime}))\cap\Lambda_{G}\ \text{with}\\\
&\pi^{u}_{G}(z)=W^{u}_{\beta}(G,z)\cap
W^{s}_{\beta}(G,h_{G}(x^{\prime\prime}))\end{aligned}$

are Lipschitz continuous.

###### Proof.

(a) See Claim $3.1$ in [41].

(b) Since $G$ satisfies (4.9), we conclude

$\displaystyle\begin{aligned}
\frac{\|DG|_{E^{u}}\|\cdot\|DG|_{E^{s}}\|}{m(DG|_{E^{u}})}&\leq
e^{4N\varepsilon}e^{N(\lambda_{s}(\mu)+2\varepsilon)}\\\
&=e^{N(\lambda_{s}(\mu)+6\varepsilon)}\\\ &\leq 1,\end{aligned}$

provide that $\lambda_{s}(\mu)+6\varepsilon<0$. By Theorem $6.3$ in [20] we
have the stable foliation is $C^{1}$. Similarly we obtain the unstable
foliation is also $C^{1}$. Then the corresponding maps are uniformly $C^{1}$
(see [35, pages 540–541] for more details), this implies the desired result.

∎

We proceed to prove Lemma 4.2. For any $y\in W^{u}_{\beta}(F,x)\cap\Lambda$,

$\displaystyle\begin{aligned}
h_{G}(\pi^{s}(y))&=h_{G}\big{(}W^{s}_{\beta}(F,y)\cap
W^{u}_{\beta}(F,x^{\prime})\big{)}\\\ &=W^{s}_{\beta}(G,h_{G}(y))\cap
W^{u}_{\beta}(G,h_{G}(x^{\prime}))\\\ &=\pi^{s}_{G}(h_{G}(y)).\end{aligned}$

For the above $\gamma$, by Claim 4.1 there exists $D_{\gamma}>0$ such that

$\pi^{s}=h_{G}^{-1}\circ\pi^{s}_{G}\circ h_{G}$

is $(D_{\gamma},\gamma)$-Hölder continuous. Using the same arguments one can
prove $(\pi^{s})^{-1}$, $\pi^{u}$ and $(\pi^{u})^{-1}$ are also
$(D_{\gamma},\gamma)$-Hölder continuous.

∎

By Lemma 4.2 and the fact $f$ is topologically mixing on
$\Lambda_{\varepsilon}$, one has $\dim_{H}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))$, $\underline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))$ and $\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))$ are independent of $\beta$ and $x$ (see the proof of
Lemma $3.4$ in [41] for more details). Let

$A_{\varepsilon,x}=(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))\times(\Lambda_{\varepsilon}\cap W^{s}_{\beta}(f,x))$

be a product space. By the properties of dimension (see Theorem $6.5$ in [33]
for details), one has

$\displaystyle\begin{aligned} &\dim_{H}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))+\dim_{H}(\Lambda_{\varepsilon}\cap W^{s}_{\beta}(f,x))\\\
\leq\ &\dim_{H}A_{\varepsilon,x}\\\ \leq\
&\underline{\dim}_{B}A_{\varepsilon,x}\\\ \leq\
&\overline{\dim}_{B}A_{\varepsilon,x}\\\ \leq\
&\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))+\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{s}_{\beta}(f,x)).\end{aligned}$ (4.10)

Let $\Phi:A_{\varepsilon,x}\to\Lambda_{\varepsilon}$ be given by

$\Phi(y,z)=W^{s}_{\beta}(f,y)\cap W^{u}_{\beta}(f,z).$

It is easy to see that $\Phi$ is a homeomorphism onto a neighborhood $V_{x}$
of $x$ in $\Lambda_{\varepsilon}$. For any $\gamma\in(0,1)$, by Lemma 4.2
there is $E_{\gamma}>0$ such that $\Phi$ and $\Phi^{-1}$ are
$(E_{\gamma},\gamma)$-Hölder continuous (see Step $2$ in the proof of Theorem
A in [41] for more details). By Lemma 2.1 and the arbitrariness of $\gamma$,
one has

$\mathrm{Dim}V_{x}=\mathrm{Dim}A_{\varepsilon,x},$

where $\mathrm{Dim}$ denotes either $\dim_{H}$ or $\underline{\dim}_{B}$ or
$\overline{\dim}_{B}$. Since $\\{V_{x}:x\in\Lambda_{\varepsilon}\\}$ is an
open cover of $\Lambda_{\varepsilon}$, one can choose a finite open cover
$\\{V_{x_{1}},V_{x_{2}},\cdots,V_{x_{k}}\\}$ of $\Lambda_{\varepsilon}$. It
follows from (4.10) that

$\displaystyle\begin{aligned} &\dim_{H}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))+\dim_{H}(\Lambda_{\varepsilon}\cap W^{s}_{\beta}(f,x))\\\
\leq\ &\dim_{H}\Lambda_{\varepsilon}=\max_{1\leq i\leq k}\dim_{H}V_{x_{i}}\\\
\leq\ &\overline{\dim}_{B}\Lambda_{\varepsilon}=\max_{1\leq i\leq
k}\overline{\dim}_{B}V_{x_{i}}\\\ \leq\
&\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{u}_{\beta}(f,x))+\overline{\dim}_{B}(\Lambda_{\varepsilon}\cap
W^{s}_{\beta}(f,x)),\end{aligned}$

for every $x\in\Lambda_{\varepsilon}$. This together with (4.7) and (4.8)
yield that

$\lim_{\varepsilon\to
0}\mathrm{Dim}\Lambda_{\varepsilon}=\frac{h_{\mu}(f)}{\lambda_{u}(\mu)}-\frac{h_{\mu}(f)}{\lambda_{s}(\mu)}=\mathrm{Dim}\mu,$

where $\mathrm{Dim}$ denotes either $\dim_{H}$ or $\underline{\dim}_{B}$ or
$\overline{\dim}_{B}$. This completes the proof of Theorem C.

### Acknowledgments

This work is partially supported by The National Key Research and Development
Program of China (2022YFA1005802). Y. Cao is partially supported by NSFC
(11790274). J. Wang is partially supported by NSFC (11501400, 12271386) and
the Talent Program of Shanghai University of Engineering Science. Y. Zhao is
partially supported by NSFC (12271386) and Qinglan project of Jiangsu
Province.

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# False Alarms Revealed in a Planet Search of TESS Light Curves

Michelle Kunimoto Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Steve Bryson
NASA Ames Research Center, Moffett Field, CA 94035, USA  Tansu Daylan LSSTC
Catalyst Fellow  Department of Astrophysical Sciences, Princeton University, 4
Ivy Lane, Princeton, NJ 08544  Jack J. Lissauer NASA Ames Research Center,
Moffett Field, CA 94035, USA  Michael R. B. Matesic Department of Physics and
Astronomy, Bishop’s University, 2600 Rue College, Sherbrooke, QC J1M 1Z7,
Canada  Susan E. Mullally Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD, 21218, USA  Jason F. Rowe Department of Physics and
Astronomy, Bishop’s University, 2600 Rue College, Sherbrooke, QC J1M 1Z7,
Canada

###### Abstract

We examined the period distribution of transit-like signatures uncovered in a
Box-Least Squares transit search of TESS light curves, and show significant
pileups at periods related to instrumental and astrophysical noise sources.
Signatures uncovered in a search of inverted light curves feature similar
structures in the period distribution. Automated vetting methods will need to
remove these excess detections, and light curve inversion appears to be a
suitable method for simulating false alarms and designing new vetting metrics.

##  1 Introduction

Exoplanet demographics provide significant insight into exoplanet formation
and evolution theory. Exoplanet catalogs, however, do not represent the true
exoplanet population. Catalogs are not complete because they miss exoplanets,
and may not be reliable because of contamination from false positives (e.g.,
nearby or bound eclipsing binaries) or false alarms (e.g., noise, systematics,
and stellar variability). The Kepler mission (Borucki et al., 2010) was
designed to produce demographic studies, yielding a catalog with well-measured
completeness and reliability (Thompson et al., 2018), making it possible to
infer the underlying planet population. The TESS mission (Ricker et al., 2015)
is finding thousands of exoplanets, but the TESS Object of Interest catalog is
not useful for demographics because it has unknown completeness and
reliability.

Exoplanet catalogs are typically produced by a software pipeline that detects
transit events which are vetted to determine if the events are caused by
planets. The most straightforward way to produce a catalog with well-measured
completeness and reliability is to use a fully automated detection and vetting
pipeline. Designing an automated vetting pipeline is a non-trivial task
requiring an understanding of false alarms found in observed data. Automated
vetting (Thompson et al., 2018) becomes particularly challenging in the low
signal-to-noise ratio (S/N) regime of smaller and/or longer period planets
because most false alarms due to noise and systematics have low S/N.

In this paper, we investigate the impact of false alarms on planet searches of
TESS data by performing a search of a large number of stars. We use light
curves obtained from multiple sources to investigate whether or not
systematics depend on how the light curves were generated. We identify periods
with excess detections, or pileups, and speculate about their causes in TESS
systematics. We also perform a preliminary exploration using light curve
inversion to simulate false alarms, needed for characterizing catalog
reliability for future TESS demographics studies.

##  2 Planet Search

We perform our investigation on M dwarfs ($R<0.6~{}R_{\odot}$,
$M<0.6~{}M_{\odot}$, $T_{\text{eff}}<3900~{}K$), which are of high interest to
exoplanet searches, especially for small planets. We selected all 92,899 M
dwarfs brighter than $T=13.5$ mag from the TESS Input Catalog (TICv8.2;
Paegert et al., 2021) that were observed by TESS across sectors 1 to 52.
Multi-sector light curves from the Quick-Look Pipeline (QLP; Huang et al.,
2020; Kunimoto et al., 2021, 2022) were obtained for each star. To compare to
QLP, light curves were also extracted using the eleanor pipeline (Feinstein et
al., 2019). Both eleanor CORR_FLUX and PCA_FLUX systematics-corrected versions
of each light curve were saved.

Long-term trends from all three sets of light curves (QLP, CORR_FLUX, and
PCA_FLUX) were removed using a biweight detrending algorithm with a 0.5-day
window length as implemented in wotan (Hippke et al., 2019). We then performed
a multi-planet search of the detrended light curves with the Box-Least Squares
(BLS) algorithm implemented in
cuvarbase111https://johnh2o2.github.io/cuvarbase/. Each star was searched from
0.5 days to 100 days or the length of the longest consecutive stretch of TESS
sectors comprising the light curve, whichever was shorter. This strategy was
chosen to minimize the challenges of searching data with long gaps between
sectors. If a detection had S/N$~{}>9$, data within one transit duration of
the measured center of the transit time is masked out and the residuals were
searched, up to five times per light curve. We define Threshold Crossing
Events (TCEs) as signals with S/N$~{}>9$ and at least two events.

The same BLS procedure was re-run on the same light curves after inversion to
produce inverted TCEs (invTCEs). Inversion should eliminate valid transits but
preserve the general noise and systematic properties seen in the original time
series, and was a strategy used by Kepler to simulate false alarms (Thompson
et al., 2018).

Figure 1: Period histograms of TCEs and invTCEs identified in various input
TESS light curves by our detection pipeline, with insets showing inverted
eleanor CORR_FLUX light curves with planet-like false alarms. Transits in the
insets are indicated by short blue vertical lines, and the red vertical dashed
lines delineate sectors. The filled histograms are TCEs and the line
histograms are invTCEs. The vertical gray lines are aliases of the 13.7-day
TESS orbital period. Most signals are false alarms. We see that, in all cases,
pileups in TCEs correspond to pileups in invTCEs, indicating that many
detections in the observed data are false alarms. Top panel: CORR_FLUX light
curves, comparing two- and three-transit thresholds for transit detection.
Middle panel: Comparing three-transit detections from CORR_FLUX and PCA_FLUX
light curves. Lower panel: Comparing three-transit detections from CORR_FLUX
and QLP light curves. The lower panel also shows “no sines” TCEs and invTCEs
after the removal of sinusoidal signals identified by the SWEET test (see
text).

##  3 Results

Figure 1 shows histograms of TCE and invTCE periods from all three light curve
sources. In all cases, we see the same general structure of pileups, which are
similar to those found by the SPOC pipeline (Jenkins et al., 2016) using
postage stamp targets222https://archive.stsci.edu/tess/tess_drn.html. While
the inverted light curves include detections of astrophysical phenomena such
as stellar flares and some eclipsing binaries, detection pileups occur at
periods not correlated with flare statistics. We conclude that light curve
inversion is a suitable method for simulating TESS false alarms in future
demographic studies.

We see two classes of pileups:

  * •

Sharp peaks with broad shoulders at aliases of the 13.7-day TESS orbital
period. Inspection of observed and inverted light curves reveals that most of
these detections are easily identified excursions at the edges of data gaps,
due to scattered light, that were not removed by detrending. Not all residual
edge effects are easily identified, however, as shown by the inset light curve
examples in Figure 1.

  * •

A general excess with a period $<$ 1 day. Inspection of the light curves
reveals that most of these are highly periodic, which we believe are due
primarily to scattered light from the Earth’s rotation, and also stellar
variability (e.g., spots and pulsations) and contact eclipsing binaries. We
find that the Kepler SWEET test (Thompson et al., 2018), which measures the
correlation of light curves with sinusoids, enables the removal of the highly
sinusoidal detections, shown by the “no sines” histograms in Figure 1.

Comparing the pileup structures for the various light curve sources, we see
little difference between eleanor CORR_FLUX and PCA_FLUX light curves. For
periods $\gtrsim 3$ days, we see little difference between eleanor and QLP,
but QLP light curves have a larger excess with periods below one day. We find
that requiring three vs. two transits for a transit detection significantly
reduces the number of detections with period $\gtrsim 3$ days, but does not
remove the pileup structure. While most excess detections below one day are
sinusoidal, some appear more transit like, as shown in the inset example in
Figure 1.

##  4 Conclusions

Creating an exoplanet catalog with well-measured reliability requires the
ability to detect false positives and false alarms. Kepler effectively used
automated vetting based on metrics that were tuned to remove detections from
inverted Kepler data, and inverted data was used to measure Kepler
reliability. The observations in this paper lead us to expect that a similar
strategy, appropriately tuned for TESS using inverted TESS data, will be
successful. Automated vetting based, in part, on inverted light curves will be
a significant step toward making TESS data useful for demographic studies.

##  5 Acknowledgements

We thank Michael Fausnaugh for useful insight into TESS systematics. This
paper utilizes data from the Quick-Look Pipeline at the TESS Science Office at
MIT. This work makes use of FFIs calibrated by TESS Image CAlibrator
(Fausnaugh et al., 2020), which are also available as High-Level Science
Products stored on the Mikulski Archive for Space Telescopes. Funding for the
TESS mission is provided by NASA’s Science Mission Directorate.

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