# Continuity properties of Lyapunov exponents
for surface diffeomorphisms

Jérôme Buzzi  J. Buzzi, Laboratoire de Mathématiques d’Orsay, CNRS - UMR 8628.
Université Paris-Saclay, 91405 Orsay, France. _E-mail address:_
<EMAIL_ADDRESS>,  Sylvain Crovisier  S. Crovisier, Laboratoire de
Mathématiques d’Orsay, CNRS - UMR 8628. Université Paris-Saclay, 91405 Orsay,
France. _E-mail address:_<EMAIL_ADDRESS>and  Omri Sarig  O.
Sarig, Weizmann Institute of Science, 234 Herzl Street, 7610001 Rehovot,
Israel. _E-mail address:_<EMAIL_ADDRESS>
###### Abstract.

We study the entropy and Lyapunov exponents of invariant measures $\mu$ for
smooth surface diffeomorphisms $f$, as functions of $(f,\mu)$. The main result
is an inequality relating the discontinuities of these functions. One
consequence is that for a $C^{\infty}$ surface diffeomorphisms, on any set of
ergodic measures with entropy bounded away from zero, continuity of the
entropy implies continuity of the exponents. Another consequence is the upper
semi-continuity of the Hausdorff dimension on the set of ergodic invariant
measures with entropy bounded away from zero. We also obtain a new criterion
for the existence of SRB measures with positive entropy.

###### Keywords:

Dynamical systems; smooth ergodic theory; entropy; Lyapunov exponents

###### 2010 Mathematics Subject Classification:

37C40, 37D30, 37A35, 37D35

J.B. was partially supported by the ISDEEC project ANR-16-CE40-0013

S.C. was partially supported by the ERC project 692925 _NUHGD_

Part of this work was done when O.S. was visiting Université Paris-Sud and
IHÉS, and he would like to thank these institutions for their hospitality and
excellent working conditions. O.S. also acknowledges partial support of ISF
grant 1149/18.

## Introduction

Entropy and Lyapunov exponents play a major role in the study of
differentiable dynamical systems, and their dependence on the measure and the
map is of great interest. This dependence is sometimes continuous, but not
always (for entropy, see [33, 35, 12, 9, 17, 10], and for Lyapunov exponents,
see [44, 20, 6, 7, 1, 47]). While there are many works relating the values of
the entropy to the values of the Lyapunov exponents [43, 39, 31, 27], the
relation between the (dis)continuity of these objects as functions of the
measure and the diffeomorphism has not yet been studied. The purpose of this
work is to fill this gap, in the smooth two-dimensional case. For instance, we
show:

######  Theorem.

Let $f$ be a $C^{\infty}$ diffeomorphism of a compact surface without
boundary. Let $\nu_{1},\nu_{2},\dots$ be ergodic measures for $f$, which
converge in the weak-$*$ topology to an ergodic measure $\mu$ with positive
entropy.

If the entropy of $\nu_{k}$ converges to the entropy of $\mu$, then the
Lyapunov exponents of $\nu_{k}$ converge to the Lyapunov exponents of $\mu$.

This has strong dynamical consequences, some of which we will discuss here,
and some of which we will discuss in a companion paper. For example, we have
the following application to the problem of the existence of SRB measures. Let
$\delta^{u}(\mu)$ denote the unstable dimension of $\mu$ in the sense of
Ledrappier and Young (see Section 1.5).

######  Corollary.

Let $f$ be a $C^{\infty}$ diffeomorphism of a compact surface without
boundary. If there exist ergodic invariant probability measures $\nu_{k}$,
with entropy uniformly bounded away from $0$, and such that
$\delta^{u}(\nu_{k})\to 1$, then $f$ admits an ergodic SRB measure with
positive entropy.

For other consequences, including the upper semi-continuity of the unstable
dimension and of the Hausdorff dimension of ergodic measures with positive
entropy, see Section 1.5.

These results follow from inequalities between the multiplicative size of the
defects in continuity of the entropy and the top Lyapunov exponent. These
inequalities, which are the main results of this work, are described in detail
in the next section.

##  1\. Main results

Throughout this paper, $M$ is a two-dimensional compact $C^{\infty}$
Riemannian manifold without boundary. Let $\operatorname{Diff}^{r}(M)$ denotes
the class of $C^{r}$ diffeomorphisms on $M$ (see §3.1).

Suppose $\mu$ is an $f$-invariant probability measure for some
$f\in\operatorname{Diff}^{1}(M)$. The Kolmogorov-Sinaĭ entropy of $\mu$ will
be denoted by $h(f,\mu)$. Almost every $x\in M$ has two well-defined Lyapunov
exponents $\lambda^{-}(f,x)\leq\lambda^{+}(f,x)$. The Lyapunov exponents of
$\mu$ are defined by

$\lambda^{-}(f,\mu):=\lambda^{-}_{\mu}:=\int\lambda^{-}(f,x)d\mu\ ,\
\lambda^{+}(f,\mu):=\lambda^{+}_{\mu}:=\int\lambda^{+}(f,x)d\mu.$

We are interested in the regularity of $(f,\mu)\mapsto h(f,\mu)$ and
$(f,\mu)\mapsto\lambda^{\pm}(f,\mu)$.

In the $C^{\infty}$ case, these functions are semi-continuous. Specifically,
suppose $f_{k},f\in\operatorname{Diff}^{\infty}(M)$, $\nu_{k}$ are ergodic
$f_{k}$-invariant measures, $f_{k}\to f$ in $C^{\infty}$ and $\nu_{k}\to\mu$
weak-$*$. Then

$\displaystyle\limsup\limits_{k\to\infty}h(f_{k},\nu_{k}$
$\displaystyle)\;\;\leq\;h(f,\mu),$ (1)
$\displaystyle\limsup\limits_{k\to\infty}\lambda^{+}(f_{k},\nu_{k})\leq\lambda^{+}(f,\mu),$
$\displaystyle\liminf\limits_{k\to\infty}\lambda^{-}(f_{k},\nu_{k})\geq\lambda^{-}(f,\mu).$

(See Section 1.6 for the history of these results.)

By (1) and Ruelle’s inequality, if $\limsup_{k}h(f_{k},\nu_{k})>0$, then
$\lambda^{+}(f,\mu)\geq h(f,\mu)\geq\limsup_{k}h(f_{k},\nu_{k})>0$, and

$\frac{\limsup\limits_{k\to\infty}\lambda^{+}(f_{k},\nu_{k})}{\lambda^{+}(f,\mu)}\
,\ \frac{\limsup\limits_{k\to\infty}h(f_{k},\nu_{k})}{h(f,\mu)}\in(0,1].$

We call these quantities the discontinuity ratios, and think of them as
measures for the difference between the two sides of the inequalities in (1).

We will provide inequalities relating the discontinuity ratio of the entropy
to the discontinuity ratio of $\lambda^{+}$. (It is enough to consider
$\lambda^{+}$, because $\lambda^{-}(f,\mu)=-\lambda^{+}(f^{-1},\mu)$.)

###  1.1. The ergodic $C^{\infty}$ case.

Our results are simplest and strongest when the maps are $C^{\infty}$ and the
limiting measure is ergodic:

######  Theorem A.

For every $k\geq 1$, let $f_{k}\in\operatorname{Diff}^{\infty}(M)$ and let
$\nu_{k}$ be an $f_{k}$-ergodic invariant measure. Suppose

  * –

$\lim_{k}\lambda^{+}(f_{k},\nu_{k})$ and $\lim_{k}h(f_{k},\nu_{k})$ exist and
are positive,

  * –

$f_{k}$ converge in the $C^{\infty}$ topology to a diffeomorphism
$f\in\operatorname{Diff}^{\infty}(M)$,

  * –

$\nu_{k}$ converge weak-$*$ to a probability measure $\mu$ (necessarily
$f$-invariant).

If $\mu$ is $f$-ergodic, then
$\displaystyle\lim_{k\to\infty}\frac{h(f_{k},\nu_{k})}{{h(f,\mu)}}\leq\lim_{k\to\infty}\frac{\lambda^{+}(f_{k},\nu_{k})}{\lambda^{+}(f,\mu)}.$

The following result is an immediate consequence of this and (1), and was the
original aim of our work:

######  Corollary 1.1.

For every $k\geq 1$, let $f_{k}\in\operatorname{Diff}^{\infty}(M)$ and let
$\nu_{k}$ be an $f_{k}$-ergodic invariant measure. Suppose $f_{k}\to f$ in the
$C^{\infty}$ topology, and $\nu_{k}\to\mu$ weak-$*$ where $\mu$ is an
$f$-ergodic invariant measure with positive entropy. If $h(f_{k},\nu_{k})\to
h(f,\mu)$, then:

$\lambda^{+}(f_{k},\nu_{k})\to\lambda^{+}(f,\mu)\text{ and
}\lambda^{-}(f_{k},\nu_{k})\to\lambda^{-}(f,\mu).$

The result mentioned in the introduction is the special case
$f_{1}=f_{2}=\cdots=f$.

###  1.2. The ergodic $C^{r}$ case

The following result extends Theorem A to the $C^{r}$ case up to an extra term
similar to what happens in Yomdin’s theory. We define the asymptotic dilation
of a $C^{1}$ map $f:M\to M$ to be

$\lambda(f):=\lim_{n\to+\infty}\frac{1}{n}\log\|Df^{n}\|_{\sup},\text{ where
}\|Df\|_{\sup}:=\sup_{x\in M}\underset{v\neq 0}{\sup_{v\in
T_{x}M}}\frac{\|Df_{x}v\|_{f(x)}}{\|v\|_{x}}.$ (2)

Since $M$ is compact, $\lambda(f)$ is independent of the choice of a
Riemannian metric $\|\cdot\|_{x}$.

######  Theorem B.

Fix $r>2$. For every $k\geq 1$, let $f_{k}\in\operatorname{Diff}^{r}(M)$ and
let $\nu_{k}$ be an $f_{k}$-ergodic invariant measure. Suppose

  * –

$\lim\limits_{k\to\infty}\lambda^{+}(f_{k},\nu_{k})$ and
$\lim\limits_{k\to\infty}h(f_{k},\nu_{k})$ exist and are positive,

  * –

$f_{k}\to f$ in the $C^{r}$-topology,

  * –

$\nu_{k}$ converge weak-$*$ to a probability measure $\mu$.

If $\mu$ is $f$-ergodic and has positive entropy, then

$\frac{\lim\limits_{k\to\infty}h(f_{k},\nu_{k})}{h(f,\mu)}-\frac{1}{h(f,\mu)}\frac{{\lambda}(f)+\lambda(f^{-1})}{r-1}\leq\;\frac{\lim\limits_{k\to\infty}\lambda^{+}(f_{k},\nu_{k})}{\lambda^{+}(f,\mu)}.$

By [10], the condition $h(f,\mu)>0$ holds once
$\lim\limits_{k\to\infty}h(f_{k},\nu_{k})>\frac{\min(\lambda(f),\lambda(f^{-1}))}{r}$.

As we will explain in Section 1.4, the smoothness index $r$ above does not
need to be an integer.

###  1.3. The non-ergodic case

The assumption that the limiting measure is ergodic is often difficult to
check, and we now explain what can be said in its absence (a more general but
also more technical result, Theorem D, will be given in section 7).

######  Theorem C.

Fix $r>2$. For every $k\geq 1$, let $f_{k}\in\operatorname{Diff}^{r}(M)$ and
let $\nu_{k}$ be an $f_{k}$-ergodic invariant measure. Suppose

  * –

$\lim\limits_{k\to\infty}\lambda^{+}(f_{k},\nu_{k})$ and
$\lim\limits_{k\to\infty}h(f_{k},\nu_{k})$ exist and are positive,

  * –

$f_{k}\to f$ in the $C^{r}$-topology,

  * –

$\nu_{k}\to\mu$ weak-$*$ for some $f$-invariant probability measure $\mu$
(perhaps non-ergodic).

If
$\lim\limits_{k\to\infty}h(f_{k},\nu_{k})>\tfrac{\lambda(f)+\lambda(f^{-1})}{r-1}$
then there exist $\beta\in(0,1]$, and two $f$-invariant probability measures
$\mu_{0},\mu_{1}$ with $h(f,\mu_{1})>0$ such that
$\mu=(1-\beta)\mu_{0}+\beta\mu_{1}$ and

$\frac{\lim\limits_{k\to\infty}h(f_{k},\nu_{k})}{h(f,\mu_{1})}\;-\;\frac{1}{h(f,\mu_{1})}\frac{{\lambda}(f)+\lambda(f^{-1})}{r-1}\;\leq\;\beta=\frac{\lim\limits_{k\to\infty}\lambda^{+}(f_{k},\nu_{k})}{\lambda^{+}(f,\mu_{1})}.$
(3)

Moreover $\lambda^{+}(f,x)>0\geq\lambda^{-}(f,x)$ for $\mu_{1}$-a.e. $x\in M$.

Note that for $C^{\infty}$ diffeomorphisms, the term
$(\lambda(f)+\lambda(f^{-1}))/(r-1)$ can be replaced by zero, because the
theorem can be applied with $r$ arbitrarily large.

The decomposition $\mu=(1-\beta)\mu_{0}+\beta\mu_{1}$ depends on the sequences
$(f_{k})_{k\geq 1}$, $(\nu_{k})_{k\geq 1}$, and not just on their limits. We
give a heuristic description of this decomposition in Section 2.2.

###  1.4. Additional comments

We now supplement Theorems A, B and C by some examples, comments,
strengthenings, and generalizations. The proofs can be found in Section 8.

####  1.4.1. Examples of discontinuities

Theorem A is sharp in the following sense:

######  Example 1.2.

For every $0<\alpha\leq\beta\leq 1$, there exist a $C^{\infty}$ surface
diffeomorphism $f$ and a sequence of ergodic and invariant measures $\nu_{k}$
converging weak-$*$ to an ergodic invariant probability measure $\mu$ such
that $\lim_{k}h(f,\nu_{k})>0$, and

$\lim_{k\to\infty}h(f,\nu_{k})/h(f,\mu)=\alpha\ ,\
\lim_{k\to\infty}\lambda^{+}(f,\nu_{k})/\lambda^{+}(f,\mu)=\beta.$

####  1.4.2. Variant inequality

Theorem A does not use the symmetry between a diffeomorphism and its inverse.
When $0<-\lambda^{-}(f,\mu)<\lambda^{+}(f,\mu)$, this symmetry yields a
sharper bound:

######  Corollary 1.3.

Under the assumptions of Theorem A,

$\lambda^{+}(f,\mu)-\lim_{k}\lambda^{+}(f_{k},\nu_{k})\leq|\lambda^{-}(f,\mu)|\left(1-\frac{\lim_{k}h(f_{k},\nu_{k})}{h(f,\mu)}\right).$
(4)

####  1.4.3. Sequences of non-ergodic measures

Our results can be extended to the case when the invariant measures $\nu_{k}$
are not ergodic, but this requires stronger assumptions on the Lyapunov
exponents of $\nu_{k}$, which in the non-ergodic case are functions and not
constants. See Corollary 8.5.

####  1.4.4. Lifted version

$f$ induces a dynamical system $\widehat{f}$ on the projective tangent bundle
$\widehat{M}$, see Section 3.2. It turns out that
$\mu=(1-\beta)\mu_{0}+\beta\mu_{1}$ is a projection of a decomposition of a
limit point $\widehat{\mu}:=\lim\widehat{\nu}_{k_{i}}$, where
$\widehat{\nu}_{k}$ are lifts of $\nu_{k}$ to $\widehat{f}$-invariant measures
on $\widehat{M}$. The decomposition of $\widehat{\mu}$ contains more
information than the decomposition of $\mu$, and leads to a stronger
statement, Theorem D, in Section 7. This strengthening is essential to the
proof of Corollary 8.5 on the case when $\nu_{k}$ are not ergodic.

####  1.4.5. Convergence of $C^{r}$-diffeomorphisms

In finite differentiability, Theorem D allows a weaker convergence assumption
(denoted by $f_{k}\overset{\scriptscriptstyle r-\text{bd}}{\longrightarrow}f$,
see Section 3.1), and $r$ does not have to be an integer.

####  1.4.6. Entropy upper semi-continuity in the $C^{r}$ case

It is well-known that for $C^{r}$-diffeomorphisms the entropy may fail to be
upper semi-continuous, but the defect in upper semi-continuity can be bounded
(see the discussion in Section 1.6 below). This bound manifests itself in
Theorems B and C in the expression $\frac{\lambda(f)+\lambda(f^{-1})}{r-1}$.
But our proof gives a slightly stronger bound
$\frac{\lambda(\widehat{f})}{r-1}$, in terms of the dynamics of $\widehat{f}$
on the projective tangent bundle, see §3.2, §3.5, and Theorem D.

In some special cases, even this stronger bound can be improved. For instance,
if $\lambda^{+}(f_{k},\nu_{k})\to\lambda^{+}(f,\mu)$, then

$\lim_{k\to\infty}h(f_{k},\nu_{k})\leq
h(f,\mu)+\frac{\min\\{\lambda(f),\lambda(f^{-1})\\}}{r},$ (5)

which is stronger than the assertion of Theorem B. When
$f_{1}=f_{2}=\cdots=f$, (5) is a refinement of a classical inequality of
Yomdin and Newhouse; it follows from bounds on the tail entropy, which were
first written explicitly in [10], and which are consequences of the
Downarowicz variational principle [16]. When the sequence $(f_{k})$ is non-
constant, it follows from a bound on robust tail entropy, which can be shown
using techniques in [11]. We thank David Burguet for explaining this to us.

###  1.5. Applications to dimension theory and to SRB measures

The Hausdorff dimension of a Borel measure $\mu$ on $M$ is defined to be the
infimum of the Hausdorff dimensions of all Borel sets of full $\mu$-measure
[49, p.115]. We denote this by $\mathrm{HD}(\mu)$.

######  Corollary 1.4.

For every $k\geq 1$, let $f_{k}\in\operatorname{Diff}^{\infty}(M)$ and let
$\nu_{k}$ be an $f_{k}$-ergodic invariant measure. Suppose $f_{k}\to f$ in the
$C^{\infty}$ topology, $\nu_{k}\to\mu$ weak-$*$, and
$\liminf\limits_{k\to\infty}h(f_{k},\nu_{k})>0$. If $\mu$ is ergodic, then
$\displaystyle{\limsup\limits_{k\to\infty}\mathrm{HD}(\nu_{k})\leq\mathrm{HD}(\mu).}$

###### Proof.

One can always take a subsequence such that $\lim_{k}h(f_{k},\nu_{k})>0$,
$\lim_{k}\lambda^{+}(f_{k},\nu_{k})>0$ and
$\lim_{k}\lambda^{-}(f_{k},\nu_{k})<0$ exist, and such that
$\mathrm{HD}(\nu_{k})$ converges to the limsup of the initial sequence. In
[49], Young gives the following formula for the Hausdorff dimension:

$\mathrm{HD}(\mu)=h(f,\mu)\left(1/\lambda^{+}(f,\mu)+1/|\lambda^{-}(f,\mu)|\right).$
(6)

Writing
$\displaystyle\lim_{k}\frac{h(f_{k},\nu_{k})}{\lambda^{+}(f,\nu_{k})}=\left(\frac{h(f,\mu)}{\lambda^{+}(f,\mu)}\right)\biggl{(}\frac{\lim_{k}h(f_{k},\nu_{k})}{h(f,\mu)}\biggr{)}\biggl{(}\frac{\lim_{k}\lambda^{+}(f_{k},\nu_{k})}{\lambda^{+}(f,\mu)}\biggr{)}^{-1}$,
we conclude from Theorem A that
$\lim_{k}\frac{h(f_{k},\nu_{k})}{\lambda^{+}(f,\nu_{k})}\leq\frac{h(f,\mu)}{\lambda^{+}(f,\mu)}$.
Working with $f_{k}^{-1}$ and $f^{-1}$ we obtain in a similar way that
$\lim_{k}\frac{h(f_{k},\nu_{k})}{|\lambda^{-}(f,\nu_{k})|}\leq\frac{h(f,\mu)}{|\lambda^{-}(f,\mu)|}$.
∎

Suppose $f\in\operatorname{Diff}^{2}(M)$. An $f$ invariant probability measure
is called a Sinai-Ruelle-Bowen (SRB) measure, if $\lambda^{+}(f,x)>0$
$\mu$-a.e., and if the conditional measures of $\mu$ obtained by
disintegrating it with respect to a measurable partition subordinated to the
lamination by unstable manifolds are a.e. absolutely continuous with respect
to the induced Riemannian measures. We recall two classical characterizations
of SRB measures from [31] and [27].

Suppose $\lambda^{+}(f,x)>0$ $\mu$-almost everywhere. The _geometric pressure
of $\mu$_ is

$P^{u}(\mu)=P^{u}(f,\mu):=h(f,\mu)-\lambda^{+}(f,\mu).$

By Ruelle’s inequality, $P^{u}(\mu)\leq 0$ for all invariant probability
measures, and by Ledrappier-Young [31], $P^{u}(\mu)=0$ iff $\mu$ is an SRB
measure.

Next suppose $\mu$ is ergodic and $\lambda^{+}(f,\mu)>0$. The unstable
dimension of $\mu$ is

$\displaystyle\delta^{u}(f,\mu):=\frac{h(f,\mu)}{\lambda^{+}(f,\mu)}.$

By Ledrappier-Young Theory [27], $\delta^{u}(f,\mu)$ is the a.s. value of the
Hausdorff dimension of the conditional measures of $\mu$ on a measurable
partition subordinated to the local unstable manifolds, and $\mu$ is an SRB
measure iff $\delta^{u}(\mu)=1$.

Theorem A immediately implies the following:

######  Corollary 1.5.

For every $k\geq 1$, let $f_{k}\in\operatorname{Diff}^{\infty}(M)$ and let
$\nu_{k}$ be an $f_{k}$-ergodic invariant measure. Suppose $f_{k}\to f$ in the
$C^{\infty}$ topology, $\nu_{k}\to\mu$ weak-$*$, and $\lim h(f_{k},\nu_{k})$
exists and is positive. If $\mu$ is ergodic, then:

  1. (1)

$\limsup\limits_{k\to\infty}\delta^{u}(\nu_{k})\leq\delta^{u}(\mu)$

  2. (2)

$\limsup\limits_{k\to\infty}P^{u}(f_{k},\nu_{k})\leq
P^{u}(f,\mu)\frac{\lim\limits_{k\to\infty}h(f_{k},\nu_{k})}{h(f,\mu)}$

  3. (3)

if $P^{u}(f_{k},\nu_{k})\to 0$ or $\delta^{u}(\nu_{k})\to 1$, then $\mu$ is an
SRB measure.

###### Proof.

The proof of Corollary 1.4 also shows (1); $(1)\Rightarrow(2)$ is a simple
algebraic manipulation; and (1)+(2)$\Rightarrow(3)$ by the Ledrappier-Young
characterizations of SRB measures as measures with zero pressure and/or
unstable dimension equal to one. ∎

We can remove the assumption that $\mu$ is ergodic, using Theorem C:

######  Corollary 1.6.

For every $k\geq 1$, let $f_{k}\in\operatorname{Diff}^{\infty}(M)$ and let
$\nu_{k}$ be an $f_{k}$-ergodic invariant measure. Suppose $f_{k}\to f$ in the
$C^{\infty}$ topology, $\nu_{k}\to\mu$ weak-$*$, and $\lim h(f_{k},\nu_{k})$
exists and is positive. Then there are ergodic components
$\mu^{\prime},\mu^{\prime\prime}$ of $\mu$ satisfying
$\lambda^{+}(\mu^{\prime}),\lambda^{+}(\mu^{\prime\prime})>0$ and

$\delta^{u}(\mu^{\prime\prime})\geq\lim_{k\to\infty}\delta^{u}(\nu_{k})\text{
, }P^{u}(f,\mu^{\prime})\geq\lim_{k\to\infty}P^{u}(f_{k},\nu_{k}).$

###### Proof.

Consider the decomposition $\mu=(1-\beta)\mu_{0}+\beta\mu_{1}$ and take
suitable ergodic components of $\mu_{1}$. ∎

The following statement implies the corollary in the introduction.

######  Corollary 1.7.

Let $f$ be a $C^{\infty}$ diffeomorphism of a compact smooth surface without
boundary, and fix some $h>0$. The following are equivalent:

  1. (i)

$f$ admits an SRB measure with entropy at least $h$;

  2. (ii)

$\sup\\{P^{u}(f,\mu):\mu\text{ ergodic measure for $f$ s.t. }h(f,\mu)\geq
h\\}=0$;

  3. (iii)

$\sup\\{\delta^{u}(\mu):\mu\text{ ergodic measure for $f$ s.t. }h(f,\mu)\geq
h\\}=1$.

###### Proof.

(i)$\Rightarrow$(ii) is due to Ledrappier & Strelcyn [29], and
(ii)$\Leftrightarrow$(iii) is trivial.

To see (ii)$\Rightarrow$ (i), we take a sequence of $f$-ergodic measures
$\nu_{k}$ with $h(f,\nu_{k})\to h^{\prime}\geq h$ and $P^{u}(f,\nu_{k})\to 0$.
We select a subsequence $(\nu_{k_{i}})_{i\geq 1}$ s.t.
$\lambda^{+}:=\lim\lambda^{+}(\nu_{k_{i}})$ and $\mu:=\lim\nu_{k_{i}}$ exist.
By Ruelle’s inequality, $\lambda^{+}\geq h^{\prime}>0$, and by Theorem C,
$\mu=\beta\mu_{1}+(1-\beta)\mu_{0}$ where

$\lim_{i\to\infty}\lambda^{+}(f,\nu_{k_{i}})=\beta\lambda^{+}(f,\mu_{1})\text{,
and }\lim_{i\to\infty}h(f,\nu_{k_{i}})\leq\beta h(f,\mu_{1}).$

It follows that $0=\lim_{i\to\infty}P^{u}(f,\nu_{k_{i}})\leq\beta
P^{u}(f,\mu_{1})\leq 0$, where the last inequality is Ruelle’s inequality for
$\mu_{1}$. Since
$\beta\lambda^{+}(f,\mu_{1})=\lim\lambda^{+}(f,\nu_{k_{i}})=\lambda^{+}\neq
0$, it must be the case that $\beta\neq 0$, and $P^{u}(f,\mu_{1})=0$. If
$\mu_{1}=\int\mu_{\xi}^{\prime}d\xi$ is the ergodic decomposition of
$\mu_{1}$, then $\int P^{u}(f,\mu_{\xi}^{\prime})d\xi=P^{u}(f,\mu_{1})=0$. By
Ruelle’s inequality, the integrand is non-positive, so
$P^{u}(f,\mu^{\prime}_{\xi})=0$ for $\mu_{1}$-a.e. ergodic component. At the
same time, $h(f,\mu_{1})\geq\beta^{-1}\lim h(f_{k},\nu_{k})>h^{\prime}$, so
some of these ergodic components must have entropy $\geq h^{\prime}$. Thus
$\mu_{1}$ has ergodic components with entropy bigger than $h$, and zero
pressure. By Ledrappier-Young Theory, these are SRB measures with entropy
bigger than $h$, and (i) is proved. ∎

Notice that it is essential in this proof to be able to deal with non-ergodic
limits, since we have no control of $\lim\nu_{k}$.

###  1.6. Related works

In this paper we relate the continuity properties of the entropy to those of
the Lyapunov exponents. The continuity of these objects has been studied
separately before in several works, which we now recall.

##### Entropy.

In general, the entropy map $(f,\mu)\mapsto h(f,\mu)$ is not lower semi-
continuous, even in the uniformly hyperbolic case. For example, it is easy to
construct sequences of atomic measures (with zero entropy) on a basic set,
which converge to limits with positive entropy.

However, for $C^{\infty}$ diffeomorphisms on compact manifolds, the entropy
map is upper semi-continuous: This is due to Newhouse [35]. For $C^{r}$
diffeomorphisms with finite $r$, even upper semi-continuity may fail (for
examples in dimension four see [33], and for examples in dimension two see
[14]). However, the (additive) defect in semi-continuity:

$D(f,\mu):=\limsup_{(g,\nu)\to(f,\mu)}h(g,\nu)-h(f,\mu)$

can be bounded from above by $\min(\lambda(f),\lambda(f^{-1}))/r$, using
Yomdin theory [35, 12, 10, 11]. A subject of more recent interest is the loss
of semi-continuity due to non-compactness. This has been studied for countable
Markov shifts [22, 23], geodesic flows on non-compact homogeneous spaces [18,
25], and geodesic flows on non-compact manifolds with negative sectional
curvatures [21, 42].

##### Lyapunov exponents.

The top Lyapunov exponent map $(f,\mu)\mapsto\lambda^{+}(f,\mu)$ varies
continuously for uniformly hyperbolic systems on surfaces. It even depends
analytically on the diffeomorphism $f$ [44]. Moreover if $\mu_{\max}$ is the
unique measure of maximal entropy of a mixing Anosov surface diffeomorphism,
then [24] (see also [41, 45]) implies that
$|\lambda^{+}(\mu_{\max})-\lambda^{+}(\nu)|\leq
c\sqrt{|h(f,\mu_{\max})-h(f,\nu)|}$, where $c$ only depends on $f$.

In the non-uniformly hyperbolic case, the situation is different. For example,
[6] proves that among conservative systems the Lyapunov exponents of the
volume measure are discontinuous when the diffeomorphism varies in the
$C^{1}$-topology, unless they vanish.

We are not aware of other general results on the continuity of the Lyapunov
exponents for general non-uniformly hyperbolic surface diffeomorphisms.

By contrast, much is known on the continuity of Lyapunov exponents of random
products of independent identically distributed $SL(2,\mathbb{R})$ matrices,
as functions of the underlying Bernoulli process, see [20, 7]. More general
Hölder continuous matrix cocycles with holonomies are considered in [2], and a
higher-dimensional extension has been announced in [47].

##### Dimension.

L.-S. Young gave the famous formula (6) for the dimension of hyperbolic
invariant measures in [49] in terms of the entropy and the Lyapunov exponents
of the measure. For further dimension theoretic properties of hyperbolic
invariant measures, see [4] and [3]. The continuity of the dimension of
invariant sets and measures for hyperbolic systems have been considered in
numerous works, for instance [37] proves that basic sets on surface have a
Hausdorff dimension which varies continuously with the diffeomorphism, [5]
proves that the supremum of the Hausdorff dimensions of ergodic measures on
such a basic set is attained by a measure of maximal dimension and [3]
discusses some non-uniformly hyperbolic cases.

##  2\. A heuristic overview of the proof

All our results follow from Theorem C, and the remainder of the paper is
dedicated to the proof of this theorem. Here we give a heuristic overview of
the proof, in the special case when $f_{1}=f_{2}=\cdots=f$ is a $C^{\infty}$
diffeomorphism.

###  2.1. The origin of the discontinuities in $\lambda^{+}$

As Furstenberg discovered, the Lyapunov exponents are easier to study in terms
of the _projective dynamics_ $\widehat{f}(x,E)=(f(x),Df_{x}(E))$ on the
projective tangent bundle

$\displaystyle\widehat{M}:=\\{(x,E):x\in M,\ E\subset T_{x}M\text{ is a one-
dimensional linear space}\\}.$

Indeed by Ledrappier’s work, a Lyapunov exponent of an $f$-ergodic measure
$\mu$ is simply the integral of the continuous function

$\varphi(x,E):=\log\|Df_{x}|_{E}\|,$

with respect to the lift of $\mu$ to the bundle of the associated Oseledets
spaces.

Suppose $\nu_{k}$ are ergodic measures with positive entropy such that
$\nu_{k}\to\mu$ weak∗, and suppose for the moment that $\mu$ is ergodic and
with positive entropy. Since $\dim(M)=2$, $\nu_{k}$ have two simple Lyapunov
exponents, and there are exactly two ergodic lifts $\widehat{\nu}^{+}_{k}$ and
$\widehat{\nu}^{-}_{k}$, one carried by the bundle $\mathcal{E}^{u}$ of
unstable Oseledets spaces, and the other carried by the bundle
$\mathcal{E}^{s}$ of the stable Oseledets spaces. (The third bundle
$\mathcal{E}^{0}$ associated to the zero exponent has measure zero for all
lifts of $\nu_{k}$.) Hence,

$\lambda^{\pm}(f,\nu_{k})=\int\varphi d\widehat{\nu}_{k}^{\pm}.$

Suppose $\widehat{\nu}^{+}_{k}$ converge weak-star on $\widehat{M}$ to an
$\widehat{f}$-invariant probability measure $\widehat{\mu}$ (this is true for
a subsequence). Since $\varphi:\widehat{M}\to\mathbb{R}$ is continuous,

$\lim_{k\to\infty}\lambda^{+}(f,\nu_{k})=\lim_{k\to\infty}\widehat{\nu}_{k}(\varphi)=\widehat{\mu}(\varphi).$

The limiting measure $\widehat{\mu}$ is a lift of $\mu$, but this does not
have to be the lift of $\mu$ to $\mathcal{E}^{u}$, $\widehat{\mu}^{+}$. If
$\widehat{\mu}(\varphi)\neq\widehat{\mu}^{+}(\varphi)$, then
$\lim_{k\to\infty}\lambda^{+}(f,\nu_{k})\neq\lambda^{+}(f,\mu)$.

It is certainly possible that $\widehat{\mu}\neq\widehat{\mu}^{+}$: The
Oseledets bundle $\mathcal{E}^{u}$ carrying the lifts $\widehat{\nu}_{k}^{+}$
is not necessarily bounded away from $\mathcal{E}^{s}$, and some mass
$0\leq\rho\leq 1$ on $\mathcal{E}^{u}$ can escape to $\mathcal{E}^{s}$.

Escape of mass to $\mathcal{E}^{s}$ is reflected in long stretches of time
when $\nu_{k}$-typical orbits do not experience the exponential growth of
$\mathcal{E}^{u}$-directions predicted by $\lambda^{+}$. Instead, they see,
temporarily, exponential decay at rate $\lambda^{-}$, cancelling some of the
previous growth. If $\widehat{\mu}^{+},\widehat{\mu}^{-}$ denote the two
ergodic lifts. we must have
$\widehat{\mu}=(1-\rho)\widehat{\mu}^{+}+\rho\widehat{\mu}^{-}$ and thus,

$\lim_{k\to\infty}\lambda^{+}(f,\nu_{k})=(1-\rho)\widehat{\mu}^{+}(\varphi)+\rho\widehat{\mu}^{-}(\varphi)=\lambda^{+}_{\mu}-\rho(\lambda^{+}_{\mu}-\lambda^{-}_{\mu}).$

In the language of Theorem C (and since $\mu=\mu_{1}$ by ergodicity), the
discontinuity ratio $\beta$ is:

$\beta:=\frac{\lim_{k}\lambda^{+}(f,\nu_{k})}{\lambda^{+}(f,\mu)}=1-\rho(1-\lambda^{-}_{\mu}/\lambda^{+}_{\mu}).$

(A different description of $\beta$ will be given below.) So if $\mu$ is
ergodic, then $\beta$ is a function of $\rho$, whence of the amount of mass
which escapes to $\mathcal{E}^{s}$.

In the case where $\mu$ is not ergodic, the different ergodic components of
$\mu$ have to be considered, and some of them may have zero Lyapunov
exponents. The way in which $\nu_{k}$-typical orbits approximate those ergodic
components determine the possible cancellations. So if $\mu$ is not ergodic,
then $\beta$ may depend on the entire sequence $(\nu_{k})$, not just on its
limit $\mu$.

###  2.2. Neutral blocks, the decomposition of $\mu$, and the parameter
$\beta$

Recall the measurable $\widehat{f}$-invariant decomposition
$\mathcal{X}=\mathcal{E}^{s}\cup\mathcal{E}^{u}\cup\mathcal{E}^{0}$ defined by
the Oseledets theorem according to the sign of the limit
$\tfrac{1}{n}\log\|Df^{n}_{x}|_{E}\|$. It has full measure with respect to any
$\widehat{f}$-invariant measure. To get quantitative estimates, we select
compact subsets $K^{*}\subset\mathcal{E}^{*}$, for $*\in\\{s,u,0\\}$, from
which the contraction, expansion, or “central” behavior of the sequence
$\|Df^{n}_{x}|_{E}\|$ are uniformly controlled, and such that the
$\widehat{\mu}$-measure of $K:=K^{s}\cup K^{u}\cup K^{0}$ is close to $1$.
Since each $\mathcal{E}^{*}$ is invariant, we can choose these compact sets to
be nearly invariant: Points in a very small neighborhood stay close for a long
time.

Hence, if $\widehat{x}_{0}$ is a $\widehat{\nu}_{k}^{+}$-typical point for
some very large $k$, its orbit under $\widehat{f}$ spends nearly all its time
close to $K$ and every visit in a small neighborhood of $K^{s}\cup K^{0}$ is
the beginning of a long period of uniform contraction (or weak
expansion/contraction). One expects no entropy creation not only during this
period, but also during the “recovery period” which follows, i.e., until the
expansion predicted by the Lyapunov exponent of $\nu_{k}$ cancels this period
of contraction (or weak expansion/contraction).

We select such long time intervals along the orbit of $\widehat{x}_{0}$ in the
following greedy way. Fixing $\alpha>0$ small and $L$ large, an _$(\alpha,L)$
–neutral block_ is a maximal interval of integers $(n_{0},\dots,n_{0}+\ell)$
such that $\ell\geq L$ and

$\|Df^{n}_{f^{n_{0}}(x_{0})}|_{E^{u}}\|\leq\exp(\alpha(n-n_{0}))\text{ for all
}0<n\leq\ell.$

We will check that indeed, there is very little if any entropy creation during
neutral blocks.

Our estimates will be in terms of the distribution of these long neutral
blocks. Let $\widehat{x}_{k}=(x_{k},E^{u}(x_{k}))$ be
$\widehat{\nu}_{k}^{+}$-generic points and let
$\mathfrak{N}_{\alpha,L}(\widehat{x}_{k})$ denote the union of all
$(\alpha,L)$-neutral blocks of the orbit of $\widehat{x}_{k}$. In Section 6 we
show that it is possible to choose a subsequence $k_{i}\to\infty$ so that
following limits make sense weak-$*$ on $\widehat{M}$ for
$(\widehat{\nu}^{+}_{k_{1}}\times\widehat{\nu}^{+}_{k_{2}}\times\cdots)$–a.e.
$(\widehat{x}_{k_{1}},\widehat{x}_{k_{2}},\ldots)$:

$\displaystyle\widehat{m}_{0}:=\underset{L\to\infty}{\lim\limits_{\alpha\to
0}}\lim_{i\to\infty}\left(\lim_{N\to\infty}\frac{1}{N}\sum_{j\in[0,N)\cap\mathfrak{N}_{\alpha,L}(\widehat{x}_{k})}\delta_{\widehat{f}^{j}(\widehat{x}_{k_{i}})}\right)$
$\displaystyle\widehat{m}_{1}:=\underset{L\to\infty}{\lim\limits_{\alpha\to
0}}\lim_{i\to\infty}\left(\lim_{N\to\infty}\frac{1}{N}\sum_{j\in[0,N)\setminus\mathfrak{N}_{\alpha,L}(\widehat{x}_{k})}\delta_{\widehat{f}^{j}(x_{k_{i}})}\right)\,.$

Notice that the sum of the two limits in the brackets is a.s.
$\widehat{\nu}_{k_{i}}^{+}$, because this is the limit of the empirical
measure of $\widehat{x}_{k_{i}}$, and $\widehat{x}_{k_{i}}$ are all a.s.
$\widehat{\nu}_{k_{i}}^{+}$–generic. So

$\widehat{m}_{0}+\widehat{m}_{1}=\lim_{k\to\infty}\widehat{\nu}_{k}^{+}=\widehat{\mu}.$

The measures $\widehat{m}_{0},\widehat{m}_{1}$ are $\widehat{f}$-invariant. We
will see that $\int\varphi d\widehat{m}_{0}=0$, and that $\widehat{m}_{1}$ is
carried by $\mathcal{E}^{u}$. The decomposition
$\mu=\beta\mu_{1}+(1-\beta)\mu_{0}$ in Theorem C is defined by

$\beta:=1-\widehat{\mu}_{0}(\widehat{M}),\;(1-\beta)\mu_{0}:=\widehat{\pi}_{*}(\widehat{\mu}_{0}),\text{
and }\beta\mu_{1}:=\widehat{\pi}_{*}(\widehat{\mu}-\widehat{\mu}_{0}),$

where $\widehat{\pi}:\widehat{M}\to M$ is the natural projection. Note that
$\beta$ is indeed the discontinuity ratio
$\lim_{k}\lambda^{+}(f,\nu_{k})/\lambda^{+}(f,\mu_{1})$ and the quantity
$1-\beta$ coincides with the fraction of the time spent in maximal neutral
blocks. The measures $\mu_{0},\mu_{1}$ and $\beta$ depend not just on
$\widehat{\mu}$, but also on the way the measures $\widehat{\nu}^{+}_{k}$
accumulate on $\widehat{\mu}$.

###  2.3. Upper bound on the entropy

To complete the proof of the theorem it remains to show that
$\displaystyle\lim_{k\to\infty}h(f,\nu_{k})\leq\beta h(f,\mu_{1})$. This is
the heart of the proof, and where most of the difficulties lie. We use
Ledrappier-Young Theory and Yomdin Theory.

  * •

_Ledrappier-Young theory_ bounds $h(f,\nu_{k})$ by the exponential rate of
growth of the minimal number of $(n,\epsilon)$-balls needed to cover a
definite fraction of a local unstable manifold $W^{u}_{loc}(x_{k})$, where
$x_{k}$ is a fixed $\nu_{k}$-typical point and the scale $\varepsilon$ tends
to zero. The “fraction” is measured using the conditional measure
$\nu^{u}_{x_{k}}$ of $\nu_{k}$ on $W^{u}_{loc}(x_{k})$. In particular, it
suffices to follow points $x\in W^{u}_{\operatorname{loc}}(x_{k})$ with
$T_{x}W^{u}_{\operatorname{loc}}(x_{k})=\mathcal{E}^{u}(x)$.

  * •

_Yomdin theory_ provides tools for controlling the number of
$(n,\varepsilon)$-balls needed for such covers, for $C^{r}$ maps. Instead of
working with $(n,\varepsilon)$-balls, one works with parametrized pieces of
unstable manifolds which lie inside $(n,\varepsilon)$-balls and which have
uniformly bounded $C^{r}$ size, and Yomdin Theory allows to bound the number
of such pieces. Here the regularity assumptions on $f$ come into play. The
expression $\frac{\lambda(f)+\lambda(f^{-1})}{r-1}$ in (3) is due to Yomdin
theory (see section 2.4).

Let us sketch our argument for the upper bound on the entropy using the
neutral blocks. Since the unstable lift of $\nu_{x_{k}}^{u}$-almost every
point is $\widehat{\nu}^{+}_{k}$-typical, neutral blocks represent roughly a
fraction $1-\beta$ of their time. During a neutral block, typical points on a
small piece of $f^{n}(W^{u}_{loc}(x_{k}))$ do not separate much, therefore
this piece remains small (or can be kept small by a subdivision into a small
exponential number of pieces). For the rest of the time, these subcurves
follow the ergodic components of $\mu_{1}$, hence they experience entropic
separation at an exponential rate given by $h(f,\mu_{1})$. Since the time
outside neutral blocks is a proportion $\beta$ of the total time, this leads
to the bound

$h(f,\nu_{k})\leq\beta h(f,\mu_{1}).$

This argument explains the link between the entropy bound and the
semicontinuity defect of the Lyapunov exponents.

This sketch glosses over several difficulties. We will only comment on the
main issue: How to use non-expansion of the linearization $Df$ at
$(x_{k},E^{u}(x_{k}))$ during a neutral block, to infer non-expansion of the
map $f$ itself on a small piece of $W^{u}_{loc}(x_{k})$ during this neutral
block. The difficulty is in controlling $Df$ on
$(x^{\prime}_{k},E^{u}(x^{\prime}_{k}))$ for $x_{k}^{\prime}$ close to
$x_{k}$.

###  2.4. Control of the expansion during neutral blocks

This is one of the most delicate points in the proof. To deduce the non-
expansion of the small piece of the unstable manifold containing this point,
we need to know that not only the diameter of this curve is small but that its
tangent is almost constant too. This forces us to work with pieces of
$W^{u}_{\operatorname{loc}}(x_{k})$ whose lifts to $\widehat{M}$ are also
small: The size in the fiber of $\widehat{M}$ measures the variability of the
tangent directions.

How small is small enough? To use information on $Df$ to control what happens
on $W^{u}_{loc}(x_{k})$, we need the fluctuations of the tangent direction
along any piece to be smaller than some $\widehat{\varepsilon}>0$, determined
(mostly) by the modulus of continuity of $Df$. Using the uniform continuity of
the measurable unstable bundle on a set of large measure, we find an
$\varepsilon>0$ such that if the diameter of the projection to $M$ is less
than $\varepsilon$, then the fluctuation of the tangent is smaller than
$\widehat{\varepsilon}$.

The price we pay for this solution is that we need to work with different
scales in $M$ and along the fibers of the bundle $\widehat{M}\to M$. This
leads us to introduce _fibered $(n,\varepsilon,\widehat{\varepsilon})$-balls_,
and to work with Yomdin theory for $\widehat{f}:\widehat{M}\to\widehat{M}$.
When dealing with diffeomorphisms of finite regularity, there is an additional
price to pay: If $f$ is $C^{r}$, then $\widehat{f}$ is only $C^{r-1}$, and
this accounts for the extra term from Yomdin theory

$\frac{\widehat{\lambda}(\widehat{f})}{r-1}\leq\frac{\lambda(f)+\lambda(f^{-1})}{r-1}\quad\text{
in \eqref{e.eqC2}}.$

###  2.5. Organisation of the paper

The different ingredients of the proof appear as follows in the text.

Section 3:

background on tangent dynamics and Lyapunov exponents.

Section 4:

results from Ledrappier-Young and Yomdin theories on the entropy in
differentiable dynamics.

Section 5:

reparametrization lemmas estimating the entropy from neutral blocks and other
time intervals.

Section 6:

neutral decomposition of typical orbits.

Section 7:

proof of the technical version of our main theorem.

Section 8:

proof of the remaining statements.

A remark on style. Our constructions, estimating entropy for a sequence of
measures converging to a nonergodic one, require many parameters. We have
chosen to make the dependences as explicit as possible to help the reader
check that there is no circular argument.

###  2.6. Standing notations for the duration of the paper

We collect here some notations that we will use frequently below.

  1. $\bullet$

$|X|$ or $\operatorname{Card}(X)$: the cardinality of a set $X$.

  2. $\bullet$

$M$ is a compact Riemannian $C^{\infty}$ manifold without boundary, with
tangent bundle $TM$, tangent spaces $T_{x}M$, and Riemannian norm
$\|\cdot\|_{x}$. Derivatives of maps $f:M\to M$ are denoted by $Df:TM\to TM$
or $Df_{x}:T_{x}M\to T_{f(x)}M$.

  3. $\bullet$

$h(f,\mu)$, $\lambda^{\pm}_{\mu}$, $\lambda^{\pm}_{x}$, $TM=E^{+}\oplus
E^{-}$: the entropy, average and pointwise Lyapunov exponents, Osededets
splitting associated to a measure $\mu$ (also denoted $\lambda^{s/u}_{\mu}$,
$E^{s/u}$ when the measure is hyperbolic of saddle type), see section 3.3.

  4. $\bullet$

$\widehat{M}$, $\widehat{f}$, $\widehat{x}=(x,E)$: the projective tangent
bundle, the lift of a diffeomorphism $f$ and of a point $x$, see section 3.2.

  5. $\bullet$

$B_{f}(x,n,\varepsilon)$, $r_{f}(n,\varepsilon,X)$: an $(n,\varepsilon)$-Bowen
ball for $f$ and the $(n,\varepsilon)$-covering number of a set $X\subset M$,
see section 4.2.

  6. $\bullet$

$\|Df\|_{\sup}$: the sup-norm of the tangent map, see section 1.2.

  7. $\bullet$

$\lambda(f):=\lim_{n\to+\infty}\frac{1}{n}\log\|Df^{n}\|_{\sup}$: the
asymptotic dilation.

  8. $\bullet$

$\|f\|_{C^{r}}$: the $C^{r}$ size of $f$, see section 3.1.

  9. $\bullet$

$Q_{r,N}(f)$: the supremum of the $C^{r}$ sizes of $f,f^{2},\dots,f^{N}$ and
of the $C^{r-1}$ semi-norm of
$\widehat{f},\widehat{f}^{2},\dots,\widehat{f}^{N}$ see section 4.6.

  10. $\bullet$

If $v$ is a vector, then $\mathbb{R}.v:=\\{tv:t\in\mathbb{R}\\}$.

##  3\. Tangent dynamics and the semi-continuity of Lyapunov exponents

Let $M$ be a smooth compact Riemannian surface without boundary.

###  3.1. Review of the $C^{r}$ size of maps

Let $U$ be an open subset of $\mathbb{R}^{n}$.

Given $k\in{\mathbb{N}}$, we say that a map $F:U\to\mathbb{R}^{d}$ is $C^{k}$
if for all $\omega\in({\mathbb{N}}\cup\\{0\\})^{n}$ such that
$|\omega|:=\omega_{1}+\cdots+\omega_{n}=k$, the partial derivative

$\partial^{\omega}F:=\frac{\partial^{\omega_{1}+\cdots+\omega_{n}}F}{\partial^{\omega_{1}}x_{1}\cdots\partial^{\omega_{n}}x_{n}}$

exists and is continuous on $U$. For any compact subset $K\subset U$, we then
define the $C^{k}$ _size_

$\|F\|_{C^{k},K}:=\max_{1\leq|\omega|\leq k}\max_{x\in
K}\|\partial^{\omega}F(x)\|.$

Given $\alpha\in(0,1)$, we say that a map $F$ is $C^{\alpha}$ if the following
quantity is finite for any compact set $K\subset U$,

$\|F\|_{C^{\alpha},K}:=\sup_{\tiny\begin{array}[]{c}{x,y\in K}\\\ x\neq
y\end{array}}\frac{\|F(x)-F(y)\|}{\|x-y\|^{\alpha}}.$

Given $r>1$ which is not an integer, we decompose it as $r=k+\alpha$, with
$k=\lfloor r\rfloor$ and $\alpha\in(0,1)$. We say that $F$ is $C^{r}$ if it is
$C^{k}$ and each partial derivative $\partial^{\omega}F$, $|\omega|=k$ is
$C^{\alpha}$. For any compact set $K\subset U$, we define the $C^{r}$ size

$\|F\|_{C^{r},K}:=\|F\|_{C^{k},K}+\max_{|\omega|=k}\|\partial^{\omega}F\|_{C^{\alpha},K}.$

Let $\Omega$ be a compact subset of $\mathbb{R}^{n}$ which is equal to the
closure of its interior (we mostly need $[0,1]^{n}$). A map
$F:\Omega\to\mathbb{R}^{d}$ is $C^{r}$ if $F$ has a $C^{r}$ extension to an
open neighborhood of $\Omega$. In this case, the $C^{r}$ size of $F$ on
$\Omega$ is

$\|F\|_{C^{r}}:=\;\sup_{K\subset\mathrm{int}(\Omega)}\;\|F\|_{C^{r},K}.$

This (finite) quantity is independent of the extension of $F$ to the
neighborhood of $\Omega$. Notice that the $C^{r}$ size of a constant function
is zero.

A $C^{r}$ structure on a smooth manifold $N$ is defined by a maximal atlas
$\mathfrak{A}$ with $C^{r}$ changes of coordinates. A smooth manifold equipped
with a $C^{r}$ structure $\mathfrak{A}$ is called a $C^{r}$ manifold. A finite
subset of $\mathfrak{A}$ which covers $N$ is called a $C^{r}$ atlas of $N$.

Let $N_{1},N_{2}$ be two compact $C^{r}$ manifolds (later this will be $M$,
$\widehat{M}$ or the circle $S^{1}$), and let $\mathcal{A}_{i}$ be finite
$C^{r}$ atlases of $N_{i}$. Let $\Omega$ be a compact subset of $N_{1}$ equal
to the closure of its interior. We say that $f\colon\Omega\to N_{2}$ is a
$C^{r}$ map if each map $\chi_{2}^{-1}\circ f\circ\chi_{1}$, where $\chi_{i}$
ranges over $\mathcal{A}_{i}$, is $C^{r}$. The _$C^{r}$ size_ of $f$ is:

$\|f\|_{C^{r}}:=\max_{\chi_{1}\in\mathcal{A}_{1},\chi_{2}\in\mathcal{A}_{2}}\|\chi_{2}^{-1}\circ
f\circ\chi_{1}\|_{C^{r}}<\infty.$

Again, the constant map has size zero.

The quantity $\|f\|_{C^{r}}$ depends on the choice of atlases
$\mathcal{A}_{i}$, but if $N_{i}$ are compact, then finite atlases induce
equivalent $C^{r}$ sizes. In case $N_{1}=S^{1}$, we will always use the
Euclidean atlas.

Suppose $f_{k},f\in\operatorname{Diff}^{r}(M)$ and $1\leq r<\infty$. We will
say that $f_{k}$ converges to $f$ uniformly in a $C^{r}$-bounded way, if
$f_{k}\to f$ uniformly, and $\sup_{k\geq 1}\|f_{k}\|_{C^{r}}<\infty.$ We write
in this case

$f_{k}\overset{\scriptscriptstyle r-\text{bd}}{\longrightarrow}f.$

If $M$ is compact, $f_{k},f\in\operatorname{Diff}^{\infty}(M)$, and $f_{k}\to
f$ in $C^{\infty}$, then $f_{k}\overset{\scriptscriptstyle
r-\text{bd}}{\longrightarrow}f$ for all $r$ finite.

The Arzela-Ascoli theorem implies the following.

######  Lemma 3.1.

Let $N_{1},N_{2}$ be compact $C^{r}$ manifolds, and $f,f_{1},f_{2},\dots\colon
N_{1}\to N_{2}$ be a collection of $C^{r}$ maps such that $(f_{k})$ converges
to $f$ uniformly, and $\sup_{k}\|f\|_{C^{r}}<\infty$. Then $(f_{k})$ converges
to $f$ in the $C^{\ell}$-topology for any $\ell<r$, $\ell\in{\mathbb{N}}$.

Thus, if for some real $r>1$ s.t. $r\not\in{\mathbb{N}}$,
$f,f_{1},f_{2},\dots\in\operatorname{Diff}^{r}(M)$ where $M$ is a compact
manifold, then $f_{k}\overset{\scriptscriptstyle
r-\text{bd}}{\longrightarrow}f$ implies that $f_{k}\to f$ in the $C^{\lfloor
r\rfloor}$-topology.

###  3.2. The projective tangent bundle

Let

$P_{x}M:=\\{E:E\text{ is a one-dimensional linear subspace of }T_{x}M\\}.$

$P_{x}M$ is the quotient of $T_{x}M\setminus\\{0\\}$ by the equivalence
relation $v\sim w$ $\iff$ $\exists\lambda\neq 0,\;v=\lambda w$. It can also be
viewed as the image of $\\{v\in T_{x}M:\|v\|_{x}=1\\}$ by the two-to-one map
$v\mapsto\mathrm{Span}\\{v\\}$. These identifications allow us to endow
$P_{x}M$ with a topology and with a smooth structure, and to identify the
tangent spaces $T_{E}(P_{x}M)$ with $\\{w\in T_{x}M:w\perp E\\}$. We can also
pull back the induced Riemannian inner product on $\\{w\in T_{x}M:w\perp E\\}$
to an inner product on $T_{E}(P_{x}M)$. This endows $P_{x}M$ with a Riemannian
structure. The resulting Riemannian distance on $P_{x}M$ is simply
${\mathrm{dist}}(E_{1},E_{2})=|\measuredangle(E_{1},E_{2})|$. With this
structure, $P_{x}M$ is isometric to the circle with perimeter $\pi$.

The projective tangent bundle (or just “projective bundle”) of $M$ is the
bundle $(\widehat{M},{\widehat{\pi}},M)$ where $\widehat{\pi}:\widehat{M}\to
M$ is the natural projection $\widehat{\pi}(x,E)=x$, and

$\widehat{M}:=\\{(x,E):x\in M,\text{$E$ is a one-dimensional linear subspace
of $T_{x}M$}\\}=\bigsqcup_{x\in M}P_{x}M.$

$\widehat{M}$ is a smooth compact three-dimensional manifold. We endow it with
the Riemannian metric $\sqrt{ds^{2}+d\theta^{2}}$, where $ds$ is the length
element on $M$ and $d\theta$ is the length element on $P_{x}M$. Points in
$\widehat{M}$ will be denoted by $\widehat{x}=(x,E)$.

Let $f:M\to M$ be a $C^{1}$ diffeomorphism. The canonical lift of $f$ is the
homeomorphism $\widehat{f}:\widehat{M}\to\widehat{M}$ given by

$\widehat{f}(x,E)=(f(x),Df_{x}(E)).$ (7)

If $f$ is of class $C^{r}$, then $\widehat{f}$ is of class $C^{r-1}$. Notice
that $\widehat{\pi}\circ\widehat{f}=f\circ\widehat{\pi}$, and

$\widehat{f}^{n}(x,E)=(f^{n}(x),(Df^{n})_{x}(E)).$

Every $\widehat{f}$-invariant probability measure $\widehat{\nu}$ on
$\widehat{M}$ projects to an $f$-invariant probability measure $\nu$ on $M$
given by

$\nu:=\widehat{\pi}_{\ast}(\widehat{\nu}):=\widehat{\nu}\circ\pi^{-1}.$

We call $\nu$ the projection of $\widehat{\nu}$, and $\widehat{\nu}$ a lift of
$\nu$. In what follows, when we “lift”, we always mean an
$\widehat{f}$-invariant lift. The following lemma is a well-known consequence
of the compactness of $\widehat{M}$.

######  Lemma 3.2.

  1. (1)

Every $f$-invariant probability measure $\nu$ has at least one lift
$\widehat{\nu}$.

  2. (2)

If $\nu$ is $f$-ergodic and $\widehat{\nu}$ lifts $\nu$, then a.e. ergodic
component of $\widehat{\nu}$ is a lift of $\nu$.

Hence every ergodic $f$-invariant probability measure has at least one ergodic
lift.

###  3.3. Review of Lyapunov exponents

We review some facts on Lyapunov exponents in dimension two (see [46, Theorems
3.12 and 3.14]). Suppose $f\in\operatorname{Diff}^{1}(M)$ and $\mu$ is an
$f$-invariant Borel probability measure. Oseledets’ theorem asserts that for
$\mu$-a.e. $x$, $\lim_{|n|\to\infty}\frac{1}{n}\log\|Df^{n}_{x}v\|$ exists for
all $v\in T_{x}M\setminus\\{0\\}$. The possible values of the limit are called
the Lyapunov exponents of $x$. There are at most two such values. We denote
them by $\lambda^{+}(f,x)$, $\lambda^{-}(f,x)$, or $\lambda^{+}_{x}$,
$\lambda^{-}_{x}$, with the convention

$\lambda^{+}(f,x)\geq\lambda^{-}(f,x).$

If $\lambda^{+}(f,x)\neq\lambda^{-}(f,x)$ then $T_{x}M=E^{+}(x)\oplus
E^{-}(x)$ where

$E^{\pm}(x):=\left\\{v\in
T_{x}M\setminus\\{0\\}:\lim_{|n|\to\infty}\frac{1}{n}\log\|Df^{n}_{x}v\|=\lambda^{\pm}(f,x)\right\\}\cup\\{0\\}.$

The decomposition $T_{x}M=E^{+}(x)\oplus E^{-}(x)$ is called the Oseledets
splitting.

If $\mu$ is ergodic, the functions $\lambda^{+}(f,x)\geq\lambda^{-}(f,x)$ are
equal $\mu$-almost everywhere to constants called the Lyapunov exponents of
$\mu$ and denoted by $\lambda^{+}(f,\mu),\lambda^{-}(f,\mu)$. If $\mu$ is not
ergodic, the Lyapunov exponents of $\mu$ are defined by

$\lambda^{+}(f,\mu):=\int\lambda^{+}(f,x)d\mu(x),\;\lambda^{-}(f,\mu):=\int\lambda^{-}(f,x)d\mu(x).$

In both cases, $\lambda^{+}(f,\mu)\geq\lambda^{-}(f,\mu)$.

By the subadditive ergodic theorem, the largest Lyapunov exponent also
satisfies

$\lambda^{+}(f,\mu)=\lim_{n\to\infty}\frac{1}{n}\int\log\|Df^{n}_{y}\|\,d\mu(y)=\inf_{n}\frac{1}{n}\int\log\|Df^{n}_{y}\|\,d\mu(y).$
(8)

Throughout this paper, an ergodic invariant probability measure $\mu$ is
called _hyperbolic_ if one of its Lyapunov exponents is positive, and the
other is negative (sometimes this is called hyperbolic of saddle-type). If
$\mu$ is hyperbolic, we sometimes write $\lambda^{u}=\lambda^{+}$,
$\lambda^{s}=\lambda^{-}$, $E^{u}=E^{+}$ and $E^{s}=E^{-}$.

###  3.4. Semi-continuity of Lyapunov exponents

We will use the dynamics of the projective tangent bundle to study the semi-
continuity properties of $(f,\mu)\mapsto\lambda^{+}(f,\mu)$ (and by symmetry
of $(f,\mu)\mapsto\lambda^{-}(f,\mu)$). The principal tool is the function

$\varphi:\widehat{M}\to{\mathbb{R}}\ ,\varphi(x,E)=\log\|Df_{x}|_{E}\|.$

Notice that if $f$ is a $C^{1}$ diffeomorphism, then $\varphi$ is bounded and
uniformly continuous. We will make frequent use of the following identity:

$\log\|Df^{n}|_{E}\|=\sum_{k=0}^{n-1}(\varphi\circ\widehat{f}^{k})(x,E)\ \ \
((x,E)\in\widehat{M}).$ (9)

This is because $E$ is a one-dimensional subspace of $T_{x}M$, and therefore
by the chain rule
$\|Df^{n}_{x}|_{E}\|=\prod_{k=0}^{n-1}\|Df_{f^{k}(x)}|_{Df^{k}_{x}(E)}\|=\prod_{k=0}^{n-1}\exp[(\varphi\circ\widehat{f}^{k})(x,E)]$.
Equation (9) presents the subadditive cocycle $\log\|Df^{n}|_{E}\|$ for $f$ as
an additive cocycle for $\widehat{f}$. See [28, Prop. 5.1 on p. 328] for a
proof of a more general fact (and [19, Lemma 8.7] for the first use of a
related idea).

######  Lemma 3.3.

Suppose $f\in\operatorname{Diff}^{1}(M)$, and $\mu$ is an ergodic
$f$-invariant probability measure. Then:

  1. (1)

$\lambda$ is a Lyapunov exponent of $\mu$ iff $\mu$ has an
$\widehat{f}$-ergodic lift $\widehat{\mu}$ s.t. $\int\varphi
d\widehat{\mu}=\lambda$;

  2. (2)

If $\mu$ has two different Lyapunov exponents, then it has exactly two ergodic
$\widehat{f}$–invariant lifts:

$\widehat{\mu}^{+}:=\int_{\widehat{M}}\delta_{(x,E^{+}(x))}d\mu(x)\text{, and
}\widehat{\mu}^{-}:=\int_{\widehat{M}}\delta_{(x,E^{-}(x))}d\mu(x).$

Moreover $\int\varphi d\widehat{\mu}^{\pm}=\lambda^{\pm}(f,\mu)$.

For any $\widehat{f}$-invariant probability measure, it will be convenient to
denote

$\widehat{\lambda}(\widehat{f},\widehat{\mu}):=\int_{\widehat{M}}\varphi
d\widehat{\mu}.$

When $\mu$ is hyperbolic, the lifts $\widehat{\mu}^{+},\widehat{\mu}^{-}$ are
called the unstable and stable lifts of $\mu$.

###### Proof.

If $\mu$ has equal Lyapunov exponents, (1) follows from eq. (9) and the
ergodic theorem. Otherwise, by Oseledets theorem, there are two a.e. defined
sections $x\mapsto E^{\pm}_{x}$ s.t. $Df_{x}E^{\pm}_{x}=E^{\pm}_{f(x)}$. Every
$\widehat{f}$-invariant probability measure carried by the graph of an
invariant section $x\mapsto E_{x}$ is ergodic, and coincides with
$\int_{M}\delta_{(x,E_{x})}\,d\mu(x)$. So (1) and (2) follow from Lemma 3.2
and (9) (see [28]). ∎

######  Corollary 3.4.

Suppose $f\in\operatorname{Diff}^{1}(M)$ and $\mu$ is an $f$-invariant
probability measure (not necessarily ergodic) s.t.
$\lambda^{+}(f,x)>\lambda^{-}(f,x)$ for $\mu$-a.e. $x\in M$. Then any
$\widehat{f}$-invariant lift $\widehat{\mu}$ of $\mu$ is carried by
$\mathrm{graph}(E^{+})\cup\mathrm{graph}(E^{-})$, and there are unique
$\widehat{f}$-invariant lifts $\widehat{\mu}^{+},\widehat{\mu}^{-}$ s.t.
$\widehat{\mu}^{+}(\mathrm{graph}(E^{+}))=1$,
$\widehat{\mu}^{-}(\mathrm{graph}(E^{-}))=1$.

###### Proof.

By a general Borel construction, the graphs of $E^{+},E^{-}$ are measurable.
There are unique lifts $\widehat{\mu}^{+},\widehat{\mu}^{-}$ of $\mu$ to
$\mathrm{graph}(E^{+}),\mathrm{graph}(E^{-})$, and it is easy to check using
the identity $Df_{x}(E^{\pm}_{x})=E^{\pm}_{f(x)}$ that $\widehat{\mu}^{\pm}$
are $\widehat{f}$-invariant.

Now let $\widehat{\mu}$ be an arbitrary lift of $\mu$ and consider its ergodic
decomposition $\widehat{\mu}=\int\widehat{\mu}_{\xi}\,dm$. For almost every
$\xi$, the ergodic measure
$\mu_{\xi}=\widehat{\mu}_{\xi}\circ\widehat{\pi}^{-1}$ has two different
exponents, hence, by Lemma 3.3, its lift $\widehat{\mu}_{\xi}$ is some
combination $a(\xi)\widehat{\mu}_{\xi}^{+}+(1-a(\xi))\widehat{\mu}_{\xi}^{-}$
of its lifts $\widehat{\mu}^{\pm}_{\xi}$ where $0\leq a(\xi)\leq 1$. Observe
that $a(\xi)=\widehat{\mu}_{\xi}(\operatorname{graph}(E^{+}))$, hence the
function $\xi\mapsto a(\xi)$ is measurable. It follows that

$\widehat{\mu}=\int
a(\xi)\widehat{\mu}_{\xi}^{+}+(1-a(\xi))\widehat{\mu}_{\xi}^{-}\,dm.$

In particular, any lift $\widehat{\mu}$ is carried by the union of the graphs
of $E^{+}$ and $E^{-}$. ∎

For each $n\geq 1$, $(f,\mu)\mapsto\int\log\|Df^{n}_{x}\|\,d\mu(x)$ is
continuous as a function on
$\operatorname{Diff}^{1}(M)\times\\{\text{probability measure on $M$}\\}$. Eq.
(8) now give us the following “folklore” fact:

######  Theorem 3.5 (Upper semicontinuity).

Let $f_{k}\in\operatorname{Diff}^{1}(M)$ be diffeomorphisms with ergodic
invariant probability measures $\nu_{k}$. If $f_{k}\overset{\scriptscriptstyle
C^{1}}{\longrightarrow}f$ and $\nu_{k}\overset{w^{*}}{\longrightarrow}\mu$,
then the largest Lyapunov exponents $\lambda^{+}(f_{k},\nu_{k})$ satisfy
$\limsup_{k}\lambda^{+}(f_{k},\nu_{k})\leq\lambda^{+}(f,\mu).$

The next result computes the defect in continuity
$\lambda^{+}(f,\mu)-\limsup\lambda^{+}(f_{k},\nu_{k})$ in terms of the
dynamics on the projective bundle. It relates the defect in continuity to the
escape of some of the mass of the lifts to $\mathrm{graph}(E^{+})$ to the
vicinity of $\mathrm{graph}(E^{-})$.

Let $\widehat{\mu}$ be an $\widehat{f}$-invariant probability measure with
ergodic decomposition $\widehat{\mu}=\int_{\Omega}\widehat{\mu}_{\xi}dm$. The
projection $\mu$ of $\widehat{\mu}$ to $M$ has ergodic decomposition
$\mu=\int_{\Omega}\mu_{\xi}\,dm$, where
$\mu_{\xi}=\widehat{\mu}_{\xi}\circ\pi^{-1}$. Note that in general, the map
$\widehat{\mu}_{\xi}\mapsto\mu_{\xi}$ is not injective.

We split the set of ergodic components $\widehat{\mu}_{\xi}$ $(\xi\in\Omega)$
by considering whether they are carried by the invariant line bundles $E^{+}$
or $E^{-}$ or by a subset of $\widehat{M}$ where these bundles are not
defined:

$\displaystyle\Omega^{-}:=\\{\xi\in\Omega:\lambda^{-}(f,\mu_{\xi})<\lambda^{+}(f,\mu_{\xi})\
,\ \widehat{\mu}_{\xi}=\widehat{\mu}_{\xi}^{-}\\}$
$\displaystyle\Omega^{+}:=\\{\xi\in\Omega:\lambda^{-}(f,\mu_{\xi})<\lambda^{+}(f,\mu_{\xi})\
,\ \widehat{\mu}_{\xi}=\widehat{\mu}_{\xi}^{+}\\}$
$\displaystyle\Omega^{0}:=\\{\xi\in\Omega:\lambda^{-}(f,\mu_{\xi})=\lambda^{+}(f,\mu_{\xi})\\}.$

######  Theorem 3.6 (Defect in continuity).

Let $M$ be a compact smooth boundaryless surface. For each $k\geq 1$, fix
$f_{k}\in\operatorname{Diff}^{1}(M)$ and ergodic $f_{k}$-invariant measures
$\nu_{k}$ with $\lambda^{-}(f_{k},\nu_{k})<\lambda^{+}(f_{k},\nu_{k})$. Let
$\widehat{\nu}_{k}^{+}$ be the ergodic lift to $\widehat{M}$ carried by the
bundle $E^{+}$. Suppose $f_{k}\overset{\scriptscriptstyle
C^{1}}{\longrightarrow}f$, $\nu_{k}\overset{w^{*}}{\longrightarrow}\mu$, and
$\widehat{\nu}_{k}^{+}\overset{w^{*}}{\longrightarrow}\widehat{\mu}.$
Considering the ergodic decompositions
$\widehat{\mu}=\int_{\Omega}\widehat{\mu}_{\xi}dm$,
$\mu=\int_{\Omega}\mu_{\xi}\,dm$ and defining
$\Omega=\Omega^{+}\cup\Omega^{-}\cup\Omega^{0}$ as above, we have

$\displaystyle\lim_{k\to\infty}\lambda^{+}(f,\nu_{k})$
$\displaystyle=\lambda^{+}(f,\mu)-\int_{\Omega^{-}}[\lambda^{+}(f,\mu_{\xi})-\lambda^{-}(f,\mu_{\xi})]dm.$

In the special case when $\mu$ is ergodic, we have the following (see Lemma
3.3):

  1. (1)

If $\lambda^{+}(f,\mu)=\lambda^{-}(f,\mu)$ or
$\widehat{\mu}=\widehat{\mu}^{+}$ , then
$\lim_{k}\lambda^{+}(f_{k},\nu_{k})=\lambda^{+}(f,\mu).$

  2. (2)

If $\lambda^{+}(f,\mu)<\lambda^{-}(f,\mu)$ and
$\widehat{\mu}\neq\widehat{\mu}^{+}$, then
$\lim_{k}\lambda^{+}(f_{k},\nu_{k})<\lambda^{+}(f,\mu)$.
More precisely, there is a unique $0<a\leq 1$ such that
$\widehat{\mu}=a\widehat{\mu}^{-}+(1-a)\widehat{\mu}^{+}$ and

$\lim_{k\to\infty}\lambda^{+}(f_{k},\nu_{k})=\lambda^{+}(f,\mu)-a(\lambda^{+}(f,\mu)-\lambda^{-}(f,\mu)).$

If $\nu_{k}$ and $\mu$ are hyperbolic (of saddle type), then $a\neq 1$.

###### Proof.

Using Lemma 3.3, we see that

$\int\varphi d\widehat{\mu}_{\xi}=\begin{cases}\int\varphi
d\widehat{\mu}_{\xi}^{+}=\lambda^{+}(f,\mu_{\xi})&\text{ if
}\xi\in\Omega^{+}\\\ \int\varphi
d\widehat{\mu}_{\xi}^{-}=\lambda^{-}(f,\mu_{\xi})&\text{ if
}\xi\in\Omega^{-}\\\ \lambda^{+}(f,\mu_{\xi})&\text{ if
}\xi\in\Omega^{0},\end{cases}$ (10)
$\lim_{k\to\infty}\lambda^{+}(f_{k},\nu_{k})=\lim_{k\to\infty}\int\varphi
d\widehat{\nu}_{k}^{+}=\int\varphi d\widehat{\mu}=\iint\varphi
d\widehat{\mu}_{\xi}dm.$ (11)

Substituting (10) in (11), we obtain

$\displaystyle\lim_{k\to\infty}\lambda^{+}(f_{k},\nu_{k})$
$\displaystyle=\int_{\Omega^{-}}\lambda^{-}(f,\mu_{\xi})dm+\int_{\Omega^{+}\cup\Omega^{0}}\lambda^{+}(f,\mu_{\xi})dm$
$\displaystyle=\int_{\Omega^{-}}\lambda^{-}(f,\mu_{\xi})dm+\int_{\Omega}\lambda^{+}(f,\mu_{\xi})dm-\int_{\Omega^{-}}\lambda^{+}(f,\mu_{\xi})dm$
$\displaystyle=\lambda^{+}(f,\mu)-\int_{\Omega^{-}}[\lambda^{+}(f,\mu_{\xi})-\lambda^{-}(f,\mu_{\xi})]dm,$

which proves the theorem for general, possibly non-ergodic, limits $\mu$.

When $\mu$ is ergodic, we apply equation (11) and Lemma 3.3:

(1) If $\lambda^{+}(f,\mu)=\lambda^{-}(f,\mu)$ or
$\widehat{\mu}=\widehat{\mu}^{+}$, then
$\int\varphi\,d\widehat{\mu}=\lambda^{+}(f,\mu)$ and item (1) follows.

(2) Otherwise, $\mu$ has two different exponents and $m(\Omega^{+})<1$.
Necessarily, $m(\Omega^{0})=0$, $a:=m(\Omega^{-})\in(0,1]$ and the lift
$\widehat{\mu}$ can be written as $(1-a)\widehat{\mu}^{+}+a\widehat{\mu}^{-}$.
So

$\displaystyle\lim_{k\to\infty}\lambda^{+}(f_{k},\nu_{k})=\int\varphi\,d\widehat{\mu}$
$\displaystyle=(1-a)\lambda^{+}(f,\mu)+a\lambda^{-}(f,\mu)$
$\displaystyle=\lambda^{+}(f,\mu)-a(\lambda^{+}(f,\mu)-\lambda^{-}(f,\mu))$
$\displaystyle<\lambda^{+}(f,\mu).$

Finally, note that, if $a=1$ and $\mu$ is hyperbolic, then
$\lim_{k}\lambda^{+}(f_{k},\nu_{k})=\lambda^{-}(f,\mu)<0$, and $\nu_{k}$ are
not hyperbolic of saddle type for $k$ large enough. ∎

Notice that the defect in continuity originates at $\Omega^{-}$, the set of
ergodic components of $\lim\widehat{\nu}_{k}^{+}$ which are carried by
$\mathrm{graph}(E^{-})$. This confirms the heuristic that discontinuity in
Lyapunov exponents is due to the asymptotic escape of mass from
$\mathrm{graph}(E^{+})$, which carries $\widehat{\nu}_{k}^{+}$, to
$\mathrm{graph}(E^{-})$, which carries $\widehat{\mu}_{\xi}$ for
$\xi\in\Omega^{-}$.

###  3.5. A bound for the asymptotic dilation of $\widehat{f}$

######  Lemma 3.7.

For any $C^{2}$ diffeomorphism $f$ of a surface $M$,
$\lambda(\widehat{f})\leq\lambda(f)+\lambda(f^{-1})$.

###### Proof.

Working locally in charts, we identify the iterates $f^{n}$ locally with
diffeomorphisms $F_{i}\colon U_{i}\to{\mathbb{R}}^{2}$ defined on open subsets
of ${\mathbb{R}}^{2}$. We choose the charts so that the change of coordinates
distorts the metric by a factor of less than $2$. The lift
$\widehat{f}^{n}\in\operatorname{Diff}^{1}(\widehat{M})$ is identified with:

$\widehat{F}_{i}(x,v)=\left(F_{i}(x),\frac{(DF_{i})_{x}v}{\|(DF_{i})_{x}v\|}\right).$

In what follows, we omit the first factor.

The differential of $\widehat{F}_{i}$ can be computed in a straightforward way
(writing $a\ast b$ for the scalar product of two vectors in $\mathbb{R}^{2}$):

$\displaystyle(D\widehat{F}_{i})_{(x,v)}(y,w)=$
$\displaystyle\frac{(DF_{i})_{x}w}{\|(DF_{i})_{x}v\|}+\frac{(D^{2}F_{i})_{x}(y).v}{\|(DF_{i})_{x}v\|}$
(12)
$\displaystyle\quad-\left(\frac{(DF_{i})_{x}v}{\|(DF_{i})_{x}v\|}\ast\frac{(DF_{i})_{x}w+(D^{2}F_{i})_{x}(y).v}{\|(DF_{i})_{x}v\|}\right)\frac{(DF_{i})_{x}(v)}{\|(DF_{i})_{x}v\|}.$

Thus,

$\|(D\widehat{F}_{i})_{(x,v)}\|\leq
2\left(\|(DF_{i})_{x}\|\,\|(DF_{i}^{-1})_{f(x)}\|+\|(D^{2}F_{i})_{x}\|\,\|(DF_{i}^{-1})_{f(x)}\|\right).$

We apply this to some iterate of $f$ on $M$, remembering the distortion in the
metric:

$\|D\widehat{f}^{n}\|_{\sup}\leq
32\|Df^{n}\|_{\sup}\|Df^{-n}\|_{\sup}+16\|Df^{-n}\|_{\sup}\|D^{2}f^{n}\|_{\sup}$

Let $0<\varepsilon<1/4$. Fix $n$ an integer so large that $32\leq
e^{\varepsilon n/4}$ and

$\|Df^{n}\|_{\sup}\leq e^{n(\lambda(f)+\varepsilon/8)}\text{ and
}\|Df^{-n}\|_{\sup}\leq e^{n(\lambda(f^{-1})+\varepsilon/8)}.$

Therefore,

$\lambda(\widehat{f})\leq\lambda(f)+\lambda(f^{-1})+\varepsilon/2+\log\left(1+e^{-\lambda(f)}\|D^{2}f^{n}\|_{\sup}^{1/n}\right)$

By dilating the metric on $M$, we can ensure that
$\|D^{2}f^{n}\|_{\sup}^{1/n}\leq(\varepsilon/4)e^{\lambda(f)}$ without
changing the asymptotic dilations. Thus
$\lambda(\widehat{f})\leq\lambda(f)+\lambda(f^{-1})+\varepsilon$ As
$\varepsilon$ is arbitrarily small, the claim follows. ∎

These computations allow the following control of the $C^{r-1}$ size of the
lift of a $C^{r}$ diffeomorphism.

######  Lemma 3.8.

For every real $2<r<\infty$, there is a constant $A=A(r)$ with the following
property. For any $g\in\operatorname{Diff}^{r}(M)$ with lift
$\widehat{g}\in\operatorname{Diff}^{r-1}(\widehat{M})$,

$\|\widehat{g}\|_{C^{r-1}}\leq
A\left(\|g\|_{C^{r}}\cdot\|Dg^{-1}\|_{\sup}\right)^{A}.$

###### Proof.

We first consider the $k$th derivative of $f$ in charts for the maximal
integer $k\leq r$. A straightforward induction on the integer $k\geq 2$ based
on eq. (12) shows that the $(k-1)$th differential of $\widehat{F}$ at some
point $(x,v)\in\widehat{M}$ can be written as a linear combination of terms:

$\frac{1}{\|D_{x}F.v\|^{p}}(D^{\alpha_{1}}F*D^{\beta_{1}}F)\dots(D^{\alpha_{j}}F*D^{\beta_{j}}F)\cdot
D^{\gamma}F$

where $p,\alpha_{1},\beta_{1}\dots,\alpha_{j},\beta_{j},\gamma$ are integers,
and $\alpha_{1},\beta_{1},\dots,\gamma\leq k$. The coefficients of this linear
combination depend only on $k$.

If $k=r$, the claim is immediate. If $\alpha:=r-k>0$, recall that the $C^{r}$
size is the sum of the $C^{k}$ size and $\alpha$-Hölder size of the $k$-th
derivative. A further computation using the above expression gives the
required bound for the Hölder constant of order $\alpha$ of
$D^{k}\widehat{g}$. ∎

##  4\. Entropy formulas and reparametrizations

We saw in last section that the defect in continuity of
$(f,\mu)\mapsto\lambda^{+}(f,\mu)$ can be described in terms of the canonical
lift $\widehat{f}:\widehat{M}\to\widehat{M}$. In this section we develop tools
for studying the entropy map $(f,\mu)\mapsto h(f,\mu)$ in terms of
$\widehat{f}:\widehat{M}\to\widehat{M}$.

Specifically, we will show that the entropy of hyperbolic measures on $M$ can
be studied in terms of the exponential rate of growth in $C^{r}$–complexity of
$\widehat{f}^{n}\circ\widehat{\sigma}$, where
$\widehat{\sigma}:[0,1]\to\widehat{M}$ is the curve
$\widehat{\sigma}(t)=(\sigma(t),\mathbb{R}.\sigma^{\prime}(t))$ and
$\sigma:[0,1]\to M$ is a smooth parameterization of a local unstable manifold.

###  4.1. Review of entropy and the ergodic decomposition

This section collects several classical facts on the entropy theory of non-
ergodic measures. For proofs and details, see [15, chap. 13].

Consider a compact metric space $X$ together with a continuous map $T$
preserving an invariant Borel probability measure $\mu$ and the
$\sigma$-algebra $\mathcal{X}$ of Borel subsets of $X$. The ergodic
decomposition of $\mu$ with respect to $T$ is:

$\mu=\int_{X}\mu_{x}\,d\mu(x)$

where $\mu_{x}:=\lim_{n\to\infty}\frac{1}{n}\sum_{0\leq k<n}\delta_{T^{k}x}$
(the weak-$*$ limit exists almost everywhere by the ergodic theorem).

The map $x\mapsto\mu_{x}$ is $\mu$-measurable with respect to the
$\sigma$-algebra $\mathcal{I}$ of invariant measurable subsets; for $\mu$-a.e.
$x\in X$, $\mu_{x}$ is a $T$-invariant and ergodic Borel probability measure
and for every Borel $\mu$-absolutely integrable $u$, $x\mapsto\mu_{x}(u)$
belongs to $L^{1}(\mu)$ and $\mu(u)=\int_{X}\mu_{x}(u)\,d\mu(x)$. The metric
entropy of $\mu$ and $\mu_{x}$ are related by

$h(T,\mu)=\int_{X}h(T,\mu_{x})d\mu(x).$

Suppose $\xi$ is a countable measurable partition of $X$ with finite mean
entropy $H_{\mu}(\xi):=-\sum_{A\in\xi}\mu(A)\log\mu(A)$. Let
$\xi_{n}:=\\{\bigcap_{i=0}^{n-1}T^{-i}A_{i}:A_{i}\in\xi\\}$. It is a classical
fact that $h(T,\mu,\xi):=\lim_{n\to\infty}\frac{1}{n}H_{\mu}(\xi_{n})$ exists.

Similarly, one defines $h(T,\mu_{x},\xi)$. The function $x\mapsto
h(T,\mu_{x},\xi)$ is defined $\mu$-a.e., is measurable with respect to the
$\mu$-completion of $\mathcal{I}$. Let $\xi_{n}(x)$ denote the atom of
$\xi_{n}$ which contains $x$. The Shannon-McMillan-Breiman theorem, in its
version for non-ergodic measures [15, (13.4)], states that,

$\lim_{n\to\infty}-\frac{1}{n}\log\mu\left(\xi_{n}(x)\right)=h(T,\mu_{x},\xi)\quad\text{both
$\mu$-a.e. and in }L^{1}(\mu).$ (13)

In addition, we have the following identity: [15, (13.3)]:
$h(T,\mu,\xi)=\int_{X}h(T,\mu_{x},\xi)\,d\mu(x).$ In particular,
$h(T,\mu,\xi)\leq\overline{h}(T,\mu,\xi)$, where

$\overline{h}(T,\mu):=\operatorname{ess-sup}_{x\in X}h(T,\mu_{x}).$ (14)

We call $\overline{h}(T,\mu)$ the essential supremum entropy of $\mu$:

###  4.2. Bowen and Katok entropy formulas

Let $T:X\to X$ be a continuous map on a compact metrix space $X$. An
$(n,\epsilon)$-Bowen ball is a set of the form

$B_{T}(x,n,\varepsilon):=\\{y\in X:\forall 0\leq
k<n,\;d(T^{k}y,T^{k}x)<\varepsilon\\}.$

The $(n,\varepsilon)$-covering number of a subset $Z\subset X$, is

$r_{T}(n,\varepsilon,Z):=\min\\{{|C|}:\bigcup_{x\in
C}B_{T}(x,n,\varepsilon)\supset Z\\}.$

Bowen [8] defined the topological entropy of a (possibly non-invariant) set
$Z\subset X$ for $T$ to be

$h_{\operatorname{top}}(T,Z)=\lim_{\varepsilon\to
0}h_{\operatorname{top}}(T,Z,\varepsilon)\text{ with
}h_{\operatorname{top}}(T,Z,\varepsilon)=\limsup_{n\to\infty}\frac{1}{n}\log
r_{T}(n,\varepsilon,Z),$ (15)

and showed that the topological entropy of $T$ is
$h_{\operatorname{top}}(T)=h_{\operatorname{top}}(T,X)$.

Katok gave a similar formula for the metric entropy of an invariant measure.
Let $\mu$ be an invariant probability measure. For every $\gamma\in(0,1)$, let

$r_{T}(n,\varepsilon,\mu,\gamma):=\inf\\{r_{T}(n,\varepsilon,Z):Z\subset
X\text{ measurable s.t. }\mu(Z)>\gamma\\}.$

He showed that if $\mu$ is ergodic, then
$\displaystyle{h}(T,\mu)=\lim_{\lambda\to 1}\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log r_{T}(n,\varepsilon,\mu,\gamma).$
Katok’s proof in [26] also works in the non-ergodic case, if we replace the
usual Shannon-McMillan-Breiman Theorem by (13). The result is that for a
general (possily non-ergodic) invariant probability measure $\mu$,

$\overline{h}(T,\mu)=\lim_{\lambda\to 1}\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log r_{T}(n,\varepsilon,\mu,\gamma).$ (16)

Here $\overline{h}(T,\mu)$ is the essential supremum entropy from (14).

###  4.3. The lift to the projective tangent bundle preserves entropy

This is a standard consequence of the following theorem [30].

######  Theorem 4.1 (Ledrappier-Walters).

Let $(X,T)$, $(Y,S)$ be continuous self-maps of compact metric spaces and let
$\pi:(Y,S)\to(X,T)$ be a topological factor map s.t.

$\forall x\in X\qquad h_{\operatorname{top}}(S,\pi^{-1}(x))=0.$ (17)

Then, for any $S$-invariant Borel probability measure $\nu$ on $Y$,
$h(S,\nu)=h(T,\pi_{*}\nu).$

Thus by the variational principle, in the setup of the theorem,
$h_{\operatorname{top}}(T)=h_{\operatorname{top}}(S)$.

######  Corollary 4.2.

Suppose $f\in\operatorname{Diff}^{1}(M)$ and $\widehat{f}$ is the canonical
extension of $f$ to the projective bundle $\widehat{M}$. Then
$h_{\operatorname{top}}(\widehat{f})=h_{\operatorname{top}}(f)$, and for every
$\widehat{f}$-invariant probability measure $\widehat{\nu}$ with projection
$\nu$, $h(\widehat{f},\widehat{\nu})=h(f,\nu)$ and
$\overline{h}(\widehat{f},\widehat{\nu})=\overline{h}(f,\nu)$.

###### Proof.

We check condition (17) for $T=f$, $S=\widehat{f}$, and apply the previous
theorem. $\widehat{M}$ is a topological bundle over $M$, and its fibers
$P_{x}M$ are homeomorphic to circles. The map $\widehat{f}:P_{x}M\to
P_{f(x)}M$ is a homeomorphism. For every $\varepsilon>0$, one can find
partitions $\xi_{x}$ of $P_{x}M$ into a bounded number of arcs with diameter
at most $\varepsilon$. It is easy to see that
$\xi_{x}\vee\widehat{f}^{-1}\xi_{\widehat{f}\widehat{x}}\vee\dots\vee\widehat{f}^{-n+1}\xi_{\widehat{f}^{n-1}\widehat{x}}$
has cardinality at most
$\sum_{k=0}^{n-1}\operatorname{Card}(\xi_{\widehat{f}^{k}\widehat{x}})=O(n)$.
It follows that $h_{top}(\widehat{f},P_{x}M)=0$ for all $x$. ∎

###  4.4. Bowen and Katok entropy formulas on the bundle $\widehat{M}$.

We need a variant of the Bowen and Katok entropy formulas which uses a
different type of Bowen balls, which are better adapted to the bundle
structure of $\widehat{M}$.

Recall the natural projection $\widehat{\pi}:\widehat{M}\to M$,
$\widehat{\pi}(x,E)=x$. The fibered ball with center
$\widehat{x}\in\widehat{M}$ and scales $\varepsilon,\widehat{\varepsilon}>0$
is the set

$B(\widehat{x},\varepsilon,\widehat{\varepsilon}):=\\{\widehat{y}\in\widehat{M}:\;d(\widehat{x},\widehat{y})<\widehat{\varepsilon}\text{
and }d(\widehat{\pi}(\widehat{x}),\widehat{\pi}(\widehat{y}))<\varepsilon\\}.$

A set $S\subset\widehat{M}$ has fibered size
$\leq(\varepsilon,\widehat{\varepsilon})$ if $S\subset
B(\widehat{x},\varepsilon,\widehat{\varepsilon})$ for some $\widehat{x}$.

Suppose $f\in\operatorname{Diff}^{1}(M)$ and $\widehat{f}$ is the canonical
extension of $f$ to $\widehat{M}$. The fibered
$(n,\varepsilon,\widehat{\varepsilon})$–Bowen ball with center
$\widehat{x}\in\widehat{M}$, size $(\varepsilon,\widehat{\varepsilon})$ and
length $n$ is the set

$\begin{split}B_{\widehat{f}}(\widehat{x},n,\varepsilon,\widehat{\varepsilon}):=\\{\widehat{y}\in\widehat{M}:\;\forall
0\leq
k<n,\;d(&\widehat{f}^{k}(\widehat{x}),\widehat{f}^{k}(\widehat{y}))<\widehat{\varepsilon}\\\
&\text{ and
}d(f^{k}(\widehat{\pi}(\widehat{x})),f^{k}(\widehat{\pi}(\widehat{y})))<\varepsilon\\}.\end{split}$
(18)

The $(n,\varepsilon,\widehat{\varepsilon})$-covering number of a subset
$Z\subset\widehat{M}$ is the minimal number of fibered
$(n,\varepsilon,\widehat{\varepsilon})$-Bowen balls whose union contains $Z$.
It is denoted by

$r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},Z).$

Clearly if $\varepsilon_{1}\leq\varepsilon_{2}$ and
$\widehat{\varepsilon}_{1}\leq\widehat{\varepsilon}_{2}$, then
$r_{\widehat{f}}(n,\varepsilon_{1},\widehat{\varepsilon}_{1},Z)\geq
r_{\widehat{f}}(n,\varepsilon_{2},\widehat{\varepsilon}_{2},Z)$.

Similarly, given an ergodic measure $\widehat{\mu}$ of $\widehat{f}$ and a
number $0<\gamma<1$, the
$(n,\varepsilon,\widehat{\varepsilon},\gamma)$-covering number of
$\widehat{\mu}$ is the minimal number of fibered
$(n,\varepsilon,\widehat{\varepsilon})$-Bowen balls whose union has
$\widehat{\mu}$-measure at least $\gamma$. It is denoted by

$r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{\mu},\gamma).$

If
$\varepsilon_{1}\leq\varepsilon_{2},\widehat{\varepsilon}_{1}\leq\widehat{\varepsilon}_{2}$,
and $\gamma_{1}\geq\gamma_{2}$, then
$r_{\widehat{f}}(n,\varepsilon_{1},\widehat{\varepsilon}_{1},\widehat{\mu},\gamma_{1})\geq
r_{\widehat{f}}(n,\varepsilon_{2},\widehat{\varepsilon}_{2},\widehat{\mu},\gamma_{2})$.

######  Proposition 4.3.

Fix $f\in\operatorname{Diff}^{1}(M)$ with canonical lift $\widehat{f}$. We
have:

  1. (1)

Bowen’s formula: For every $\widehat{\varepsilon}>0$,

$h_{\operatorname{top}}(f)=\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log
r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{M})=\lim_{\varepsilon\to
0}\liminf_{n\to\infty}\frac{1}{n}\log
r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{M})$

  2. (2)

Katok’s formula: Suppose $\mu$ is an $f$-ergodic invariant measure, and let
$\widehat{\mu}$ be an $\widehat{f}$-ergodic lift of $\mu$. Then for every
$\widehat{\varepsilon}>0$ and $0<\gamma<1$,

$h(f,\mu)=\lim_{\varepsilon\to 0}\limsup_{n\to\infty}\frac{1}{n}\log
r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{\mu},\gamma),$

If $\mu$ is $f$-invariant, but possibly not ergodic, then for every lift
$\widehat{\mu}$,

$\overline{h}(f,\mu)=\lim_{\gamma\to 1}\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log
r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{\mu},\gamma),$

###### Proof.

We follow the proof of [8, Thm 17]. We need the following claim.

######  Claim 4.4.

For every $\widehat{\varepsilon},\alpha>0$, there are $C,\varepsilon_{*}>0$
such that

$\forall\varepsilon\in(0,\varepsilon_{*}),\;\forall n\geq 1,\;\forall x\in
M,\quad
r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{\pi}^{-1}B_{f}(x,n,\varepsilon))\leq
Ce^{\alpha n}.$

Proof of the Claim. Let $\widehat{\varepsilon},\alpha>0$. Note first that

$r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{\pi}^{-1}B_{f}(x,n,\varepsilon)){\leq}r_{\widehat{f}}(n,\widehat{\varepsilon},\widehat{\pi}^{-1}B_{f}(x,n,\varepsilon)),$

hence it is enough to bound the latter. Since
$h_{\operatorname{top}}(\widehat{f},\widehat{\pi}^{-1}(x))=0$, for each $x\in
M$ there is a smallest integer $n_{x}\geq 1$ such that
$r_{\widehat{f}}(n_{x},\widehat{\varepsilon}/2,\widehat{\pi}^{-1}(x))\leq
e^{\alpha n_{x}}$.

Recall that a set-valued function $F$ from a topological space $X$ to the set
of subsets of a topological set $Y$ is called upper semi-continuous, if for
every $E\subset Y$ closed, $\\{x\in X:F(x)\cap E\neq\varnothing\\}$ is closed.
The continuity of $\widehat{\pi}$ and the compactness of $\widehat{M}$ implies
that $x\mapsto\widehat{\pi}^{-1}(x)$ is upper semi-continuous.

It follows that if $\widehat{\pi}^{-1}(x)$ is contained in some open set $U$
(say the union of a minimal cover by fibered Bowen balls), then
$\widehat{\pi}^{-1}(y)$ is contained in $U$ for all $y$ sufficiently close to
$x$. Hence there is an $r_{x}>0$ such that

$r_{\widehat{f}}(n_{x},\widehat{\varepsilon}/2,\widehat{\pi}^{-1}B(x,r_{x}))\leq
r_{\widehat{f}}(n_{x},\widehat{\varepsilon}/2,\widehat{\pi}^{-1}(x))\leq
e^{\alpha n_{x}}.$ (19)

Using a compactness argument, we see that $n_{*}:=\sup\\{n_{x}:x\in M\\}$ is
finite and that one can arrange for ${\varepsilon_{*}}:=\inf\\{r_{x}:x\in
M\\}$ to be positive.

Let $\varepsilon\in(0,\varepsilon_{*})$, $n\geq 1$ and $x\in M$. Define
$t_{0}(x):=0$ and $t_{i+1}(x):=t_{i}(x)+n_{f^{t^{i}(x)}x}$. Choose $i\geq 0$
maximal such that $t_{i}(x)\leq n$. Note that $0\leq n-t_{i}(x)<n_{*}$. Thus,
setting $x_{j}:=f^{t_{j}(x)}x$ and $n_{j}:=n_{x_{j}}$,

$B_{f}(x,n,\varepsilon)=\bigcap_{j=0}^{i-1}f^{-t_{j}(x)}B_{f}(x_{j},n_{j},{\varepsilon})\cap
f^{-t_{i}(x)}B(x_{i},n-t_{i}(x),{\varepsilon}).$

Let $\\{z_{1},\dots,z_{C}\\}$ be a $(n_{*}-1,\widehat{\varepsilon}/2)$-cover
of $\widehat{M}$ with cardinality $C$. For each $0\leq j<i$, let
$\\{y_{j1},\dots,y_{jm}\\}$ with $m\leq e^{\alpha n_{j}}$ be the
$(n_{j},\widehat{\varepsilon}/2)$-cover of
$\widehat{\pi}^{-1}B(x_{j},n_{j},\varepsilon)$ implied by eq. (19). Then
$\widehat{\pi}^{-1}B_{f}(x,n,\varepsilon)$ is contained in the following
union:

$\bigcup_{\tiny\begin{array}[]{c}k,k_{0},\dots,k_{i-1}\\\ k_{j}\leq\exp\alpha
n_{j}\\\ k\leq
C\end{array}}\bigcap_{j=0}^{i-1}\widehat{f}^{-t_{j}(x)}B_{\widehat{f}}(y_{jk_{j}},n_{j},\widehat{\varepsilon}/2)\cap\widehat{f}^{-t_{i}(x)}B_{\widehat{f}}(z_{k},n-t^{i}(x),\widehat{\varepsilon}/2).$

Any two points contained in the same term of this union are not
$(n,\widehat{\varepsilon})$-separated. So the cardinality of any
$(n,\widehat{\varepsilon})$-separated subset of
$\widehat{\pi}^{-1}B_{f}(x,n,\varepsilon)$ is bounded by

$C\times\prod_{j=0}^{i-1}e^{\alpha n_{j}}\leq Ce^{\alpha n}.$

Since a maximal $(n,\widehat{\varepsilon})$–separated subset is the set of
centers of a cover by $(n,\widehat{\varepsilon})$–Bowen balls,
$r_{\widehat{f}}(n,\widehat{\varepsilon},\widehat{\pi}^{-1}B_{f}(x,n,\varepsilon))\leq
Ce^{\alpha n}$, which proves the claim.

The claim implies that for any $\widehat{\varepsilon},\alpha>0$, there are
$C,\varepsilon_{*}>0$ such that, for any $0<\varepsilon\leq\varepsilon_{*}$
and any ${Z}\subset M$,

$r_{f}(n,\varepsilon,{Z})\leq
r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{\pi}^{-1}{Z})\leq
Ce^{\alpha n}r_{f}(n,\varepsilon,{Z}).$

The proposition now follows from the classical identities (15) and (16). ∎

###  4.5. Curves and $C^{r}$ reparametrizations

The entropy of a diffeomorphism can be related to the exponential rate of
growth in $C^{r}$ complexity of the iterates of a local unstable manifold. To
do this we need to control the curvature, and for this purpose it is useful to
lift the curve to projective bundle $\widehat{M}$ and study its iterations
there. Here we develop the tools needed for doing this.

######  Definition 4.5.

A $C^{r}$ curve $\sigma:[0,1]\to M$ is _regular_ if its derivative
$\sigma^{\prime}(t)$ never vanishes. In this case, it has a _canonical lift_
$\widehat{\sigma}:[0,1]\to\widehat{M}$ defined by

$\widehat{\sigma}(t):=(\sigma(t),{\mathbb{R}}.\sigma^{\prime}(t)).$

Here and throughout,
$\mathbb{R}.\sigma^{\prime}(t)\equiv\mathrm{span}\\{\sigma^{\prime}(t)\\}$ is
a linear subspace of $T_{x}M$. By regularity, $\mathbb{R}.\sigma^{\prime}(t)$
is one-dimensional.

######  Definition 4.6.

Fix $r\geq 2$ and let $\varepsilon,\widehat{\varepsilon}$ be two positive
numbers. A regular $C^{r}$ curve $\sigma:[0,1]\to M$ has _$C^{r}$ size_ (or
just size) less than $(\varepsilon,\widehat{\varepsilon})$ if

$\|\sigma\|_{C^{r}}{<}\varepsilon\text{ and
}\|\widehat{\sigma}\,\|_{C^{r-1}}{<}\widehat{\varepsilon}.$

The curve has _diameter_ less than $(\varepsilon,\widehat{\varepsilon})$ if

${\operatorname{diam}}_{M}(\sigma([0,1]))<\varepsilon\text{ and
}{\operatorname{diam}}_{\widehat{M}}(\widehat{\sigma}([0,1]))<\widehat{\varepsilon}.$

To say that a curve has finite $C^{r}$ size implies that it is regular and
$C^{r}$.

######  Remark 4.7.

If a curve $\sigma$ is parametrized by length (i.e.,
$\|\sigma^{\prime}(t)\|=1$ for all $t$), then it has size at most
$(\|\sigma\|_{C^{r}},C\|\sigma^{\prime}\|_{C^{r-1}})$ for some constant $C>0$
which depends on the choice of $C^{r-1}$ atlas of $\widehat{M}$ used to define
$C^{r-1}$ size (see section 3.1).

Suppose $\sigma:[0,1]\to M$ is a curve. If we cut $[0,1]$ into small intervals
$[a_{i},b_{i}]$ and reparametrize $\sigma|_{[a_{i},b_{i}]}$ by
$\sigma\circ\psi_{i}$ where $\psi_{i}[0,1]\to[a_{i},b_{i}]$ is an affine
bijection, then
$\|\sigma\circ\psi_{i}\|_{C^{r}}\leq\kappa\|\sigma\|_{C^{r}},\|\widehat{\sigma}\circ\psi_{i}\|_{C^{r}}\leq\kappa\|\widehat{\sigma}\|_{C^{r}}$,
where $\kappa=|a_{i}-b_{i}|<1$. Cutting sufficiently finely, we can obtain
covering by pieces with affine reparametrizations with $C^{r}$ size as small
as we wish.

Yomdin measured the $C^{r}$ complexity of a curve (more generally a set) by
counting how many reparametrized pieces with $C^{r}$ size less than $1$ are
needed to cover it. We adapt this to the projective dynamics:

######  Definition 4.8.

Let $\sigma:[0,1]\to M$ be a $C^{r}$ curve. A _reparametrization of $\sigma$_
is a non-constant affine map $\psi\colon[0,1]\to[0,1]$.
A _family of reparametrizations of $\sigma$ over a subset $T\subset[0,1]$_ is
a collection $\mathcal{R}$ of reparametrizations such that
$T\subset\bigcup_{\psi\in\mathcal{R}}\psi([0,1])$.

Let $f\in\operatorname{Diff}^{r}(M)$ and let $\sigma$ be a regular $C^{r}$
curve. We will be interested in families of reparametrizations which remain
bounded in $C^{r}$ size after application of $f^{n}$ for certain $n$.
Specifically, fix numbers $\varepsilon,\widehat{\varepsilon}>0$, an integer
$N\geq 1$, and $T\subset[0,1]$.

######  Definition 4.9.

A reparametrization $\psi$ of $\sigma$ is
$(C^{r},f,N,\varepsilon,\widehat{\varepsilon})$-admissible up to time $n$, if
there exists an increasing sequence $(n_{0},n_{1},\dots,n_{\ell})$ such that

  * •

$n_{0}=0$, $n_{\ell}=n$, and $n_{j}-n_{j-1}\leq N$ for each $1\leq j\leq\ell$,

  * •

for each $0\leq j\leq\ell$ the curve $f^{n_{j}}\circ\sigma\circ\psi$ has
$C^{r}$ size less than $(\varepsilon,\widehat{\varepsilon})$.

We call the integers $n_{j}$ the admissible times.

A family $\mathcal{R}$ of reparametrizations of $\sigma$ over $T$ is
_$(C^{r},f,N,\varepsilon,\widehat{\varepsilon})$ -admissible up to time_ $n$,
if each $\psi\in\mathcal{R}$ is
$(C^{r},f,N,\varepsilon,\widehat{\varepsilon})$-admissible up to time $n$.

######  Lemma 4.10 (Concatenation of reparametrizations).

Let $f\in\operatorname{Diff}^{r}(M)$, $r\geq 2$, and consider a regular
$C^{r}$ curve $\sigma$ on $M$, and $T\subset[0,1]$. Suppose that

  1. $\bullet$

$\mathcal{R}$ is a family of reparametrizations of $\sigma$ over $T$;

  2. $\bullet$

$\mathcal{R}$ is $(C^{r},f,N,\varepsilon,\widehat{\varepsilon})$-admissible up
to time $n$;

  3. $\bullet$

for each $\psi\in\mathcal{R}$, there is a family $\mathcal{R}_{\psi}$ of
reparametrizations of $f^{n}\circ\sigma\circ\psi$ over $\psi^{-1}(T)$, which
is $(C^{r},f,N,\varepsilon,\widehat{\varepsilon})$-admissible up to time
$n^{\prime}$.

Then
$\mathcal{R}^{\prime}:=\bigcup_{\psi\in\mathcal{R}}\\{\psi\circ\phi:\phi\in\mathcal{R}_{\psi}\\}$
is a family of reparameterizations of $\sigma$ over $T$, and
$\mathcal{R}^{\prime}$ is
$(C^{r},f,N,\varepsilon,\widehat{\varepsilon})$-admissible up to time
$n+n^{\prime}$.

###### Proof.

Since $\psi,\phi:[0,1]\to[0,1]$ are non-constant and affine,
$\psi\circ\phi:[0,1]\to[0,1]$ is non-constant and affine. Next,

$\bigcup_{\psi\in\mathcal{R}}\bigcup_{\phi\in\mathcal{R}_{\psi}}(\psi\circ\phi)[0,1]=\bigcup_{\psi\in\mathcal{R}}\psi\bigl{(}\bigcup_{\phi\in\mathcal{R}_{\psi}}\phi[0,1]\bigr{)}\supseteq\bigcup_{\psi\in\mathcal{R}}\psi(\psi^{-1}(T))=\bigcup_{\psi\in\mathcal{R}}T\cap\psi[0,1]=T,$

So $\mathcal{R}^{\prime}$ is a family of reparametrizations of
$(\sigma,\widehat{\sigma})$ over $T$.

To see that $\mathcal{R}^{\prime}$ is admissible, fix
$\psi\in\mathcal{R},\phi\in\mathcal{R}_{\psi}$, and choose admissible times
$0=n_{0}<n_{1}<\cdots<n_{\ell}=n$ and
$0=n_{0}^{\prime}<n_{1}^{\prime}<\cdots<n_{m}^{\prime}=n^{\prime}$ for $\psi$
and $\phi$. Let

$n_{\ell+k}:=n+n_{k}^{\prime}\ \ (k=0,\ldots,m).$

We claim that $0=n_{0}<n_{1}<\cdots<n_{\ell+m}=n+n^{\prime}$ are admissible
times for $\psi\circ\phi$. That the gaps are no larger than $N$ is clear. If
$j=\ell+1,\ldots,\ell+m$, then $f^{n_{j}}\circ\psi\circ\phi$ is a regular
curve with size $<(\varepsilon,\widehat{\varepsilon}),$ because
$\phi\in\mathcal{R}_{\psi}$ and $\mathcal{R}_{\psi}$ is admissible. If
$j\leq\ell$, then using the fact that $\phi^{\prime}=c$ with $c$ a constant
s.t. $|c|\leq 1$, we find that
$\|f^{n_{j}}\circ\sigma\circ\psi\circ\phi\|_{C^{r}}\leq\|f^{n_{j}}\circ\sigma\circ\psi\|_{C^{r}}\cdot|c|\leq\|f^{n_{j}}\circ\sigma\circ\psi\|_{C^{r}}$.
This is less than $\varepsilon$, because of the admissibility of
$\mathcal{R}$. Similarly
$\|\widehat{f}^{n_{j}}\circ\widehat{\sigma}\circ\psi\circ\phi\|_{C^{r-1}}\leq\|\widehat{f}^{n_{j}}\circ\widehat{\sigma}\circ\psi\|_{C^{r-1}}<\widehat{\varepsilon}$.
∎

######  Lemma 4.11 (Length of reparametrizations).

Let $f\in\operatorname{Diff}^{r}(M)$, $r\geq 2$. For any $\eta>0$ and $N\geq
1$, there exist $\varepsilon_{*}>0$ and a $C^{2}$-neighborhood
$\mathcal{U}_{*}$ of $f$ in $\operatorname{Diff}^{r}(M)$ with the following
property.

Consider $g\in\mathcal{U}_{*}$,
$\varepsilon,\widehat{\varepsilon}\in(0,\varepsilon_{*})$, a regular $C^{r}$
curve $\sigma$, and a reparametrization $\varphi$ of $\sigma$ which is
$(C^{r},g,N,\varepsilon,\widehat{\varepsilon})$-admissible up to time $n$.
Then for any $(x,E)\in\widehat{\sigma}[0,1]$,

$\operatorname{Length}(g^{n}\circ\sigma\circ\varphi)<e^{\eta
n/10}\|Dg^{n}_{x}|_{E}\|\operatorname{Length}(\sigma\circ\varphi).$

###### Proof.

$\widehat{M}$ is compact, so one can find $\varepsilon_{*}>0$ and a small
$C^{2}$-neighborhood $\mathcal{U}_{*}$ of $f$ in $\operatorname{Diff}^{r}(M)$
such that for all $g\in\mathcal{U}_{*}$ and
$\widehat{x},\widehat{y}\in\widehat{M}$ satisfying
$d(\widehat{x},\widehat{y})<\varepsilon_{*}$,

$\log\|Dg|_{\widehat{g}^{k}(\widehat{x})}\|\leq\log\|Dg|_{\widehat{g}^{k}(\widehat{y}}\|)+\tfrac{\eta}{10}\qquad(0\leq
k\leq N).$

Let us consider $g\in\mathcal{U}_{*}$, a curve $\sigma$ and a
reparametrization $\varphi$ as in the statement, with admissibility times
$n_{0},\dots,n_{\ell}$. One gets for any $0\leq i<\ell$,

$\forall(x,E),(y,F)\in\widehat{g}^{n_{i}}\circ\widehat{\sigma}[0,1],\qquad\|Dg^{n_{i+1}-n_{i}}_{y}|_{F}\|<e^{(n_{i+1}-n_{i})\eta/10}\|Dg^{n_{i+1}-n_{i}}_{x}|_{E}\|,$

which immediately implies the conclusion of the lemma. ∎

Admissible families of reparametrizations yield covers by fibered Bowen balls
with size of the same order of magnitude.

######  Lemma 4.12 (Bowen covers from admissible reparametrizations).

Let $f\in\operatorname{Diff}^{r}(M)$, $r\geq 2$, $T\subset[0,1]$, and $\sigma$
be a regular $C^{r}$ curve. Fix $\varepsilon_{*},\widehat{\varepsilon}_{*}>0$
and $N\geq 1$. Let $\mathcal{R}$ be a family of reparametrizations of $\sigma$
over $T$ which is
$(C^{r},f,N,\varepsilon_{*},\widehat{\varepsilon}_{*})$-admissible up to time
$n$. Then, for every $\varepsilon,\widehat{\varepsilon}>0$,

$r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{\sigma}(T))\leq\frac{2\widehat{\varepsilon}_{\ast}\|D\widehat{f}\|_{\sup}^{N}}{\min(\varepsilon,\widehat{\varepsilon})}|\mathcal{R}|.$

###### Proof.

Let $L:=\|D\widehat{f}\|_{\sup}$. $L\geq 1$ since $\widehat{f}$ is surjective.

Let
${\displaystyle\rho:=\frac{\min(\varepsilon,\widehat{\varepsilon})}{\widehat{\varepsilon}_{\ast}L^{N}}}$,
and construct a $\rho$–dense set $C_{\rho}\subset[0,1]$ s.t.
$|C_{\rho}|\leq\frac{2}{\rho}$. Set
$C_{\rho}(\mathcal{R}):=\\{(\widehat{\sigma}\circ\psi)(t^{\prime}):t^{\prime}\in
C_{\rho},\psi\in\mathcal{R}\\}$ .

Fix $\psi\in\mathcal{R}$ with admissible times
$0=n_{0}<n_{1}<\cdots<n_{\ell}=n$. Then $n_{j+1}-n_{j}\leq N$ and the regular
curve $f^{n_{j}}\circ\sigma\circ\psi$ has $C^{r}$ size at most
$(\varepsilon_{\ast},\widehat{\varepsilon}_{\ast})$, whence

$\|D\widehat{f}^{k}\circ\sigma\circ\psi\|_{\sup}\leq\widehat{\varepsilon}_{\ast}L^{k-n_{j}}\
\text{ for }k\in[n_{j},n_{j+1}).$

In particular,
$\|D\widehat{f}^{k}\circ\sigma\circ\psi\|_{\sup}\leq\widehat{\varepsilon}_{\ast}L^{N}\text{
for all }0\leq k\leq n.$

For every $\widehat{x}\in\widehat{\sigma}(T\cap\psi[0,1])$ with
$\widehat{x}=\widehat{\sigma}\circ\psi(t)$, there is
$\widehat{y}=\widehat{\sigma}\circ\psi(t^{\prime})$ with $t^{\prime}\in
C_{\rho}$ such that $|t-t^{\prime}|<\rho$, whence

$d(\widehat{f}^{k}(\widehat{x}),\widehat{f}^{k}(\widehat{y}))<\rho\widehat{\varepsilon}_{\ast}L^{N}\text{
and
}d(f^{k}(\widehat{\pi}(\widehat{x})),f^{k}(\widehat{\pi}(\widehat{y})))<\rho\widehat{\varepsilon}_{\ast}L^{N}\text{
for all }k=0,\ldots,n.$

(the second inequality follows from the first). Notice that $\widehat{y}\in
C_{\rho}(\mathcal{R})$.

All this shows that
$\displaystyle\widehat{\sigma}(T)\subset\bigcup_{\widehat{y}\in
C_{\rho}(\mathcal{R})}B_{\widehat{f}}(\widehat{y},{n,}\rho\widehat{\varepsilon}_{\ast}L^{N},\rho\widehat{\varepsilon}_{\ast}L^{N})\subset\bigcup_{\widehat{y}\in
C_{\rho}(\mathcal{R})}B_{\widehat{f}}(\widehat{y},n,\varepsilon,\widehat{\varepsilon}).$
So
$r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{\sigma}(T))\leq|C_{\rho}(\mathcal{R})|\leq|\mathcal{R}|\cdot|C_{\rho}|\leq\frac{2}{\rho}|\mathcal{R}|\leq\frac{2\widehat{\varepsilon}_{\ast}}{\min(\varepsilon,\widehat{\varepsilon})}\|D\widehat{f}\|_{\sup}^{N}|\mathcal{R}|$.
∎

###  4.6. Yomdin estimates

In this section we discuss a converse to Lemma 4.12: Covers by Bowen balls
generate admissible reparametrizations with cardinality of the same order of
magnitude. This result is much more delicate than Lemma 4.12, and requires
Yomdin’s Theorem [48]. Here is the tool we need from Yomdin’s work, in a form
adapted to our setup. Let

$Q_{r}(g):=\max(\|g\|_{C^{r}},\|\widehat{g}\|_{C^{r-1}})\ ,\
Q_{r,N}(g):=\max_{n=1,2,\dots,N}Q_{r}(g^{n}).$ (20)

######  Theorem 4.13 (Yomdin).

Given real numbers $2\leq r<\infty$ and $Q>0$, there are
$\Upsilon=\Upsilon(r)>0$ and $\varepsilon_{Y}=\varepsilon_{Y}(r,Q)>0$ with the
following properties. For every

  1. $\bullet$

$C^{r}$ diffeomorphism $g:M\to M$ such that $Q_{r}(g)\leq Q$

  2. $\bullet$

regular $C^{r}$ curve $\sigma$ with $C^{r}$ size at most
$(\varepsilon,\widehat{\varepsilon})$ with
$0<\varepsilon,\widehat{\varepsilon}\leq\varepsilon_{Y}$,

  3. $\bullet$

$\widehat{x}\in\widehat{\sigma}[0,1]$, and
$T:=\\{t\in[0,1]:\widehat{g}(\widehat{\sigma}(t))\in
B(\widehat{g}(\widehat{x}),\varepsilon,\widehat{\varepsilon})\\}.$

there exists a family $\mathcal{R}$ of reparametrizations of $\sigma$ over $T$
such that

  1. (1)

for every $\psi\in\mathcal{R}$, $g\circ\sigma\circ\psi$ is a regular $C^{r}$
curve with $C^{r}$ size at most $(\varepsilon,\widehat{\varepsilon})$,

  2. (2)

$|\mathcal{R}|\leq\Upsilon\|D\widehat{g}\|_{\sup}^{1/(r-1)}$.

###### Proof.

The reader who will compare this theorem to Yomdin’s original statement in
[48] will find that the two results are nearly the same, except for the
following differences: (1) Yomdin considered the more general case of
$\sigma:[0,1]^{\ell}\to M$ whereas we restrict to $\ell=1$; (2) Yomdin did not
specify that all reparametrizations are affine as we do; (3) Our result allows
$r$ to be real, not just an integer; and (4) We use
$(n,\varepsilon,\widehat{\varepsilon})$ balls in $\widehat{M}$, whereas Yomdin
used $(n,\varepsilon)$ balls in $M$.

In the special case of curves $\ell=1$, Yomdin’s proof works verbatim with
affine reparametrizations, see, e.g., [48, p. 297–298]. The extension to non-
integer smoothness is also simple and well-known, see, e.g., [13, p.133].

To deal with (4), we apply the Yomdin’s original theorem twice, first for
$(g,\sigma)$ on $M$ and then for the lift $(\widehat{g},\widehat{\sigma})$ on
$\widehat{M}$. This yields numbers $\Upsilon_{i}=\Upsilon_{i}(r)$ $(i=1,2)$
and ${\varepsilon_{Y}}=\varepsilon_{Y}(r,Q)>0$ as follows. Suppose
$0<\varepsilon,\widehat{\varepsilon}\leq\varepsilon_{Y}$,
$g\in\operatorname{Diff}^{r}(M)$, $Q_{r}(g)\leq Q$, and $\sigma:[0,1]\to M$
has $C^{r}$ size at most $(\varepsilon,\widehat{\varepsilon})$. Then

(1) There is a family $\mathcal{R}_{1}$ of reparametrizations of
$g\circ\sigma$ over $T$ s.t.
$|\mathcal{R}_{1}|\leq\Upsilon_{1}\|Dg\|_{\sup}^{1/r}$, and so that each
$\psi\in\mathcal{R}_{1}$ is affine, contracting, and
$\|g\circ\sigma\circ\psi\|_{C^{r}}\leq\varepsilon$.

(2) There is a family $\mathcal{R}_{2}$ of reparametrizations of
$\widehat{g}\circ\widehat{\sigma}$ over $T$ s.t.
$|\mathcal{R}_{2}|\leq\Upsilon_{2}\|D\widehat{g}\|_{\sup}^{\frac{1}{r-1}}$ and
so that each $\psi\in\mathcal{R}_{2}$ is affine, contracting, and
$\|\widehat{g}\circ\widehat{\sigma}\circ\psi\|_{C^{r}}\leq\widehat{\varepsilon}$.

Each family of reparametrizations $\mathcal{R}_{i}$ generates a cover of $T$
by the intervals $\psi([0,1])$, $\psi\in\mathcal{R}_{i}$. Without loss of
generality, the interiors of these intervals are pairwise disjoint (otherwise
discard some of them and shrink the rest by composing the reparametrizations
by affine contractions). We define

$\mathcal{R}:=\\{\phi_{\psi_{1},\psi_{2}}\mid(\psi_{1},\psi_{2})\in\mathcal{R}_{1}\times\mathcal{R}_{2}\text{
s.t. }\psi_{1}((0,1))\cap\psi_{2}((0,1))\neq\varnothing\\}$

where $\phi_{\psi_{1},\psi_{2}}:[0,1]\to\psi_{1}([0,1])\cap\psi_{2}([0,1])$ is
an affine diffeomorphism. The image of this new family of reparametrizations
contains the intersection of the images of $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$, hence $\mathcal{R}$ is a family of reparametrizations of
$\sigma$ over $T$.

As the reparametrizations are affine and contracting, we see that

$\|g\circ\sigma\circ\phi_{\psi_{1},\psi_{2}}\|_{C^{r}}\leq\|g\circ\sigma\circ\psi_{1}\|_{C^{r}}{<}\varepsilon$

Likewise for $\widehat{g}\circ\widehat{\sigma}\circ\phi_{\psi_{1},\psi_{2}}$.
Since $\widehat{g}\circ\widehat{\sigma}\circ\phi_{\psi_{1},\psi_{2}}$
coincides with the lift of $g\circ\sigma\circ\phi_{\psi_{1},\psi_{2}}$, item
(1) of the theorem holds.

Next, using the order structure on the interval, it is not difficult to show
that
$|\mathcal{R}|\leq|\mathcal{R}_{1}|+|\mathcal{R}_{2}|-1\leq\Upsilon_{1}\|Dg\|_{\sup}^{1/r}+\Upsilon_{2}\|D\widehat{g}\|_{\sup}^{\frac{1}{r-1}}.$
Obviously $\|Dg\|_{\sup}^{1/r}\leq\|D\widehat{g}\|_{\sup}^{1/(r-1)}$, so item
(2) holds with $\Upsilon:=\Upsilon_{1}+\Upsilon_{2}$. ∎

######  Corollary 4.14 (Existence of admissible reparametrizations).

For $2\leq r<\infty$, $Q>0$, let $\Upsilon(r)$, ${\varepsilon_{Y}}(r,Q)$ be
the constants from Theorem 4.13. Suppose

  1. $\bullet$

$g\in\operatorname{Diff}^{r}(M)$ and $Q_{r,N}(g)<Q$,

  2. $\bullet$

$\sigma:[0,1]\to M$ is a regular curve with $C^{r}$ size
$\leq(\varepsilon,\widehat{\varepsilon})$, where
$0<\varepsilon,\widehat{\varepsilon}<\varepsilon_{Y}$,

  3. $\bullet$

$N,n\geq 1$, and $T\subset[0,1]$.

Then there exists a family $\mathcal{R}$ of reparametrizations of $\sigma$
over $T$, which is $(C^{r},g,N,\varepsilon,\widehat{\varepsilon})$-admissible
up to the time $n$, and with cardinality

$|\mathcal{R}|\leq
C(r,\widehat{g})r_{\widehat{g}}(n,\varepsilon,\widehat{\varepsilon},\widehat{g}\circ\widehat{\sigma}(T)),$

where
$C(r,\widehat{g})=\Upsilon(r)^{\lceil\frac{n}{N}\rceil}\|D\widehat{g}^{N}\|_{\sup}^{\frac{\lfloor\frac{n}{N}\rfloor}{r-1}}\|D\widehat{g}^{n-N\lfloor\frac{n}{N}\rfloor}\|_{\sup}^{\frac{1}{r-1}}.$

###### Proof.

Fix $n\geq 1$ and divide with remainder $n=qN+p$, $q\geq 0$, $p=0,\ldots,N-1$.

Let
${\ell}:=r_{\widehat{g}}(n,\varepsilon,\widehat{\varepsilon},\widehat{g}\circ\widehat{\sigma}(T))$,
then there exists a cover of ${\widehat{g}\circ}\widehat{\sigma}(T)$ by
${\ell}$ fibered $(n,\varepsilon,\widehat{\varepsilon})$–Bowen balls
$B_{i}:=B(\widehat{g}(\widehat{x}_{i}),{n,}\varepsilon,\widehat{\varepsilon})$.
For each $B_{i}$, we will construct an admissible family of reparametrizations
of $\sigma$ over $T_{i}:=(\widehat{g}\circ\widehat{\sigma})^{-1}(B_{i}),$ and
then take the union over $i$.

Step 0. If $p=0$ move to step 1. Otherwise proceed as follows.

Fix $1\leq i\leq{\ell}$. Yomdin’s theorem for $g^{p}$, ${\sigma}$ and
$\widehat{x}_{i}$ gives a family of reparametrizations $\mathcal{R}_{0}$ of
$\sigma$ over

$T_{i}^{0}:=\\{t\in[0,1]:\widehat{g}^{p}(\widehat{\sigma}(t))\in
B(\widehat{g}^{p}(\widehat{x}_{i}),\varepsilon,\widehat{\varepsilon})\\},$

which is $(C^{r},{g,}N,\varepsilon,\widehat{\varepsilon})$-admissible up to
time $p$ (the admissible times are $0,p$), and such that
$|\mathcal{R}_{0}|\leq\Upsilon\|D\widehat{g}^{p}\|^{1/(r-1)}_{{\sup}}.$ Notice
that $T_{i}^{0}\supset T_{i}$. If $q=0$, we have a reparametrization up to
time $n$, and we stop.

Step 1. Fix $\psi\in\mathcal{R}_{0}$ and apply Yomdin’s theorem to $g^{N}$,
$g^{p}\circ\sigma\circ\psi$, and $\widehat{g}^{p}(\widehat{x}_{i})$. The
result is a family $\mathcal{R}_{\psi}$ of reparametrizations of
$g^{p}\circ\sigma\circ\psi$ over

$T_{i}^{1}(\psi):=\\{t\in[0,1]:\widehat{g}^{N+p}[(\widehat{\sigma}\circ\psi)(t)]\in
B(\widehat{g}^{N+p}(\widehat{x}_{i}),\varepsilon,\widehat{\varepsilon})\\}$

which is $(C^{r},{g,}N,\varepsilon,\widehat{\varepsilon})$-admissible up to
time $N$ (the admissible times are $0,N$), and with cardinality
$|\mathcal{R}_{\psi}|\leq\Upsilon\|D\widehat{g}^{N}\|^{1/(r-1)}_{{\sup}}$.

Notice that $T_{i}^{1}(\psi)\supset\psi^{-1}(T_{i})$, therefore the
concatenation

$\mathcal{R}_{1}:=\\{\psi\circ\phi:\psi\in\mathcal{R}_{0},\phi\in\mathcal{R}_{\psi}\\}$

is an admissible family of reparametrizations of $\sigma$ over $T_{i}$ up to
time $N+p$, with admissible times $0,p,N+p$ and cardinality

$|\mathcal{R}_{1}|\leq\Upsilon^{2}\|D\widehat{g}^{N}\|^{1/(r-1)}_{{\sup}}\|D\widehat{g}^{p}\|^{1/(r-1)}_{{\sup}}.$

If $q=1$, we have a reparametrization up to time $n$, and we stop.

Otherwise we continue as before to a “step 2” which applies Yomdin’s theorem
to $g^{N}$, $g^{N+p}\circ\sigma\circ\psi$ and
$\widehat{g}^{N+p}(\widehat{x}_{i})$.

Eventually, at step $q$, we arrive to a family of reparametrizations
$\mathcal{R}_{q}$ over $T_{i}$ which is admissible up to time $qN+p=n$, and
which has cardinality

${|\mathcal{R}_{q}|\leq\Upsilon^{q+1}\|D\widehat{g}^{N}\|^{q/(r-1)}_{{\sup}}\|D\widehat{g}^{p}\|^{1/(r-1)}_{{\sup}}\equiv\Upsilon^{\lceil
n/N\rceil}\|D\widehat{g}^{N}\|_{\sup}^{\frac{\lfloor
n/N\rfloor}{r-1}}\|D\widehat{g}^{n-N\lfloor n/N\rfloor}\|_{\sup}^{1/(r-1)}.}$

Taking the union over $i=1,\ldots,{\ell}$, we obtain the family of
reparametrizations over $\bigcup T_{i}\supset T$, as required. ∎

###  4.7. Entropy and growth of $C^{r}$ complexity of unstable manifolds

In this section $f$ is a $C^{r}$ diffeomorphism, $r>1$, of a surface $M$ and
$\nu$ is an ergodic hyperbolic probability measure of saddle type, ie
$\lambda^{s}:=\lambda^{-}(f,\nu)$ is strictly negative, and
$\lambda^{u}:=\lambda^{+}(f,\nu)$ is strictly positive. Pesin’s Unstable
Manifold Theorem says that $\nu$–a.e. $x$ belongs to an unstable manifold
$W^{u}(x)$, which is an injectively immersed $C^{r}$ curve and is
characterized as:

$W^{u}(x)=\\{y\>:\>\limsup_{n\to+\infty}\frac{1}{n}\log
d(f^{-n}(x),f^{-n}(y))<0\\}.$

A measurable partition $\xi$ is _subordinated_ to the unstable lamination
$W^{u}$ of $\nu$ if for $\nu$-almost every $x\in M$, the atom $\xi(x)$ is a
neighborhood of $x$ inside the curve $W^{u}(x)$ and $\xi$ is increasing: Every
atom of $f(\xi)$ is a union of atoms of $\xi$. By [29], such measurable
partitions exist.

Since $\xi$ is measurable, Rokhlin’s disintegration theorem applies, and for
$\nu$–a.e. $x$ there exists a probability measure $\nu^{u}_{x}$ on $\xi(x)$ so
that

$\nu=\int\nu^{u}_{x}d\nu(x).$

The family $\\{\nu^{u}_{x}\\}$ is not unique, but given $\xi$, any two
families like that are equal outside a set of $x$ of measure zero. Therefore
it is not a serious abuse of terminology to call $\nu^{u}_{x}$ the conditional
measure on $\xi(x)$.

In this section we use the entropy theory of Ledrappier and Young [27] and
especially the following corollary established by Zang (see [50, Remark 1.8])
to show that the entropy of $\nu$ can be bounded by the exponential rate of
growth of the $C^{r}$ complexity of the curve $f^{n}(W^{u}_{loc}(x))$, as
quantified in the previous section using admissible $C^{r}$ reparametrizations
up to time $n$.

######  Theorem 4.15 (Y. Zang).

Let us consider $f\in\operatorname{Diff}^{r}(M)$ with $r>1$, an ergodic
hyperbolic probability measure $\nu$, and a system of conditional measures
$\\{\nu^{u}_{x}\\}$ on local unstable manifolds. Then for $\nu$-a.e. $x\in M$,
the measure $\nu^{u}_{x}$ satisfies:

$h(f,\nu)=\inf_{\gamma>0}\lim_{\varepsilon\to
0}\liminf_{n\to\infty}\frac{1}{n}\log
r_{f}(n,\varepsilon,\nu^{u}_{x},\gamma).$

The difference between this result and Ledrappier-Young theory is that Zang
assumes $C^{r}$ smoothness for some $r>1$ and hyperbolicity, whereas
Ledrappier and Young assume $C^{2}$ smoothness, but no hyperbolicity.

######  Corollary 4.16.

Let $f\in\operatorname{Diff}^{2}(M)$, and let $\nu$ be an ergodic hyperbolic
measure with a system of conditional measures $\\{\nu^{u}_{x}\\}$ on local
unstable manifolds. For any $F\subset M$ with positive $\nu$-measure, for
$\nu$-a.e. $x_{0}\in F$, and for any choice of

  * •

$\sigma\colon[0,1]\to W^{u}(x_{0})$, a regular $C^{r}$ curve,

  * •

$T\subset[0,1]$, a set such that $\nu^{u}_{x_{0}}(\sigma(T)\cap F)>0$,

the following holds. If $\mathcal{R}_{n}$ $(n\geq 1)$ are families of
reparametrizations of $\sigma$ over $T$ which are
$(C^{r},f,N,\varepsilon_{*},\widehat{\varepsilon}_{*})$-admissible up to time
$n$ for some (any) $\varepsilon_{\ast},\widehat{\varepsilon}_{\ast}>0,N\geq 1$
independent of $n$, then $\displaystyle
h(f,\nu)\leq\liminf_{n\to\infty}\frac{1}{n}\log|\mathcal{R}_{n}|.$

###### Proof.

Fix $\varepsilon_{\ast},\widehat{\varepsilon}_{\ast},N>0$ and let
$\mathcal{R}_{n}$ be families of admissible reparametrizations as in the
statement. Let $\widehat{\sigma}:[0,1]\to\widehat{M}$ be the canonical lift of
the regular curve $\sigma$. Fix $\varepsilon,\widehat{\varepsilon}>0$
arbitrarily small. By Lemma 4.12,

$r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{\sigma}(T))\leq
C|\mathcal{R}_{n}|,$

where
$C:=C(f,\varepsilon,\widehat{\varepsilon},\widehat{\varepsilon}_{\ast},N):={\frac{2\widehat{\varepsilon}_{\ast}\|D\widehat{f}\|_{\sup}^{N}}{\min(\varepsilon,\widehat{\varepsilon})}}$
is independent of $n$.

By the definition of
$r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{\sigma}(T))$,
there exist $\widehat{x}_{1},\ldots,\widehat{x}_{\ell}\in\widehat{M}$ with
$\ell\leq C|\mathcal{R}_{n}|$ such that
$\bigcup_{i=1}^{\ell}{B_{\widehat{f}}}(\widehat{x}_{i},n,\varepsilon,\widehat{\varepsilon})\supset\widehat{\sigma}(T)$.
Necessarily

$\bigcup_{i=1}^{\ell}{B_{f}}(\pi(\widehat{x}_{i}),n,\varepsilon)\supset\sigma(T).$

In particular, $r_{f}(n,\varepsilon,\sigma(T))\leq
r_{\widehat{f}}(n,\varepsilon,\widehat{\varepsilon},\widehat{\sigma}(T))\leq
C|\mathcal{R}_{n}|,$ whence by Zang’s Theorem,
$h(f,\nu)\leq\lim\limits_{\varepsilon\to
0}\liminf\limits_{n\to\infty}\frac{1}{n}\log
r_{f}(n,\varepsilon,{\sigma(T)\cap
F})\leq\liminf\limits_{n\to\infty}\frac{1}{n}\log|\mathcal{R}_{n}|.$ ∎

##  5\. Main reparametrization lemmas

This section collects our main technical results on the existence of
admissible families of reparametrizations of pieces of unstable manifolds.

The point is to produce families with cardinality as small as possible. The
first result provides admissible families of reparametrizations of local
unstable manifolds, with cardinality controlled in terms of the entropy. The
second result, which is much more subtle, produces much smaller families of
reparametrizations for the subset of the local unstable stable where there is
little expansion up to some iterate, see Definition 6.1.

###  5.1. Statements

Throghout this section $M$ is a compact smooth surface without boundary,
$f:M\to M$ is a diffeomorphism, $\widehat{f}:\widehat{M}\to\widehat{M}$ is the
canonical lift (7), and $\widehat{\mu}$ is a (possibly non-ergodic!)
$\widehat{f}$-invariant probability measure, which projects to an
$f$-invariant measure $\mu$. $Q_{r,N}(f)$ is given by (20) in the previous
section, $\overline{h}$ is the essential entropy (14), and
$\lambda(\widehat{f})$ is the asymptotic dilation of $\widehat{f}$, see (2)
and §3.5.

The statements of the following two propositions should be formally understood
as stating the existence of functions $N_{1},\overline{n}_{1},$ $\gamma_{0}$,
and $N_{0}$ with values in $(0,\infty)$ such that the following stated
properties hold.

######  Proposition 5.1.

Let us consider $f\in\operatorname{Diff}^{r}(M)$ with $2\leq r<\infty$, an
$\widehat{f}$-invariant probability $\widehat{\mu}$, some real numbers
${Q},\eta,\gamma,\varepsilon,\widehat{\varepsilon}>0$, and integers $N,n$.
Assume that:

  * –

$N\geq N_{1}(r,f,{\eta})$,

  * –

$0<\varepsilon,\widehat{\varepsilon}\leq{\varepsilon_{Y}(r,{Q})}$ (the
constant in Yomdin’s theorem 4.13), and

  * –

$n\geq\overline{n}_{1}(f,\widehat{\mu},\eta,\gamma,N,\varepsilon,\widehat{\varepsilon})$.

Then there are:

  * –

a $C^{2}$ neighborhood
$\mathcal{U}_{1}(f,\eta,\gamma,\varepsilon,\widehat{\varepsilon},N,n)$ of $f$
in $\operatorname{Diff}^{r}(M)$,

  * –

an open set
$\widehat{U}_{1}(f,\widehat{\mu},\eta,\gamma,\varepsilon,\widehat{\varepsilon},n)\subset\widehat{M}$
with $\widehat{\mu}(\widehat{U}_{1})>1-\gamma^{2}$ and
$\widehat{\mu}(\partial\widehat{U}_{1})=0$,

such that the following property holds:

  1. (*)

For any $g\in\mathcal{U}_{1}$ with $Q_{r,N}(g)<{Q}$ and for any regular curve
$\sigma$ with $C^{r}$ size at most $(\varepsilon,\widehat{\varepsilon})$,
there is a family $\mathcal{R}$ of reparametrizations over
$\widehat{\sigma}^{-1}(\widehat{U}_{1})$ s.t.

     * –

$\mathcal{R}$ is $(C^{r},g,N,\varepsilon,\widehat{\varepsilon})$-admissible up
to time $n$,

     * –

$|\mathcal{R}|\leq\exp\left[n\left(\overline{h}({f},\mu)+\frac{\lambda({\widehat{f}})}{r-1}+\eta\right)\right]$,

The following and key estimate applies to the part of the local unstable
manifold which does not (initially) see much expansion. More precisely,
suppose $g$ is a diffeomorphism with canonical lift $\widehat{g}$, and let
$\alpha>0$. An _orbit segment_ with length $n$ is a string
$\vartheta:=(\widehat{x},\widehat{g}(\widehat{x}),\ldots,\widehat{g}^{n-1}(\widehat{x}))$.

######  Definition 5.2.

An orbit segment
$\vartheta=(\widehat{x},\widehat{g}(\widehat{x}),\ldots,\widehat{g}^{n-1}(\widehat{x}))$
is $\alpha$-neutral, if, denoting $\widehat{x}=(x,E)$, we have
$\|Dg_{x}^{m}|_{E}\|\leq e^{\alpha m}\text{ for every }1\leq m\leq n.$

######  Proposition 5.3.

Let us consider $f\in\operatorname{Diff}^{r}(M)$ with $2\leq r<\infty$, an
$\widehat{f}$-invariant probability $\widehat{\mu}$, some real numbers
${Q},\eta,\gamma,\delta>0$, and an integer $N$. Assume that:

  * –

$0<\gamma\leq\gamma_{0}(r,f,\eta)$,

  * –

$N\geq N_{0}(r,f,\eta,\gamma)$.

Then there are:

  * –

$0<\varepsilon,\widehat{\varepsilon}\leq\delta$.

  * –

a $C^{2}$ neighborhood
$\mathcal{U}_{0}(f,\widehat{\mu},{Q},\eta,\gamma,\delta,N)$ of $f$ in
$\operatorname{Diff}^{r}(M)$,

  * –

an open set
$\widehat{U}_{0}(f,\widehat{\mu},{Q},\eta,\gamma,\delta,N)\subset\widehat{M}$
with $\widehat{\mu}(\widehat{U}_{0})>1-\gamma^{2}$ and
$\widehat{\mu}(\partial\widehat{U}_{0})=0$,

  * –

an integer
$\overline{n}_{0}:=\overline{n}_{0}(f,\widehat{\mu},{Q},\eta,\gamma,N,\delta)\geq
1$,

such that the following property holds:

  1. (**)

For any $g\in\mathcal{U}_{0}$ with $Q_{r,N}(g)<{Q}$, any regular curve
$\sigma$ with $C^{r}$size at most $(\varepsilon,\widehat{\varepsilon})$, and
any $n\geq\overline{n}_{0}$ there is a family $\mathcal{R}_{n}$ of
reparametrizations over

$T:=\widehat{\sigma}^{-1}\bigg{\\{}\widehat{x}\colon\text{
$(\widehat{x},\dots,\widehat{g}^{n-1}(\widehat{x}))$ is
$\textstyle\frac{\eta}{10}$-neutral and
$\frac{1}{n}\sum_{j=0}^{n-1}\delta_{\widehat{g}^{j}(\widehat{x})}(\widehat{U}_{0})>1-\gamma$}\bigg{\\}},$
(21)

such that

     * –

$\mathcal{R}_{n}$ is
$(C^{r},g,N,\varepsilon,\widehat{\varepsilon})$-admissible up to time $n$,

     * –

$|\mathcal{R}_{n}|\leq\exp\left[n\left(\frac{\lambda({\widehat{f}})}{r-1}+\eta\right)\right]$.

Unlike Proposition 5.1, here the upper bound has no entropic term. Indeed, in
the $C^{\infty}$ case, the exponential rate of growth tends to zero with
$\eta$. This low complexity is due to the neutrality of the piece of $\sigma$
we are parametrizing.

The proofs of these two propositions may be skipped at the first reading.

###  5.2. Proof of Proposition 5.1

Let $r\in[2,\infty)$, $f\in\operatorname{Diff}^{r}(M)$, ${Q},\eta,\gamma>0$,
and consider a $\widehat{f}$-invariant probability measure $\widehat{\mu}$.
Note that the lift $\widehat{f}\in\operatorname{Diff}^{r-1}(\widehat{M})$ is
uniquely defined by $f$ and depends continuously on
$f\in\operatorname{Diff}^{r}(M)$. Recall the number $\Upsilon:=\Upsilon(r)$
given by Yomdin’s Theorem 4.13.

– Fix an integer $N_{1}=N_{1}(r,f,\eta)\geq 1$ such that for all $N\geq
N_{1}$,

$\Upsilon<\exp(\tfrac{\eta N}{10})\;\;\text{ and
}\;\;\tfrac{1}{N}\log\|D\widehat{f}^{N}\|_{\sup}<\lambda(\widehat{f})+\tfrac{\eta}{10}.$
(22)

This is possible since
$\lambda(\widehat{f})=\lim_{N\to+\infty}\frac{1}{N}\log\|D\widehat{f}^{N}\|_{\sup}$.

– Fix $N\geq N_{1}$.

– Set ${\varepsilon_{Y}}:={\varepsilon_{Y}}(r,{Q})$ as in Yomdin’s Theorem
4.13.

– Let $\varepsilon,\widehat{\varepsilon}$ be arbitrary in
$(0,\varepsilon_{Y})$.

– Pick an integer
$\overline{n}_{1}=\overline{n}_{1}(f,\widehat{\mu},\eta,\gamma,N,\varepsilon,\widehat{\varepsilon})>N$
using Proposition 4.3(2) such that for any $n\geq\overline{n}_{1}$,

$\tfrac{1}{n}\log
r_{\widehat{f}}(n,\tfrac{\varepsilon}{2},\tfrac{\widehat{\varepsilon}}{2},\widehat{\mu},1-\gamma^{2})<\overline{h}({\widehat{f}},\widehat{\mu})+\tfrac{\eta}{4},$
(23) $\log\|D\widehat{f}\|_{\sup}^{N}<\tfrac{n\eta}{10}.$ (24)

– Let $n$ be some integerlarger than $\overline{n}_{1}$.

– Let
$\mathcal{U}_{1}=\mathcal{U}_{1}(f,\eta,N,n,\varepsilon,\widehat{\varepsilon})\subset\operatorname{Diff}^{r}(M)$
be a small enough $C^{2}$ neighborhood of $f$ in $\operatorname{Diff}^{r}(M)$
such that the lift $\widehat{g}$ of any $g\in\mathcal{U}_{1}$ satisfies:

$\|D\widehat{g}^{N}\|_{\sup}\leq e^{\eta
N/10}\|D\widehat{f}^{N}\|_{\sup},\quad\|D\widehat{g}\|_{\sup}\leq
e^{\eta/10}\|D\widehat{f}\|_{\sup},$ (25)

and so that every fibered
$(n,\frac{\varepsilon}{2},\frac{\widehat{\varepsilon}}{2})$-Bowen ball for
$\widehat{f}$ is contained in a fibered
$(n,{\varepsilon},{\widehat{\varepsilon}})$-Bowen ball for $\widehat{g}$.

By the regularity of Borel measures, there is a compact set
$\widehat{K}=\widehat{K}(f,\widehat{\mu},\eta,\gamma,n,\varepsilon,\widehat{\varepsilon})$
with $\widehat{\mu}(\widehat{K})>1-\gamma^{2}$. By eq. (23) there is a
neighborhood $\widehat{U}_{1}$ of $\widehat{K}$, such that
$\widehat{g}(\widehat{U}_{1})$ is contained in the union of a collection
$\mathcal{C}$ of fibered
$(n,\frac{\varepsilon}{2},\frac{\widehat{\varepsilon}}{2})$-Bowen balls for
$\widehat{f}$ with cardinality at most
$\exp(n(\overline{h}(f,\widehat{\mu})+\frac{\eta}{4}))$. Passing to a smaller
open set containing $\widehat{K}$, we can ensure that
$\widehat{\mu}(\partial\widehat{U}_{1})=0$.

Suppose $g\in\mathcal{U}_{1}$, $Q_{r,N}(g)<{Q}$, and let $\sigma$ be a regular
$C^{r}$ curve with $C^{r}$ size at most $(\varepsilon,\widehat{\varepsilon})$.
By construction, we can cover
$\widehat{g}(\widehat{\sigma}[0,1]\cap\widehat{U}_{1})$ using only the
$(n,\frac{\varepsilon}{2},\frac{\widehat{\varepsilon}}{2})$-Bowen balls for
$\widehat{f}$ from the collection $\mathcal{C}$. Every ball in $\mathcal{C}$
is contained in some ${(n,\varepsilon,\widehat{\varepsilon})}$–Bowen ball for
$\widehat{g}$. Thus,

$r_{\widehat{g}}(n,\varepsilon,\widehat{\varepsilon},{\widehat{g}(\widehat{\sigma}[0,1]\cap\widehat{U}_{1})})\leq\exp(n(\overline{h}({f},\widehat{\mu})+{\textstyle\frac{\eta}{4}})).$

Since $\varepsilon,\widehat{\varepsilon}<{\varepsilon_{Y}}$, we can apply
Yomdin theory in the form of Corollary 4.14. The result is a family
$\mathcal{R}$ of reparametrizations of $\sigma$ over
$\widehat{\sigma}^{-1}\widehat{U}_{1}$ which is
$(C^{r},g,N,\varepsilon,\widehat{\varepsilon})$–admissible up to time $n$, and
such that

$|\mathcal{R}|\leq\Upsilon^{\lceil
n/N\rceil}\|D\widehat{g}^{N}\|_{\sup}^{\frac{\lfloor
n/N\rfloor}{r-1}}\|D\widehat{g}\|_{\sup}^{\frac{N}{r-1}}\exp(n(\overline{h}(f,\widehat{\mu})+{\textstyle\frac{\eta}{4}})).$

Using (22), (24) and (25), one gets
$|\mathcal{R}|\leq\exp(n(\overline{h}(f,\widehat{\mu})+\frac{\lambda(\widehat{f})}{r-1}+\eta))$.
$\Box$

###  5.3. Proof of Proposition 5.3

The proof splits into two parts: In steps 0-6 we select the parameters
$\gamma_{0},N_{0},\varepsilon,\widehat{\varepsilon},\mathcal{U}_{0},\widehat{U}_{0},\overline{n}_{0}$;
In steps 7-11 we build admissible families of reparametrizations as in the
statement, and estimate their cardinality.

Fix $r\in[2,\infty)$, $f\in\operatorname{Diff}^{r}(M)$, ${Q},\eta,\delta>0$,
and consider an $\widehat{f}$-invariant probability measure $\widehat{\mu}$.
Let $\Upsilon:=\Upsilon(r)$ given by Yomdin’s Theorem 4.13. Let

$H(t):=t\ln\tfrac{1}{t}+(1-t)\ln\tfrac{1}{1-t}.$ (26)

### Step 0 (Preliminary choices)

– Choose $\gamma_{0}=\gamma_{0}(r,f,\eta)>0$ such that:

$\forall 0<\gamma\leq\gamma_{0},\quad
3\Upsilon\leq\exp(\tfrac{\eta}{10\gamma}),\
H(4\gamma)<\tfrac{\eta}{10},\;\|D\widehat{f}\|_{\sup}<\exp\big{(}\tfrac{\eta}{10\gamma}\big{)}.$
(27)

– Let $0<\gamma<\gamma_{0}$ be arbitrary.

– Fix an integer
$N_{0}=N_{0}(r,f,\gamma,\eta)>\max(\tfrac{1}{\gamma},\tfrac{10}{\eta})$ such
that (cf. (2), §3.5):

$\forall N\geq
N_{0},\quad\tfrac{1}{N}\log\|D\widehat{f}^{N}\|_{\sup}<{\lambda(\widehat{f})}+\tfrac{\eta}{10}.$
(28)

– Let $N\geq N_{0}$ be arbitrary.

– Fix ${\varepsilon_{Y}}={\varepsilon_{Y}}(r,{Q})$ as in Yomdin’s Theorem
4.13.

– Let ${\mathcal{V}}={\mathcal{V}}(f,\eta,N)$ be a $C^{2}$ neighborhood of $f$
in $\operatorname{Diff}^{r}(M)$ such that for $g\in{\mathcal{V}}$,

$\|D\widehat{g}^{k}\|_{\sup}\leq
e^{\eta/10}\|D\widehat{f}^{k}\|_{\sup}\qquad(k=1,\ldots,N).$ (29)

### Step 1 (Decomposition of $\widehat{\mu}$)

$\widehat{\mu}$ projects to an $f$-invariant probability measure. Let
$\lambda_{x}^{\pm}:=\lambda^{\pm}(f,x)$ denote the Lyapunov exponents. These
are well-defined $\mu$-a.e., but since we are assuming nothing on
$\widehat{\mu}$ they could be equal on a set of positive measure. Consider the
invariant measurable subset

$M_{\\#}:=\\{x\in M\mid\lambda_{x}^{-}\text{ and }\lambda_{x}^{+}\text{ are
defined and distinct}\\}.$

We decompose $\widehat{\mu}$ as a barycenter of two invariant probability
measures:

$\widehat{\mu}=a\widehat{\mu}_{\\#}+(1-a)\widehat{\mu}_{0},$

where $a:=\mu(M_{\\#})$,
$a\widehat{\mu}_{\\#}:={\widehat{\mu}(\cdot\cap\widehat{\pi}^{-1}(M_{\\#}))}$
and
$(1-a)\widehat{\mu}_{0}:={\widehat{\mu}(\cdot\cap\widehat{\pi}^{-1}(M\setminus
M_{\\#}))}$. In the following we assume $a\in(0,1)$. Indeed, in the special
case where $a=0$ (resp. $a=1$), we simply write
$\widehat{\mu}=\widehat{\mu}_{0}$ (resp. $\widehat{\mu}=\widehat{\mu}_{\\#}$)
and the proof of Proposition 5.3 becomes simpler and can be easily obtained by
adapting the general case.

Let $\mu_{0}$ and $\mu_{\\#}$ be the projections of
$\widehat{\mu}_{0},\widehat{\mu}_{\\#}$ to $M$. These are $f$-invariant
measures; $\mu_{0}$–a.e. $x$ has two equal Lyapunov exponents; and
$\mu_{\\#}$–a.e. $x$ has two different Lyapunov exponents.

### Step 2 (Compact subsets $\widehat{K}_{+},\widehat{K}_{-}$ approximating
$\widehat{\mu}_{\\#}$)

As in section 3.3, the Oseledets splitting induces two $\widehat{f}$-invariant
measurable sections $x\mapsto(x,E^{\pm}(x))$, $M\to\widehat{M}$, defined
$\mu_{\\#}$-a.e.

– By Lusin’s theorem there exists a compact set
$K_{\\#}=K_{\\#}(f,\widehat{\mu},\gamma,N)$ inside $M_{\\#}$ such that
$\mu_{\\#}(K_{\\#})>1-\gamma^{2}$, and so that the functions

$x\mapsto\bigl{(}x,E^{+}(f^{k}(x))\bigr{)}\ ,\
x\mapsto\bigl{(}x,E^{-}(f^{k}(x))\bigr{)}\qquad(0\leq k\leq N)$

are continuous on $K_{\\#}$. By Lemma 3.3, $\widehat{\mu}_{\\#}$ is carried by
$\mathrm{graph}(E^{+})\cup\mathrm{graph}(E^{-})$. Therefore, the sets

$\widehat{K}^{+}:=\operatorname{graph}(E^{+}|_{K_{\\#}})\text{ and
}\widehat{K}^{-}:=\operatorname{graph}(E^{-}|_{K_{\\#}})$

are compact and disjoint, and
$\widehat{\mu}_{\\#}(\widehat{K}^{+}\cup\widehat{K}^{-})=\mu_{\\#}(K_{\\#})>1-\gamma^{2}$.
We set

$\widehat{K}_{\\#}:=\widehat{\pi}^{-1}(K_{\\#})=\widehat{K}_{-}\sqcup\widehat{K}_{+}.$

Every $z\in K_{\\#}$ has two lifts: $\widehat{z}^{+}\in\widehat{K}^{+}$, and
$\widehat{z}^{-}\in\widehat{K}^{+}$. It is easy to see that
$\widehat{f}^{k}(\widehat{z}^{\pm})=(f^{k}(z),E^{\pm}(f^{k}(z)))$.

### Step 3 (Control of $N$ iterates starting near $\widehat{K}_{\\#}$)

– There exist ${\varepsilon_{*}}={\varepsilon_{*}}(r,f,\eta,N)>0$ and a
$C^{2}$ neighborhood ${\mathcal{U}_{*}}={\mathcal{U}_{*}}(f,\eta,N)$ of $f$ in
$\operatorname{Diff}^{r}(M)$ such that the conclusion of Lemma 4.11 holds.

– Let
$\widehat{\varepsilon}=\widehat{\varepsilon}(r,f,{Q},\eta,\delta,N,K_{\\#}):=\tfrac{1}{10}\min(\delta,{\varepsilon_{Y}},{\varepsilon_{*}},{\mathrm{dist}}(\widehat{K}_{+},\widehat{K}_{-})).$
Note that the latter distance is positive since $\widehat{K}_{+}$ and
$\widehat{K}_{-}$ are disjoint compact sets.

– By construction of $K_{\\#}$, there is
$\varepsilon=\varepsilon(f,\widehat{\varepsilon},K_{\\#},N)\in(0,\widehat{\varepsilon})$
such that if $x,y\in K_{\\#}$ satisfy $d(x,y)\leq\varepsilon$, then

$\forall 0\leq k\leq N,\quad
d(\widehat{f}^{k}(\widehat{x}^{+}),\widehat{f}^{k}(\widehat{y}^{+}))<\tfrac{\widehat{\varepsilon}}{2}\text{
and }\
d(\widehat{f}^{k}(\widehat{x}^{-}),\widehat{f}^{k}(\widehat{y}^{-}))<\tfrac{\widehat{\varepsilon}}{2}.$

By the choice of $\widehat{\varepsilon}$, if
$\widehat{x},\widehat{y}\in\widehat{K}_{\\#}$ satisfy
$d(\widehat{x},\widehat{y})\leq\widehat{\varepsilon}$, then either
$\widehat{x}=\widehat{x}^{+},\widehat{y}=\widehat{y}^{+}$ or
$\widehat{x}=\widehat{x}^{-},\widehat{y}=\widehat{y}^{-}$. Consequently, for
any $\widehat{x},\widehat{y}\in\widehat{M}$,

$\left.\begin{array}[]{c}\widehat{x},\widehat{y}\in\widehat{K}_{\\#}\\\
d(\widehat{x},\widehat{y})\leq\widehat{\varepsilon},\;\;d(x,y)\leq\varepsilon\end{array}\right\\}\Longrightarrow\;\;\forall
0\leq k\leq N,\quad
d(\widehat{f}^{k}(\widehat{x}),\widehat{f}^{k}(\widehat{y}))<\tfrac{\widehat{\varepsilon}}{2}.$

Below, $B(S,\beta)$ denotes the $\beta$-neighborhood of a subset $S$.

– Since $\widehat{M}$ is compact, there exist
${\delta_{\ast\ast}}={\delta_{\ast\ast}}(f,N,\varepsilon,\widehat{\varepsilon})>0$
and a $C^{2}$ neighborhood
${\mathcal{U}_{**}}={\mathcal{U}_{**}}(f,N,\varepsilon,\widehat{\varepsilon})$
of $f$ in $\operatorname{Diff}^{r}(M)$ such that

$\left.\begin{array}[]{c}g\in{\mathcal{U}_{**}}\\\ \widehat{x},\widehat{y}\in
B(\widehat{K}_{\\#},{\delta_{\ast\ast}})\\\
d(x,y)\leq\varepsilon,\;\;d(\widehat{x},\widehat{y})\leq\widehat{\varepsilon}\end{array}\right\\}\Longrightarrow\;\;\forall
0\leq k\leq N,\quad
d(\widehat{g}^{k}(\widehat{x}),\widehat{g}^{k}(\widehat{y}))<\widehat{\varepsilon}.$
(30)

– Choose an open set
$\widehat{W}_{\\#}=\widehat{W}_{\\#}(\widehat{K}_{\\#},{\delta_{\ast\ast}})$
such that $\widehat{K}_{\\#}\subset\widehat{W}_{\\#}\subset
B(\widehat{K}_{\\#},{\delta_{\ast\ast}})$ and
$\widehat{\mu}(\partial\widehat{W}_{\\#})=0$. By (30), if $\sigma$ is a
regular curve with $C^{r}$ size at most $(\varepsilon,\widehat{\varepsilon})$,
then for all $g\in{\mathcal{U}_{**}}$,

$\forall 0\leq k\leq
N\quad{\operatorname{diam}}(\widehat{g}^{k}(\widehat{\sigma}\cap\widehat{W}_{\\#}))<\widehat{\varepsilon}.$
(31)

### Step 4 (Control of $n_{*}$ iterates starting near $\widehat{\mu}_{0}$)

Recall that $\mu_{0}$ has equal Lyapunov exponents almost everywhere. By
Ruelle’s inequality $\mu_{0}$ must have zero entropy. It follows that
$\overline{h}(f,\mu_{0})=0$.

– By Proposition 4.3, $\displaystyle\lim_{\overline{\gamma}\to
1}\lim_{\overline{\varepsilon}\to 0}\limsup_{n\to\infty}\frac{1}{n}\log
r_{\widehat{f}}(n,\overline{\varepsilon},\tfrac{\widehat{\varepsilon}}{2},\widehat{\mu}_{0},\overline{\gamma})=0$.
If $0<\overline{\varepsilon}<\frac{\varepsilon}{2}$ and
$\gamma<\overline{\gamma}<1$, then
$r_{\widehat{f}}(n,\frac{\varepsilon}{2},\frac{\widehat{\varepsilon}}{2},\widehat{\mu},{\gamma})\leq
r_{\widehat{f}}(n,\overline{\varepsilon},\frac{\widehat{\varepsilon}}{2},\widehat{\mu}_{0},\overline{\gamma})$.
It follows that

$\limsup_{n\to\infty}\frac{1}{n}\log
r_{\widehat{f}}(n,\tfrac{\varepsilon}{2},\tfrac{\widehat{\varepsilon}}{2},\widehat{\mu}_{0},\overline{\gamma})=0.$

So we can find a large integer
$n_{*}=n_{*}(f,\widehat{\mu},\eta,\gamma,\varepsilon,\widehat{\varepsilon},N)$
that is a multiple of $N$ and a compact set
$\widehat{K}_{0}:=\widehat{K}_{0}(\widehat{f},\widehat{\mu},\eta,\gamma,\varepsilon,\widehat{\varepsilon},n_{*})\subset\widehat{M}$
with $\widehat{\mu}_{0}(\widehat{K}_{0})>1-\gamma^{2}$, s.t.

$\tfrac{1}{n_{*}}\log
r_{\widehat{f}}(n_{*},\tfrac{\varepsilon}{2},\tfrac{\widehat{\varepsilon}}{2},{\widehat{f}}(\widehat{K}_{0}))<\tfrac{\eta}{10}.$

– By continuity, we can choose a neighborhood
$\widehat{W}_{0}=\widehat{W}_{0}(f,\eta,\varepsilon,\widehat{\varepsilon},n_{*},\widehat{K}_{0})$
of $\widehat{K}_{0}$ with $\widehat{\mu}(\partial\widehat{W}_{0})=0$, and a
$C^{2}$ neighborhood
$\mathcal{U}_{***}=\mathcal{U}_{***}(f,\eta,\varepsilon,\widehat{\varepsilon},n_{*},\widehat{K}_{0})$
of $f$ in $\operatorname{Diff}^{r}(M)$ so that:

$\forall g\in\mathcal{U}_{***},\quad\tfrac{1}{n_{*}}\log
r_{\widehat{g}}(n_{*},\varepsilon,\widehat{\varepsilon},\widehat{g}(\widehat{W}_{0}))<\tfrac{\eta}{10}.$
(32)

### Step 5 ($\widehat{U}_{0}$ and a decomposition of long typical orbit
segments)

Let

$\widehat{U}_{0}:=\widehat{W}_{0}\cup\widehat{W}_{\\#}.$

Then
$\widehat{U}_{0}=\widehat{U}_{0}(r,f,\widehat{\mu},{Q},\eta,\gamma,\delta,N)$,
$\widehat{\mu}(\widehat{U}_{0})>1-\gamma^{2}$ and
$\widehat{\mu}(\partial\widehat{U}_{0})=0$.

######  Claim 5.4.

Suppose $n>{n_{*}/\gamma}$. Any orbit segment
$(\widehat{x},\widehat{g}(\widehat{x})\dots,\widehat{g}^{n-1}(\widehat{x}))$
which spends a proportion of time larger than $1-\gamma$ in $\widehat{U}_{0}$
can be decomposed into:

  1. (a)

orbit segments of length $n_{*}$ with initial point in $\widehat{W}_{0}$,

  2. (b)

orbit segments of length $N$ with initial point in $\widehat{W}_{\\#}$,

  3. (c)

orbit segments of length $1$, of total number less than $2\gamma n$.

($\widehat{W}_{0}$ and $\widehat{W}_{\\#}$ are not necessarily disjoint, so
the decomposition may not be unique.)

###### Proof.

A decomposition as in the statement is completely characterized by the
increasing sequence of times $(n_{0},n_{1},\dots,n_{\ell})$, where
$\widehat{g}^{n_{i}}(\widehat{x})$ is the initial point of the $i$-th segment.
(So $n_{0}=0$, $n_{\ell}=n$.)

We set $n_{0}=0$ and define the sequence inductively. Assuming that $n_{i}<n$
has already been defined, we set

  1. (a)

$n_{i+1}:=n_{i}+n_{*}$ if $n_{i}+n_{*}\leq n$ and
$\widehat{g}^{n_{i}}(\widehat{x})\in\widehat{W}_{0}\setminus\widehat{W}_{\\#}$,

  2. (b)

$n_{i+1}:=n_{i}+N$ if $n_{i}+N\leq n$ and
$\widehat{g}^{n_{i}}(\widehat{x})\in\widehat{W}_{\\#}$,

  3. (c)

$n_{i+1}:=n_{i}+1$ otherwise.

We stop when $n_{i+1}=n$.

Since $n_{*}\geq N$, the times $n_{i}$ such that
$\widehat{g}^{n_{i}}(\widehat{x})\in\widehat{U}_{0}$ but which are not
associated to case (a) or (b) must satisfy ${n_{i}}>n-n_{*}$. Since
$n>n_{*}/\gamma$ and since
$(\widehat{x},\dots,\widehat{g}^{n-1}(\widehat{x}))$ spends a proportion of
time larger than $1-\gamma$ in $\widehat{U}_{0}$, the set of times $n_{i}$
corresponding to case (c) has size smaller than $2\gamma n$. ∎

The sequence of times $\theta:=(n_{0},n_{1},\dots,n_{\ell})$ obtained in the
previous claim is called _type of a decomposition_. Recall that
$H(t)=t\ln\frac{1}{t}+(1-t)\ln\frac{1}{1-t}$.

######  Claim 5.5.

There exists $n_{H}:=n_{H}(\gamma)$ such that for all $n>n_{H}$, the number of
possible types $\theta$ is less than $\exp(H(4\gamma)n)$.

###### Proof.

By our choices of $N,N_{0}$ and $n_{\ast}$, we have $n_{*},N>1/\gamma$. Hence
there can be at most $\gamma n$ times $n_{i}$ such that
$n_{i+1}-n_{i}\in\\{n_{\ast},N\\}$. Since there are also at most $2\gamma n$
times $n_{i}$ such that $n_{i+1}-n_{i}=1$, we must have $\ell\leq\lfloor
3\gamma n\rfloor.$

The number of types is thus bounded by $\sum_{\ell=1}^{\lfloor 3\gamma
n\rfloor}{{n}\choose\ell-1}$. Since $3\gamma<\frac{1}{2}$, this is less than
$3\gamma n{n\choose\lfloor{3\gamma n}\rfloor}$, which by de Moivre’s
approximation is less than $\exp[nH(3\gamma)+o(n)]$. Since
$4\gamma<\frac{1}{2}$, we have $H(3\gamma)<H(4\gamma)$, and the claim follows.
∎

### Step 6 (Definition of $\mathcal{U}_{0},\overline{n}_{0}$)

We fix the last parameters of our construction. Recall the $C^{2}$
neighborhoods of $f$ in $\operatorname{Diff}^{r}(M)$ introduced in Lemma 4.11
and eqs. (29), (30), (32).

– Let
$\mathcal{U}_{0}:=\mathcal{U}_{0}(r,f,\widehat{\mu},{Q},\eta,\gamma,\varepsilon,\widehat{\varepsilon},n_{*},N):={\mathcal{V}}\cap\mathcal{U}_{*}\cap\mathcal{U}_{**}\cap\mathcal{U}_{***}$.

– Define
$\overline{n}_{0}=\overline{n}_{0}(f,\widehat{\mu},\eta,\gamma,\delta,N):=\max\\{{n_{*}/\gamma},n_{H}\\}$.

### Step 7 (An inductive scheme)

Now we fix some $g\in\mathcal{U}_{0}$ with $Q_{r,N}(g)<{Q}$, a regular $C^{r}$
curve $\sigma$ with $C^{r}$ size at most $(\varepsilon,\widehat{\varepsilon})$
and $n>\overline{n}_{0}$. We need to bound the minimal cardinality of a family
$\mathcal{R}_{n}$ of reparametrizations of $\sigma$ which are
$(C^{r},g,N,\varepsilon,\widehat{\varepsilon})$-admissible up to time $n$,
over the set

$T=\widehat{\sigma}^{-1}\bigg{\\{}\widehat{x}\colon\text{
$(\widehat{x},\widehat{g}(\widehat{x}),\dots,\widehat{g}^{n-1}(\widehat{x}))$
is $\textstyle\frac{\eta}{10}$-neutral and
$\frac{1}{n}\sum_{j=0}^{n-1}\delta_{\widehat{g}^{j}(\widehat{x})}(\widehat{U}_{0})\geq
1-\gamma$}\bigg{\\}}.$

For each type $\theta=(n_{0},\ldots,n_{\ell})$, we introduce the corresponding
subset

$T_{\theta}:=T\cap\widehat{\sigma}^{-1}\bigg{\\{}\widehat{x}\colon\text{
$(\widehat{x},\widehat{g}(\widehat{x}),\dots,\widehat{g}^{n-1}(\widehat{x}))$
has type $\theta$}\bigg{\\}}.$

Then $T$ is the union of $T_{\theta}$ over all possible type $\theta$.

Fixed some type $\theta=(n_{0},\ldots,n_{\ell})$. We will build by induction a
family $\mathcal{R}_{n_{i}}^{\theta}$ of reparametrizations $\psi$ of $\sigma$
over $T_{\theta}$ satisfying the following properties:

  1. (i)

admissibility: $\mathcal{R}_{n_{i}}^{\theta}\text{ is
$(C^{r},g,N,\varepsilon,\widehat{\varepsilon})$-admissible up to time
$n_{i}$};$

  2. (ii)

small cardinality: if $i\geq 1$,

$\displaystyle|\mathcal{R}^{\theta}_{n_{i}}|$
$\displaystyle\leq\exp\left((\tfrac{\lambda(\widehat{f})}{r-1}+\tfrac{7\eta}{10})n_{i}\right)|\mathcal{R}^{\theta}_{n_{i-1}}|\qquad\text{when
}n_{i}-n_{i-1}\geq N,$ $\displaystyle|\mathcal{R}^{\theta}_{n_{i}}|$
$\displaystyle\leq\exp(\eta/\gamma)\;|\mathcal{R}^{\theta}_{n_{i-1}}|\qquad\;\quad\text{otherwise;}$

  3. (iii)

small length: for each $\psi\in\mathcal{R}_{n_{i}}^{\theta}$ and any
$(x,E)\in\widehat{\sigma}\circ\psi([0,1])$,

$\operatorname{Length}(g^{n_{i}}\circ\sigma\circ\psi)<\varepsilon
e^{-\frac{\eta}{10}n_{i}}\|Dg^{n_{i}}_{x}|_{E}\|.$

At the end of the construction, one obtains a family
$\mathcal{R}_{n}^{\theta}:=\mathcal{R}_{n_{\ell}}^{\theta}$ over $T_{\theta}$
which is admissible up to time $n$. Then one can take the union over all
$\theta$ and finish the construction.

We begin the construction by defining
$\mathcal{R}_{0}^{\theta}:=\\{\operatorname{Id}\\}$: This meets our
requirements because $n_{0}=0$ and $\sigma$ has $C^{r}$ size at most
$(\varepsilon,\widehat{\varepsilon})$.

Now we assume by induction that $\mathcal{R}_{n_{i}}^{\theta}$ has been
constructed, and we build $\mathcal{R}_{n_{i+1}}^{\theta}$. The construction
uses the concatenation procedure described in Lemma 4.10. For each
$\psi\in\mathcal{R}^{\theta}_{n_{i}}$ we have to build a family
$\mathcal{R}_{\psi}$ of reparametrizations of the curve
$g^{n_{i}}\circ\sigma\circ\psi$ over $\psi^{-1}(T_{\theta})$ with the
following properties:

  1. (i’)

$\mathcal{R}_{\psi}$ is
$(C^{r},g,N,\varepsilon,\widehat{\varepsilon})$-admissible up to time
$n_{i+1}-n_{i}$,

  2. (ii’)

$\log|\mathcal{R}_{\psi}|$ is bounded by
$(\tfrac{\lambda(\widehat{f})}{r-1}+\tfrac{7\eta}{10})(n_{i+1}-n_{i})$ if
$n_{i+1}-n_{i}\geq N$ and by $\tfrac{\eta}{\gamma}$ otherwise,

  3. (iii’)

for each $\varphi\in\mathcal{R}_{\psi}$ and
$(x,E)\in\widehat{\sigma}\circ\psi\circ\varphi([0,1])$,

$\operatorname{Length}(g^{n_{i+1}}\circ\sigma\circ\psi\circ\varphi)<\varepsilon
e^{-\frac{\eta}{10}n_{i+1}}\|Dg^{n_{i+1}}_{x}|_{E}\|.$

The family
$\mathcal{R}^{\theta}_{n_{i+1}}:=\\{\psi\circ\varphi,\;\psi\in\mathcal{R}^{\theta}_{n_{i}}\;\varphi\in\mathcal{R}_{\psi}\\}$
then satisfies (i–iii) above.

– Given a type $\theta=(n_{0},\ldots,n_{\ell})$, an integer
$i\in\\{0,\dots,\ell-1\\}$ and a reparametrization
$\psi\in\mathcal{R}^{\theta}_{n_{i}}$, the construction of the families
$\mathcal{R}_{\psi}$ depends on which of the following cases from Claim 5.4
holds for $n_{i}$:

  1. Case (a):

$\widehat{g}^{n_{i}}(\widehat{x})\in\widehat{W}_{0}$ and
$n_{i+1}-n_{i}=n_{\ast}$.

  2. Case (b):

$\widehat{g}^{n_{i}}(\widehat{x})\in\widehat{W}_{\\#}$ and $n_{i+1}-n_{i}=N$.

  3. Case (c):

$n_{i+1}-n_{i}=1$.

The three cases are discussed in steps 8-10 below.

In order to simplify the notations, we set

$\sigma^{\prime}:=g^{n_{i}}\circ\sigma\circ\psi\text{ and