2.1.1 Abstract construction

Consider $(X, d)$ a compact metric space, and a map ${f: X \to X}$ that is continuous and surjective. Then, f has an invertible extension ${\tilde {f}: X_f \to X_f}$ , that is, there exists $\pi : X_f \to X$ continuous and surjective such that $\pi \circ \tilde {f} = f \circ \pi $ . This extension is called inverse limit or natural extension and it is defined as follows:

• • $X_f = \{\tilde {x} = (x_k) \in X^{\mathbb {Z}}: x_{k+1} = f(x_k) { } \text { for all } k \in \mathbb {Z}\}$ ;

• • $(\tilde {f}(\tilde {x}))_k = (x_{k+1}) \text { for all } k \in \mathbb {Z}$ and $\text { for all } \tilde {x} \in \tilde {X}$ ;

• • $\pi : X_f \to X$ is the projection on the zeroth coordinate.

Given $\tilde {x},\tilde {y}\in X_f$ , we define $n(\tilde {x},\tilde {y}) \stackrel {\scriptscriptstyle \textrm { def.}}{=} \max \{n\in {\mathbb Z}\; : \; x_i=y_i\:\text {for all}\:|i|<n\}$ . Then, by defining

$$ \begin{align*} d(\tilde{x},\tilde{y})\stackrel{\scriptscriptstyle\textrm {def.}}{=} 2^{-n(\tilde{x},\tilde{y})}, \end{align*} $$

we obtain a metric in $X_f$ which makes it a compact metric space.

Notice that the shift map $\tilde {f}$ and the projection $\pi $ are both continuous. If f is invertible, then $\pi $ is a homeomorphism between $X_f$ and X. We consider in this work only the case in which f is not invertible.

One can prove that the metric d generates the same topology as the open cylinders given by

$$ \begin{align*}[A_{-k}, \ldots A_0] \stackrel{\scriptscriptstyle\textrm {def.}}{=} \{\tilde{x} \in X_f : x_i \in A_i \text{ for } i \in \{-k, \ldots 0\} \},\end{align*} $$

where $A_i \subseteq X$ are open sets. The same cylinders generate the Borel $\sigma $ -algebra for $(X_f, \tilde {d})$ .

All measures considered in this work are over the Borel $\sigma $ -algebra for the given space and, given a space X, we denote by ${\mathcal M}(X)$ the probability measures on X, endowed with the weak $^*$ topology. Given a discrete time dynamical system $f:X\to X$ , we denote by ${\mathcal M}_f(X)$ the subset of f-invariant probability measures. Finally, we denote by ${\mathcal M}^{\operatorname {erg}}_f(X)$ the ergodic elements in ${\mathcal M}_f(X)$ .

It is easy to check that if $\tilde {\mu }$ is an $\tilde {f}$ -invariant measure on $X_f$ , then $\pi _*\tilde {\mu }$ is an f-invariant measure on X. Since $\pi $ is surjective, for all such $\mu $ , there is a $\tilde {\mu }$ that projects on it. Actually, $\pi _*: \mathcal {M}_{\tilde {f}}(X_f) \to \mathcal {M}_f(X)$ is a bijection, as proved for instance in [Reference Qian, Xie and ZhuQXZ09, Proposition I.3.1].

The uniqueness on the above result is given by the fact that $\tilde {\mu }$ is defined by the values it attributes to cylinders.

The topological structure of $(X_f, \tilde {d})$ is quite interesting if f is not invertible. If X is connected and f is a self-covering map with positive degree, then, for each $x \in X$ , $\pi ^{-1}(x)$ is compact, totally disconnected, and has no isolated points; thus, it is a Cantor set.

Notation 2.4. Herein, we use the notation

$$ \begin{align*}\Sigma(\tilde{x}) \stackrel{\scriptscriptstyle\textrm {def.}}{=} \pi^{-1}(\pi(\tilde{x}))\end{align*} $$

to refer to the fiber of $\tilde {x}$ on $(X_f, X, \pi , \Sigma )$ .

2.1.2 Topological dynamics on $X_f$

Recall that a continuous map $f: X \to X$ is said to be strongly transitive if for all $U \subseteq X$ non-empty open sets, there is $N \in {\mathbb N}$ such that $\bigcup _{n=0}^N f^n(U) = X$ . Every uniformly expanding map on a compact manifold is strongly transitive, and thus, it is transitive. The following simple result and its corollary are fundamental in allowing us to have ‘quantitative’ information about points which have ‘many’ different unstable directions (Lemma 4.3).

2.2.1 Some examples

To get a more concrete view of the variety of behaviors one can encounter among partially hyperbolic elements $f\in \operatorname {End}^1({\mathbb {T}^2})$ , we discuss some simple examples.

Both simple examples presented above are special. However, most partially hyperbolic endomorphisms are not special. More precisely, the set of non-special partially hyperbolic endomorphisms on a manifold X is $C^1$ open and dense [Reference Costa and MicenaCM22, Theorem B]. We remark that their result requires additional assumptions because they are considering systems with three non-trivial directions in the splitting and they are proving that both the center and the unstable directions are not unique for a given point. Nonetheless, the result holds, in particular, in dimension two as follows.

The proof of denseness is achieved by, starting with a special partially hyperbolic endomorphism, perturbing it to generate new directions, as we do for completeness with the following example.

Example 2.15. Recall that $f_A$ from Example 2.12 is a special partially hyperbolic uniformly expanding endomorphism with the strong unstable direction $E^u_A$ being horizontal at every point, and the center direction $E^c_A$ being vertical at every point.

Consider $p = (0, 0)$ the unique fixed point of $f_A$ and $q=(2/3,1/2) \in f_A^{-1}(\{p\})$ . Since $f_A$ is a self-covering map, there is $\tau> 0$ such that, if $f_A(y) = f_A(z)$ and $y \neq z$ , then $d(y,z)> \tau $ . Take $\delta \in (0, \tau /2)$ and consider $a \in \mathbb {R}$ such that the square ${R = [-a,a]^2 \in \mathbb {R}^2}$ , when projected under $\exp _q: T_q{\mathbb {T}^2} \to {\mathbb {T}^2}$ , is contained in $B_{\delta }(q)$ . Making an abuse of notation by identifying R and $\exp _q(R)$ , we define

$$ \begin{align*}\varphi(x,y) = \begin{cases} \bigg(x, y + \varepsilon \; a \; \psi_1\bigg( \dfrac{x}{a} \bigg) \psi_2\bigg( \dfrac{y}{a} \bigg) \bigg) & \text{ if } (x,y) \in R,\\ \operatorname{Id} & \text{ otherwise, } \end{cases}\end{align*} $$

where $\varepsilon> 0$ is as small as we want, and $\psi _1: \mathbb {R} \to [-2,2]$ and $\psi _2: \mathbb {R} \to [0,1]$ are $C^\infty $ functions satisfying:

• • $\operatorname {supp} \psi _i \subseteq [-1,1]$ ;

• • $\psi _1(1/2) = 1$ and $\psi _1(-1/2) = -1$ ;

• • $\psi _1(-1) = \psi _1(0) = \psi _1(1) = 0$ ;

• • $\psi _1 |_{[-1/2,1/2]}$ is linear;

• • $\psi _2 |_{[-1/2,1/2]} \equiv 1$ .

We have that

$$ \begin{align*}D\varphi_{(x,y)} = \begin{pmatrix} 1 & 0 \\ \varepsilon \; \psi_1'\bigg( \dfrac{x}{a} \bigg) \psi_2\bigg( \dfrac{y}{a} \bigg) & 1 + \varepsilon \; \psi_1\bigg( \dfrac{x}{a} \bigg) \psi_2'\bigg( \dfrac{y}{a} \bigg) \end{pmatrix},\end{align*} $$

in R, which is easily checked to be invertible for all $(x,y)$ , provided that $\varepsilon $ is sufficiently small.

Additionally, $\varphi $ is $C^\infty $ and satisfies $\varphi (q) = q$ , $\varphi |_{\overline {R^\complement }} = {\operatorname {Id}}$ , $\varphi (R) = R$ , and

$$ \begin{align*}D\varphi_{q} \cdot E^u_A(q) = \begin{pmatrix} 1 & 0 \\ 2 \varepsilon & 1 \end{pmatrix}\begin{pmatrix} \alpha \\ 0 \end{pmatrix} = \begin{pmatrix} \alpha \\ 2 \varepsilon \alpha \end{pmatrix} \neq E^u_A(p),\end{align*} $$

meaning that $\varphi $ does not preserve the unstable directions of $f_A$ .

Let $f\stackrel {\scriptscriptstyle \textrm {def.}}{=} f_A\circ \varphi $ . Since f is $C^1$ close to $f_A$ , we have that f is a partially hyperbolic uniformly expanding endomorphism. Additionally, since $(1,0)$ is an eigenvector for $D\varphi _{(x,y)}$ , f has the same central manifolds as $f_A$ . The same argument as in [Reference Costa and MicenaCM22] gives us that f is non-special as follows.

By the definition of $\delta $ , we have that any point $x\in {\mathbb {T}^2}$ has a past orbit $\tilde {x}\in \Sigma (x)$ such that $x_{-k}\notin R$ for every $k\in {\mathbb N}$ . Thus, along this orbit, we have that $f=f_A$ , which implies that $E^u(\tilde {x})=E^u_A$ . In particular, there exists $\tilde {q}\in \Sigma (q)$ such that $E^u(\tilde {q})$ is horizontal. Now, by invariance, this implies that for $\tilde {p}= \tilde {f}(\tilde {q}) = (\ldots q_{-1}q_0,p,p,p,\ldots )$ , we have that

$$ \begin{align*} E^u(\tilde{p})=Df(q)E^u(\tilde{q})\neq E^u_A. \end{align*} $$

However, for $\hat {p}=(\ldots p,p,p,\ldots )$ , one has that $f=f_A$ all along this fixed orbit and thus $E^u(\hat {p})=E^u_A$ . This proves that p has more than one unstable direction and, thus, f is not u-special.

Similarly, we can do a perturbation of Example 2.13.

2.2.2 Invariant foliations

Let $f\in \operatorname {End}^r({\mathbb {T}^2})$ be partially hyperbolic and let ${\tilde {f}:{\mathbb {T}_f^2}\to {\mathbb {T}_f^2}}$ denote its lift to the inverse limit space. By viewing the direction of the fibers as a stable direction and the ‘manifold direction’ as a center-unstable direction, we can see the dynamics on the inverse limit space as partially hyperbolic as follows.

As we mentioned above, one can show that each center-unstable manifold $\widetilde {{\mathcal W}}^{cu}(\tilde {x})$ is homeomorphic to the universal cover ${\mathbb R}^2$ of ${\mathbb {T}^2}$ . Notice that $\tilde {f}$ acts permuting the components. Thus, $\{\widetilde {{\mathcal W}}^{cu}(\tilde {x})\}_{\tilde {x}\in {\mathbb {T}_f^2}}$ is an $\tilde {f}$ -invariant lamination of the space ${\mathbb {T}_f^2}$ . This lamination is analogous to the center unstable foliation of a three-dimensional partially hyperbolic diffeomorphism. Indeed, notice that the fibers $\Sigma (\tilde {x})$ are contracted by the action of $\tilde {f}$ .

Thus, the fibers $\Sigma $ of the space ${\mathbb {T}_f^2}$ play the same role as local stable manifolds in three-dimensional partially hyperbolic diffeomorphisms and they are ‘transverse’ to the center-unstable manifolds $\widetilde {{\mathcal W}}^{cu}$ . Clearly, there is a notorious difference, for here, the fibers are totally disconnected compact spaces and not embedded curves as in the case of diffeomorphisms.

Nonetheless, the graph transform method applies to the dynamics of $\tilde {f}$ on ${\mathbb {T}_f^2}$ and one can show the following proposition (see [Reference Hirsch, Pugh and ShubHPS77, Reference Qian, Xie and ZhuQXZ09]).

The existence of a center foliation is a more delicate issue. In our setting, we have the following consequence of [Reference Hall and HammerlindlHH22, Theorem B].

In this work, we use consistently the following notation: $\widetilde {{\mathcal W}}^{*}(\tilde {x})$ , $* \in \{u, c, cu \}$ , is the leaf through the point $\tilde {x}\in {\mathbb {T}_f^2}$ , while ${\mathcal W}^{*}(\tilde {x})\stackrel {\scriptscriptstyle \textrm {def.}}{=}\pi (\widetilde {{\mathcal W}}^u(\tilde {x}))\subset {\mathbb {T}^2}$ .

2.2.3 The unstable foliation of Example 2.15

The unstable foliation of Example 2.12 descends to a foliation by (horizontal) compact circles, while the center foliation is the foliation by vertical circles. In Example 2.15, as the map is not special, the unstable foliation on ${\mathbb {T}_f^2}$ does not descend to a foliation of ${\mathbb {T}^2}$ , while the center foliation remains the same as that of $f_A$ .

Moreover, we can choose the parameter $a>0$ small enough so that there exists some (meager) compact set $K\subset \mathbb {S}^1$ , which is forward invariant under $x\mapsto 2x\mod 1$ , and $(\mathbb {S}^1\times K)\cap R=\emptyset $ . Thus, after perturbation, the set $\Gamma =\mathbb {S}^1\times K$ remains invariant under the perturbed map f, as $f=f_A$ is restricted to $\Gamma $ . In particular, every horizontal unstable curve through a point $x\in \Gamma $ has non-dense forward orbit. Moreover, we have that there are points $\tilde {x}\in {\mathbb {T}_f^2}$ such that the set $\bigcup _{i\in {\mathbb Z}}{\mathcal W}^u(x_i)$ is not dense in ${\mathbb {T}^2}$ . Indeed, for all $x \in \Gamma $ , there is ${\tilde {x}\in \Sigma (x)}$ such that $x_{-k}\in \Gamma $ for every $k\in {\mathbb N}$ , which implies that $E^u_f(\tilde {x}) = E^u_A$ . Additionally, the uniform expansion of f allows us to make this choice of past orbit coherently along the compact curves in $\Gamma $ , in the sense that ${\mathcal W}^u(x_i) \subseteq \Gamma $ for all $i \leq 0$ .

The behavior of the foliation in this example is still not well understood. We believe that every unstable curve which is not horizontal at some place should have a dense orbit, but this claim has not yet been confirmed.

2.3.1 u-Gibbs measures for the examples

Example 2.12 has many u-Gibbs measures: for any $\nu $ a ( $2x\ \pmod 1$ )-invariant measure on $\mathbb {S}^1$ , $\operatorname {Leb}_{\mathbb {S}^1} \times \nu $ is a u-Gibbs measure. Since the doubling map $2x\ \pmod 1$ is semi-conjugated to a full shift in two symbols, we know this system has infinitely many u-Gibbs measures. This is a particular case because the system is reducible.

In contrast, Example 2.13 has only $\operatorname {Leb}_{\mathbb {T}^2}$ as a u-Gibbs measure. A brief sketch of this argument goes as follows: up to a constant, the density of $\tilde {\mu }$ along some unstable leaf $\widetilde {{\mathcal W}}^u(\tilde {x})$ is given by

$$ \begin{align*} \lim_{n\to-\infty}\frac{\unicode{x3bb}^u_{\tilde{y}}(n)}{\unicode{x3bb}^u_{\tilde{x}}(n)}. \end{align*} $$

Since the map is linear, this shows that the density is constant and, thus, the conditional measure is invariant by translations along leaves. The invariance of conditional measures by translations along leaves implies invariance of the measure $\mu $ under the unity flow along leaves. However, the unity flow along leaves is a Diophantine flow, which is uniquely ergodic. One deduces that $\mu =\text {Leb}_{{\mathbb {T}^2}}$ .

Both Examples 2.12 and 2.13 are special, and one might wonder how the situation changes after perturbation. Example 2.15 gives us a non-special endomorphism, close to $f_A$ , which still has infinitely many u-Gibbs measures. Indeed, as we discussed above in §2.2.3, the example can be built with a forward invariant set $\Gamma =\mathbb {S}^1\times K$ which is made of horizontal compact unstable leaves. Moreover, K can be chosen so that it contains infinitely many $x\mapsto 2x\mod 1$ periodic orbits. Given $x_1,\ldots ,x_k$ such periodic orbits, the measure

$$ \begin{align*} \frac{1}{k}\sum_{i=1}^k\delta_{x_i}\times\text{Leb}_{\mathbb{S}^1} \end{align*} $$

is a u-Gibbs measure for f.

However, Example 2.16 is also non-special, but since we can apply Corollary 1.2, we deduce that the map has only one u-Gibbs measure which is the unique absolutely continuous invariant measure.

2.3.2 Uniform density estimate

For a Lebesgue measurable set $A\subset \widetilde {{\mathcal W}}^u(\tilde {x})$ , we denote, for simplicity, $|A|\stackrel {\scriptscriptstyle \textrm {def.}}{=}\text {Leb}^u_{\tilde {x}}(A)$ the Lebesgue measure of A. Given a u-Gibbs measure, the conditional measures are ‘comparable’ with respect to Lebesgue, but only up to a measurable function. To be precise, the density $({\tilde {\mu }_{\tilde {x}}}/{\text {Leb}^u_{\tilde {x}}})$ is a measurable function. By restricting to compact sets of large measure, we get a uniform comparison, as made clear by the next result.

Proof. From the definition of u-Gibbs measure, it follows that for $\tilde {\mu }$ -a.e. $\tilde {x}\in {\mathbb {T}_f^2}$ , there exists a density function $\rho ^u_{\tilde {x}}$ such that

$$ \begin{align*} \tilde{\mu}^u_{\tilde{x}}(A)=\int_A\rho^u_{\tilde{x}}(t)\,d\text{Leb}^u_{\tilde{x}}(t). \end{align*} $$

Moreover, up to a positive constant depending measurably on $\tilde {x}$ , we have that the density $\rho ^u_{\tilde {x}}(t)$ , $t=\pi (\tilde {y})$ , is given by

$$ \begin{align*} \rho^u_{\tilde{x}}(t) = \lim_{n\to-\infty}\frac{\unicode{x3bb}^u_{\tilde{x}}(n)}{\unicode{x3bb}^u_{\tilde{y}}(n)}. \end{align*} $$

By Lusin’s theorem, given $\delta>0$ , there exists a compact set ${\mathcal L}$ with measure $\tilde {\mu }({\mathcal L})>1-\delta $ such that the assignment $\tilde {x}\in {\mathcal L}\mapsto \rho ^u_{\tilde {x}}$ is continuous. In particular, there exists a constant C such that $ C\geq \rho ^u_{\tilde {x}}(t)\geq C^{-1}$ for every $\tilde {x}\in {\mathcal L}$ and every $t=\pi (\tilde {y})$ , where $\tilde {y}$ belongs to the atom containing $\tilde {x}$ .

2.4.1 Center Lyapunov exponent

The (pointwise) center Lyapunov exponent of f is defined for all points in the f-invariant regular set $\Gamma $ , where $\Gamma $ has full $\mu $ -measure for any f-invariant measure $\mu $ . It is given by

$$ \begin{align*}L^c(x) = \lim_{n \to \infty} \dfrac{1}{n} \log \Vert Df^n(x) v \Vert\quad \text{where } v \in E^c(x).\end{align*} $$

The integrated center Lyapunov exponent with respect to $\mu $ is given by

$$ \begin{align*} L^c(\mu) &= \int L^c(x) \,d\mu = \int \lim_{n \to \infty} \dfrac{1}{n} \log \Vert Df(x) v \Vert\, d\mu\quad \text{where } v \in E^c(x)\\ &= \int \log \vert {Df}\vert_{E^c} \vert\, d\mu, \end{align*} $$

using the fact that $E^c$ is one-dimensional and Birkhoff ergodic theorem. Since $L^c(x)$ is f-invariant and since we work only with ergodic measures, then $L^c(x) = L^c(\mu )$ for $\mu $ -almost every point and non-uniform expansion is given by $L^c(x)> 0$ for $\mu $ -almost every point.

The differential structure of f pulled back to ${\mathbb {T}_f^2}$ allows us to have an Oseledec’s theorem for the inverse limit space, and the Lyapunov exponents are the same as in the original manifold and they do not depend on the past orbit, that is, they are constant along $\Sigma (\tilde {x})$ , see [Reference Qian, Xie and ZhuQXZ09, Proposition I.3.5]. In §2.4, we explore Pesin theory further.

From now on, we consider a fixed ergodic f-invariant measure $\mu $ . We also assume that the center Lyapunov exponent of $\mu $ satisfies $L^c(\mu )>0$ . In this setting, we fix $0<\unicode{x3bb} <L^c(\mu )<\hat {\unicode{x3bb} }$ . Then, there exists a measurable function $C:{\mathbb {T}_f^2}\to [1,+\infty )$ such that

$$ \begin{align*} C(\tilde{x})^{-1}e^{\unicode{x3bb} n}\leq\unicode{x3bb}^c_{\tilde{x}}(n)\leq C(\tilde{x})e^{\hat{\unicode{x3bb}} n} \end{align*} $$

for every $n>0$ and

$$ \begin{align*} C(\tilde{x})^{-1}e^{\unicode{x3bb} n}\geq\unicode{x3bb}^c_{\tilde{x}}(n)\geq C(\tilde{x})e^{\hat{\unicode{x3bb}} n} \end{align*} $$

for every $n<0$ . Moreover, the function C has slow exponential growth:

$$ \begin{align*} C(\tilde{f}^n(\tilde{x}))\leq e^{\varepsilon n} C(\tilde{x}), \end{align*} $$

where $\varepsilon =(\hat {\unicode{x3bb} }-\unicode{x3bb} )/2$ .

2.4.2 Pesin center manifolds

We now briefly recall the properties of the Pesin center manifolds in our setting. There exists a pair of measurable functions $r:{\mathbb {T}_f^2}\to (0,1)$ and $\gamma :{\mathbb {T}_f^2}\to [1,+\infty )$ such that whenever $\tilde {y},\tilde {z}\in \widetilde {{\mathcal W}}^c_{r(\tilde {x})}(\tilde {x})$ , then

(2.1) $$ \begin{align} d(\tilde{f}^{-n}(\tilde{y}),\tilde{f}^{-n}(\tilde{z})\leq\gamma(\tilde{x})e^{-n\unicode{x3bb}}\,d(\tilde{y},\tilde{z}). \end{align} $$

The global Pesin manifolds, which we denote by $\widetilde W^c(\tilde {x})$ , are defined by the classical formula

$$ \begin{align*} \widetilde W^c(\tilde{x})\stackrel{\scriptscriptstyle\textrm{ def.}}{=}\bigcup_{n=0}^{+\infty}\tilde{f}^n(\widetilde W^c_{{\operatorname{loc}}}(\tilde{x}_{-n})). \end{align*} $$

Thus, $\widetilde W^c(\tilde {x})\subseteq \widetilde {{\mathcal W}}^c(\tilde {x})\subset \widetilde {{\mathcal W}}^{cu}(\tilde {x})$ is an immersed line. As a priori, we assume no uniform expansion along $E^c$ , the inclusion $\widetilde W^c(\tilde {x})\subseteq \widetilde {{\mathcal W}}^c(\tilde {x})$ could be strict. We define the global Pesin center unstable manifolds by

$$ \begin{align*} \widetilde W^{cu}(\tilde{x})=\bigcup_{\tilde{y}\in \widetilde W^c(\tilde{x})}\widetilde{{\mathcal W}}^u(\tilde{x}). \end{align*} $$

As in [Reference Alvarez, Leguil, Obata and SantiagoALOS24, Lemma 2.4], we have the following lemma.

2.4.3 The SRB property

Consider now a measurable partition ${\mathcal P}^{cu}$ subordinate to the Pesin center unstable foliation $\widetilde W^{cu}(\tilde {x})$ . This means a measurable partition whose atoms are entirely contained in a Pesin center-unstable manifold and project bijectively onto measurable subsets of ${\mathbb {T}^2}$ which contain a ball. More precisely, we give the following definition.

The construction of subordinate partitions in [Reference Ledrappier and StrelcynLS82] applies in our case.

Proof. Let us assume by contradiction that $\mu $ is not absolutely continuous with respect to $\text {Leb}_{{\mathbb {T}^2}}$ . Then, there exists a Borel measurable set $A\subset {\mathbb {T}^2}$ with $\mu (A)>0$ , but ${\text {Leb}_{{\mathbb {T}^2}}(A)=0}$ . Then, $\widetilde {A}\stackrel {\scriptscriptstyle \textrm { def.}}{=}\pi ^{-1}(A)$ satisfies $\tilde {\mu }(\widetilde {A})>0$ . By the regularity of $\mu $ , we may assume that A is compact and contained in an open set $U=B_r(x)$ for some $x\in A$ , such that $\pi |_{\pi ^{-1}(U)\cap \widetilde {{\mathcal W}}^{cu}(\tilde {y})}$ is a bijection for every $\tilde {y}\in \pi ^{-1}(x)$ . We may also choose the point x so that for $\mu _x$ -a.e. $\tilde {y}\in \pi ^{-1}(x)$ , the measure $\pi _*\tilde {\mu }^{{\mathcal P}}_{\tilde {y}}$ is absolutely continuous with respect to $\text {Leb}_{{\mathbb {T}^2}}$ . In particular, we have that

$$ \begin{align*} \tilde{\mu}^{{\mathcal P}}_{\tilde{y}}(\widetilde{A})=0 \end{align*} $$

for $\mu _x$ -a.e. $\tilde {y}\in \pi ^{-1}(x)$ . However, this implies that

$$ \begin{align*} \mu(A)=\tilde{\mu}(\widetilde{A})=\int\tilde{\mu}^{{\mathcal P}}_{\tilde{y}}(\widetilde{A})\,d\mu_x(\tilde{y})=0, \end{align*} $$

which gives a contradiction.

2.4.4 Distortion estimates

Given $\delta>0$ , we can choose a compact set ${\mathcal L}\subset {\mathbb {T}_f^2}$ with $\tilde {\mu }({\mathcal L})>1-\delta $ and such that the functions $C,r,\gamma $ are continuous when restricted to ${\mathcal L}$ . For this section, we fix $\delta>0$ and ${\mathcal L}$ .

The following result is a classical consequence of uniform expansion along unstable manifolds or center Pesin manifolds.

Proof. We write the proof for condition (2) and the center leaf, the others being analogous. If $\tilde {x} \in {\mathcal L}$ and $\tilde {y} \in \widetilde W^c_{r(\tilde {x})}(\tilde {x})$ , then

$$ \begin{align*}d(\tilde{f}^{-n}(\tilde{x}), \tilde{f}^{-n}(\tilde{y})) \leq \gamma_0 e^{-n\unicode{x3bb}}\, d(\tilde{x}, \tilde{y})\end{align*} $$

for all $n \in \mathbb {N}$ , where $\gamma _0=\max \gamma |_{{\mathcal L}}$ .

We have that

$$ \begin{align*} \bigg\vert\! \sum_{l=0}^{n-1} \varphi(\tilde{f}^{-l}(\tilde{x})) - \varphi(\tilde{f}^{-l}(\tilde{y})) \bigg\vert \leq K \sum_{l=0}^{n-1} d(\tilde{f}^{-l}(\tilde{x}), \tilde{f}^{-l}(\tilde{y}))^{\alpha}, \end{align*} $$

where $\varphi $ is $\alpha $ -Hölder with constant K. This is less or equal to

$$ \begin{align*} K \gamma_0\, d(\tilde{x}, \tilde{y}) \sum_{l=0}^{n-1} e^{-l\unicode{x3bb}} \leq K \gamma_0 \operatorname{diam}({\mathbb{T}_f^2}) \sum_{l=0}^{\infty} e^{-l\unicode{x3bb}}= C. \end{align*} $$

We immediately have the following result.

Lemma 2.29. There exists a constant $C=C(f)>1$ such that if $\tilde {f}^n(\tilde {y})\in {\mathcal W}^u_{1}(\tilde {f}^n(\tilde {x}))$ , then for every $\ell \leq n$ ,

$$ \begin{align*} \frac{\unicode{x3bb}^u_{\tilde{x}}(\ell)}{\unicode{x3bb}^u_{\tilde{y}}(\ell)}\asymp_{C}1. \end{align*} $$

Also, as a direct consequence of our choice of $\mathcal {L}$ , we have similar estimates for the center direction.

Lemma 2.30. Given a Lusin set ${\mathcal L}$ , there exists a constant $C=C({\mathcal L})>1$ such that if $\tilde {f}^n(\tilde {x})\in {\mathcal L}$ and $\tilde {f}^n(\tilde {y})\in \widetilde W^c_{r(\tilde {f}^n(\tilde {x}))}(\tilde {f}^n(\tilde {x}))$ , then for every $\ell \leq n$ ,

$$ \begin{align*} \frac{\unicode{x3bb}^c_{\tilde{x}}(\ell)}{\unicode{x3bb}^c_{\tilde{y}}(\ell)}\asymp_{C}1. \end{align*} $$

2.5.1 Growth estimate for norm $\|.\|_{L,\tilde {x}}$

When we measure the length of a vector ${v\in E^c(\tilde {x})}$ using the norm $\|.\|_{L,\tilde {x}}$ , we see expansion in a uniform fashion. The drawback is that the assignment $x\mapsto \|.\|_{L,\tilde {x}}$ is only measurable. In the following, we use the notation $\tilde {x}_n=\tilde {f}^n(\tilde {x})$ .

Lemma 2.32. There exists a constant $\sigma =\sigma (f)>1$ such that for $\tilde {\mu }$ -a.e. $\tilde {x}\in {\mathbb {T}_f^2}$ and every $v\in E^c(\tilde {x})$ , it holds that

$$ \begin{align*} e^{n\unicode{x3bb}}\|v\|_{L,\tilde{x}}\leq\|D\tilde{f}^n(\tilde{x})v\|_{L,\tilde{x}_n}\leq e^{n\sigma}\|v\|_{L,\tilde{x}}\quad\text{for every}\:\:n>0 \end{align*} $$

Proof. Take $v\in E^c(\tilde {x})\setminus \{0\}$ and $n>0$ . Then,

$$ \begin{align*} \|D\tilde{f}^n(\tilde{x})v\|^2_{L,\tilde{x}_n}=\sum_{k\leq 0}\|D\tilde{f}^k(\tilde{f}^n(\tilde{x}))D\tilde{f}^n(\tilde{x})v\|^2e^{-2\unicode{x3bb} k}=\sum_{k\leq 0}\|D\tilde{f}^{n+k}(\tilde{x})v\|^2e^{-2\unicode{x3bb} k}. \end{align*} $$

By making the change $\ell \rightarrow k+n$ in the sum, we obtain

$$ \begin{align*} \|D\tilde{f}^n(\tilde{x})v\|^2_{L,\tilde{x}_n}&=\sum_{\ell\leq 0}\|D\tilde{f}^{\ell}(\tilde{x})v\|^2e^{-2\unicode{x3bb}(\ell-n)}+\sum_{\ell=1}^{n-1}\|D\tilde{f}^{\ell}(\tilde{x})v\|^2e^{-2\unicode{x3bb}(\ell-n)}\nonumber \\ &\stackrel{\scriptscriptstyle\textrm {def.}}{=} \textrm{I}+\textrm{II}.\nonumber \end{align*} $$

Term I is equal to

$$ \begin{align*} \textrm{I}=e^{2\unicode{x3bb} n}\sum_{\ell\leq 0}\|D\tilde{f}^\ell(\tilde{x})v\|^2e^{-2\unicode{x3bb}\ell}=e^{2\unicode{x3bb} n}\|v\|^2_{L,\tilde{x}}. \end{align*} $$

As term II is non-negative, this gives immediately the lower bound. For the upper bound, take $\xi>\max \{\log \|D\tilde {f}(\tilde {x})\|;\tilde {x}\in {\mathbb {T}_f^2}\}>1$ . Notice that $n<e^{n\xi }$ for every $n>0$ . By definition, for every $\ell =1,\ldots ,n$ , we have

$$ \begin{align*} \|D\tilde{f}^\ell(\tilde{x})v\|^2\leq e^{2\xi\ell}\|v\|^2\leq e^{2\xi n}\|v\|^2. \end{align*} $$

This allows us to estimate term II above by

$$ \begin{align*} \textrm{II}&\leq\|v\|^2\sum_{\ell=1}^{n-1}e^{2\xi n}e^{-2\unicode{x3bb}(\ell-n)}\leq\|v\|^2\sum_{\ell=1}^{n-1}e^{4\xi n}e^{-2\unicode{x3bb} \ell}\nonumber\\ &\leq e^{4\xi n}\|v\|^2\sum_{\ell=1}^{n-1}e^{-2\unicode{x3bb}\ell}\leq ne^{4\xi n}\|v\|^2\nonumber\\ &\leq e^{5\xi n}\|v\|^2\leq e^{5\xi n}\|v\|_{L,\tilde{x}}^2\nonumber. \end{align*} $$

Thus,

$$ \begin{align*} \textrm{I}+\textrm{II}\leq(e^{2\unicode{x3bb} n}+e^{5\xi n})\|v\|_{L,\tilde{x}}\leq e^{7\xi n}\|v\|^2_{L,\tilde{x}}. \end{align*} $$

This establishes the upper bound and completes the proof.

2.5.2 u-saturation of the well-behaved set for $\|.\|_{L,\tilde {x}}$

Proof. As $\tilde {x}\mapsto \log \|D\tilde {f}(\tilde {x})|_{E^c}\|$ is Hölder continuous, it follows from Lemma 2.28 that

$$ \begin{align*} \frac{\unicode{x3bb}^c_{\tilde{x}}(k)}{\unicode{x3bb}^c_{\tilde{x}^u}(k)}\asymp_{C}1 \end{align*} $$

for some constant $C>1$ (which depends only on f and $d(\tilde {x},\tilde {x}^u)$ ). Since $E^c$ is one-dimensional, we can therefore write

$$ \begin{align*} \|v\|^2_{L,\tilde{x}^u}&=\sum_{k\leq 0}\|D\tilde{f}^k(\tilde{x}^u)v\|^2e^{-2\unicode{x3bb} k}\nonumber\\ &=\|v\|^2\sum_{k\leq 0}\unicode{x3bb}^c_{\tilde{x}^u}(k)^2e^{-2\unicode{x3bb} k}\nonumber\\ &\asymp_{C^2}\|v\|^2\sum_{k\leq 0}\unicode{x3bb}^c_{\tilde{x}}(k)^2e^{-2\unicode{x3bb} k}=\|v\|_{L,\tilde{x}}, \end{align*} $$

which implies the result. The proof for $\tilde {y}\in \widetilde {{\mathcal W}}^c_{r(\tilde {x})}(\tilde {x})$ is identical. Indeed, the argument for the distortion Lemma 2.28 shows that

$$ \begin{align*} \frac{\unicode{x3bb}^c_{\tilde{x}}(-n)}{\unicode{x3bb}^c_{\tilde{y}}(-n)}\asymp_C 1 \end{align*} $$

for every $n>0$ , where $C=C(\tilde {x})>1$ . It follows as before that $\|v\|^2_{L,\tilde {y}}\asymp _{C^2}\|v\|_{L,\tilde {x}}^2$ .

2.5.3 Distortion estimates

We consider now a Lusin set ${\mathcal L}\subset {\mathbb {T}_f^2}$ of measure $\tilde {\mu }({\mathcal L})>1-\delta $ .

Notation 2.34. Given $\tilde {x}\in {\mathbb {T}_f^2}$ and $n\in {\mathbb N}$ , we define $\hat {\unicode{x3bb} }^c_{\tilde {x}}(n) \stackrel {\scriptscriptstyle \textrm {def.}}{=} ({\|D\tilde {f}^n(\tilde {x})v\|_{L,\tilde {x}_n}})/({\|v\|_{L,\tilde {x}}})$ for $v\in E^c(\tilde {x})\setminus \{0\}$ . It is easy to check that

(2.2) $$ \begin{align} \hat{\unicode{x3bb}}^c_{\tilde{x}}(m+n) = \hat{\unicode{x3bb}}^c_{\tilde{f}^n(\tilde{x})}(m) \hat{\unicode{x3bb}}^c_{\tilde{x}}(n). \end{align} $$

Remark 2.35. Observe from Lemma 2.32, the following estimate:

$$ \begin{align*} e^{n\unicode{x3bb}}\leq\hat{\unicode{x3bb}}^c_{\tilde{x}}(n)\leq e^{n\sigma} \end{align*} $$

for every $n\in {\mathbb Z}$ and for a constant $0<\unicode{x3bb} <L^c(\mu )$ , which is fixed, and for $\sigma =\sigma (f)$ , the constant given in Lemma 2.32. This estimation only reflects the fact that when measuring growth with the Lyapunov norm, we see immediate expansion along $E^c$ . This will be of crucial importance later when we establish our quasi-isometric and synchronization estimates for stopping times.

We begin with a distortion result along Pesin center manifolds.

Lemma 2.36. There exists a constant $C=C({\mathcal L})>1$ such that, given $n>0$ , if $\tilde {x}_n\in {\mathcal L}$ and $\tilde {y}_n\in \widetilde W^c_{r(\tilde {x})}(\tilde {x}_n)$ , then

$$ \begin{align*} \frac{\hat{\unicode{x3bb}}^c_{\tilde{x}}(n)}{\hat{\unicode{x3bb}}^c_{\tilde{y}}(n)}\asymp_C 1. \end{align*} $$

Proof. Notice that we are under the same assumptions of Lemma 2.30. Let $C=C({\mathcal L})$ be the constant given by that lemma. Then, given any non-zero vector $v\in E^c(\tilde {x})$ , we have that

$$ \begin{align*} \hat{\unicode{x3bb}}^c_{\tilde{x}}(n)^2&=\frac{\|v\|^2}{\|v\|^2_{L,\tilde{x}}}\sum_{k\leq 0}\unicode{x3bb}^c_{\tilde{x}}(n+k)^2e^{-2\unicode{x3bb} k}\nonumber\\ &\asymp_{C^2}\frac{\|v\|^2}{\|v\|^2_{L,\tilde{x}}}\sum_{k\leq 0}\unicode{x3bb}^c_{\tilde{y}}(n+k)^2e^{-2\unicode{x3bb} k}\nonumber\\ &=\frac{\|v\|^2_{L,\tilde{y}}}{\|v\|^2_{L,\tilde{x}}}\hat{\unicode{x3bb}}^c_{\tilde{y}}(n)^2.\nonumber \end{align*} $$

The proof of Lemma 2.33 shows that $\|v\|_{L,\tilde {y}}/\|v\|_{L,\tilde {x}}\asymp _{C} 1$ . We deduce that

$$ \begin{align*} \hat{\unicode{x3bb}}^c_{\tilde{x}}(n)^2\asymp_{C^4}\hat{\unicode{x3bb}}^c_{\tilde{y}}(n)^2, \end{align*} $$

which gives the desired result.

We turn now to the case of two points on the same fiber of ${\mathbb {T}_f^2}$ . In this case, we need to assume that both points belong to the Lusin set. Indeed, for the next lemma, we demand another property of the Lusin set.

Lemma 2.37. Given a Lusin set ${\mathcal L}\subset {\mathbb {T}_f^2}$ there exists a constant $C=C({\mathcal L})>1$ such that whenever $\tilde {x},\tilde {y}\in {\mathcal L}$ with $\tilde {y}\in \Sigma (\tilde {x})$ , we have that

$$ \begin{align*} \frac{\hat{\unicode{x3bb}}^c_{\tilde{x}}(n)}{\hat{\unicode{x3bb}}^c_{\tilde{y}}(n)}\asymp_C1\quad\text{for all } n>0. \end{align*} $$

Proof. The function $\xi $ of Lemma 2.31 is uniformly bounded in ${\mathcal L}$ . Also, given any tangent vector $v\in E^c(\tilde {x})$ , there exists a unique (up to orientation) vector $\hat {v}\in E^c(\tilde {y})$ such that $\|v\|=\|\hat {v}\|$ .

Now, write as in the proof of Lemma 2.32 that

$$ \begin{align*} \|D\tilde{f}^n(\tilde{x})v\|^2_{L,\tilde{x}_n}=e^{2\unicode{x3bb} n}\|v\|^2_{L,\tilde{x}}+\sum_{j=1}^n\|D\tilde{f}^j(\tilde{x})v\|^2e^{-2\unicode{x3bb}(j-n)} \end{align*} $$

for all $v\in E^c(\tilde {x})$ and for all $n>0$ . Obviously, the same formula holds with $\tilde {x}$ and v replaced by $\tilde {y}$ and $\hat {v}$ . Since $\tilde {x},\tilde {y}\in {\mathcal L}$ , we have that

$$ \begin{align*} \|v\|^2_{L,\tilde{x}}\asymp_C\|v\|=\|\hat{v}\|\asymp_C\|\hat{v}\|^2_{L,\tilde{y}}. \end{align*} $$

The basic distortion lemma gives that $\|D\tilde {f}^j(\tilde {x})v\|\asymp _C\|D\tilde {f}^j(\tilde {y})\hat {v}\|$ . Up to enlarging the constant C, we get $\|D\tilde {f}^n(\tilde {x})v\|_{L,\tilde {x}}\asymp _C\|D\tilde {f}^n(\tilde {y})\hat {v}\|_{L,\tilde {y}}$ . Therefore, up to enlarging C once more, we have

$$ \begin{align*} \hat{\unicode{x3bb}}^c_{\tilde{x}}(n)\asymp_C\frac{\|v\|_{L,\tilde{x}}}{\|\hat{v}\|_{L,\tilde{y}}}{\hat{\unicode{x3bb}}^c_{\tilde{y}}(n)}\asymp_C\hat{\unicode{x3bb}}^c_{\tilde{y}}(n). \\[-42pt] \end{align*} $$

The following distortion result concerns points in the same unstable dynamical ball. The proof follows from the same calculations we have done in this section, so we refrain from repeating the details.

Lemma 2.38. There exists a constant $C=C(f)>1$ depending only on f such that for every $\tilde {\mu }$ generic point $\tilde {x}\in {\mathbb {T}_f^2}$ and every $\tilde {a}\in \widetilde {{\mathcal W}}^u_1(\tilde {x})$ , if $\tilde {a}_n\in \widetilde {{\mathcal W}}^u_1(\tilde {x}_n)$ for some $n\in {\mathbb N}$ , then

$$ \begin{align*} \frac{\hat{\unicode{x3bb}}^c_{\tilde{a}}(j)}{\hat{\unicode{x3bb}}^c_{\tilde{x}}(j)}\asymp_C 1\quad \text{for every}\:j=1,\ldots,n. \end{align*} $$

3.3.1 Proof of Lemma 3.1

Lemma 3.6 allows us to define the leaf-wise quotient measures, by considering $n \in {\mathbb N}$ sufficiently large,

$$ \begin{align*}\hat{\nu}^c_{\tilde{x}}(B) \stackrel{\scriptscriptstyle\textrm {def.}}{=} \dfrac{\nu^{cu}_{n, \tilde{x}}(I \times B)}{\nu^{cu}_{n, \tilde{x}}(I \times I_1)} = \dfrac{\tilde{\mu}^{cu}_{n, \tilde{x}}(\Phi_{\tilde{x}}(I \times B))}{\tilde{\mu}^{cu}_{n, \tilde{x}}(\Phi_{\tilde{x}}(I \times I_1))},\end{align*} $$

which gives us indeed a measure on ${\mathbb R}$ . Notice that $\hat {\nu }^c_{\tilde {x}}$ coincides with $\nu ^c_{n,x}$ on $Q^c_{n}(\tilde {x})$ (up to normalization). Thus, if $\hat {\nu }^c_{\tilde {x}}$ is absolutely continuous, so is $\nu ^c_{n,\tilde {x}}$ . By Lemma 3.5, this implies that $\nu ^{cu}_{n,\tilde {x}}$ is absolutely continuous with respect to the Lebesgue measure of ${\mathbb R}^2$ . Since $\Phi _{\tilde {x}}$ is $C^1$ , this proves that $\tilde {\mu }^{cu}_{n,\tilde {x}}$ is absolutely continuous with respect to the Lebesgue measure of the Pesin center manifold $W^{cu}(\tilde {x})$ . Thus, Lemma 3.1 follows from Lemma 2.27.

4.3.1 Proof of Lemma 4.5

Assume by contradiction the existence of a $\tilde {\mu }$ positive measure set $\tilde {A}$ such that $\tilde {\mu }_x({\mathcal P}(\tilde {x}))>0$ for every $\tilde {x}\in \tilde {A}$ . Consider the measurable function

$$ \begin{align*} \tilde{x}\mapsto\tilde{\mu}_x({\mathcal N}(\tilde{x})), \end{align*} $$

which is positive almost everywhere since $\tilde {\mu }(B)=0$ . By Lusin’s theorem, for every $\delta>0$ , there exists $c>0$ such that $\tilde {\mu }_x({\mathcal N}(\tilde {x}))\geq c$ if $\tilde {x}\in \tilde {K}$ for some compact set $\tilde {K}$ with $\tilde {\mu }_x(\tilde {K})>1-\delta $ . Take $\delta $ small enough such that $\tilde {\mu }(\tilde {K}\cap \tilde {A})>0$ and take $\tilde {x}\in \tilde {K}\cap \tilde {A}.$ By Rokhlin’s theorem and ergodicity, we can assume the existence of a point $\tilde {y}\in {\mathcal P}(\tilde {x})$ so that $\tilde {f}^{-n}(\tilde {y})\in \tilde {K}$ for infinitely many values of n. By Lemma 4.6, we can further assume that $\tilde {y}$ is a density point of ${\mathcal P}(\tilde {x})$ with respect to $\tilde {\mu }_x$ .

Observe that, in the metric given in Lemma 4.6, balls are cylinders. Therefore, there exists some cylinder $\tilde {y}\in I(\tilde {y})\subset \Sigma (\tilde {x})$ such that

$$ \begin{align*} \frac{\tilde{\mu}_x(I(\tilde{y})\cap{\mathcal P}(\tilde{x}))}{\tilde{\mu}_x(I(\tilde{y}))}>1-c. \end{align*} $$

By (4.1), we can take n large enough such that the following properties hold:

• • $\tilde {f}^{-n}(I(\tilde {y}))\supseteq \Sigma (\tilde {y}_{-n})$ ;

• • ${\mathcal P}(\tilde {f}^{-n}(\tilde {y}))=\tilde {f}^{-n}({\mathcal P}(\tilde {y}) \cap I(\tilde {y}))\cap \Sigma (\tilde {y}_{-n})$ ;

• • $\tilde {y}_{-n}\stackrel {\scriptscriptstyle \textrm {def.}}{=}\tilde {f}^{-n}(\tilde {y})\in \tilde {K}$ .

The first two properties above imply that

$$ \begin{align*} \tilde{\mu}_{\tilde{f}^{-n}(\tilde{x})}({\mathcal P}(\tilde{y}_{-n})) = \frac{\tilde{\mu}_{\tilde{f}^{-n}(\tilde{x})}(\tilde{f}^{-n}(I(\tilde{y})\cap{\mathcal P}(\tilde{x})))}{\tilde{\mu}_{\tilde{f}^{-n}(\tilde{x})}(\tilde{f}^{-n}(I(\tilde{y})))} = \frac{\tilde{\mu}_x(I(\tilde{y}\cap{\mathcal P}(\tilde{x})))}{\tilde{\mu}_x(I(\tilde{y}))}>1-c, \end{align*} $$

which, in turn, implies that $\tilde {\mu }_{\tilde {f}^{-n}(\tilde {y})}({\mathcal N}(\tilde {f}^{-n}(\tilde {y})))<c$ . By the definition of $\tilde {K}$ , this contradicts the third property above and completes the proof.

5.2.1 Center deviation along coupled configurations

When the points $\tilde {x},\tilde {x}^u,\tilde {y},\tilde {y}^u$ generate a quadrilateral, we say that the associated configurations $Y(\tilde {x},\tilde {x}^u,\ell ,\varepsilon )$ and $Y(\tilde {y},\tilde {y}^u,\ell ,\varepsilon )$ are coupled. This notion is devised so that one has a very precise control (up to uniform constants) on the displacement along central leaves after iteration by the stopping times.

The result below follows a similar strategy as [Reference Alvarez, Leguil, Obata and SantiagoALOS24, Lemma 9.2], but we need to be much more careful in our argument due to the lack of uniform expansion on the center. Notice that we also need to require more assumptions regarding which points belong to ${\mathcal L}$ . In particular, a subtle point is that we discuss a property of configurations whose upper time lengths m, $\hat {m}$ are the first and second stopping times, respectively.

Lemma 5.3. Given ${\mathcal L}\subset {\mathbb {T}_f^2}$ a compact Lusin set, $\beta>1$ (see Figures 3 and 4), there exist $\hat {\beta }=\hat {\beta }(\beta ,{\mathcal L})>1$ and $\ell _0=\ell _0({\mathcal L},\beta ,f)$ such that, if $\ell \geq \ell _0({\mathcal L})$ and $\varepsilon>0$ are such that there exists some length $\ell Y$ -configuration $Y(\tilde {x},\tilde {x}^u,\ell ,\varepsilon )$ which is coupled with another Y-configuration $Y(\tilde {y},\tilde {y}^u,\ell ,\varepsilon )$ so that $\tilde {x},\tilde {x}^u,\tilde {y},\tilde {y}^u$ generate a $(\beta ,\ell )$ quadrilateral and such that $\tilde {x}^u,\tilde {x}^u_\tau \in {\mathcal L}$ , where $\tau =\tau (\tilde {x},\tilde {x}^u,\ell ,\varepsilon )$ is the first stopping time, then

$$ \begin{align*} \varepsilon\hat{\beta}^{-1}\leq|{\mathcal R}^c_{\tilde{x}^u_\tau}(\tilde{y}^u_\tau)|\leq\varepsilon\hat{\beta}. \end{align*} $$

Figure 3

Coupled configurations on ${\mathbb {T}^2}$ (colour online).

Figure 4

Coupled configurations on the inverse limit space ${\mathbb {T}_f^2}$ (colour online).

Proof. From the definition of a quadrilateral, we know that $d^u(\tilde {x},\tilde {x}^u)<1$ . By the distortion estimate (Lemma 2.38), if we denote by J the segment contained in $\widetilde {{\mathcal W}}^u(\tilde {x})$ with end points $\tilde {x}$ and $\tilde {x}^u$ , we have, for every $\tilde {z}\in J$ , that

(5.6) $$ \begin{align} \unicode{x3bb}^u_{\tilde{z}_{-\ell}}(j)\asymp_C \unicode{x3bb}^u_{\tilde{x}_{-\ell}}(j) \end{align} $$

for every $j=1,\ldots ,\ell $ , where $C=C(f)>1$ is a constant depending only on f. One deduces that $d^u(\tilde {x}_{-\ell },\tilde {x}^u_{-\ell })\asymp _C(\unicode{x3bb} ^u_{\tilde {x}_{-\ell }}(\ell ))^{-1}$ . Thus, the unstable segment $f^{-\ell }(J)$ has length $\asymp _C(\unicode{x3bb} ^u_{\tilde {x}_{-\ell }}(\ell ))^{-1}$ . Since $\alpha (\tilde {x}_{-\ell },\tilde {y}_{-\ell })>\beta ^{-1}$ , a simple geometrical argument, using the continuity of the unstable bundle $E^u$ on the space ${\mathbb {T}_f^2}$ yields $d^c(\tilde {x}^u_{-\ell },\tilde {y}^u_{-\ell })\asymp _C(\unicode{x3bb} ^u_{\tilde {x}_{-\ell }}(\ell ))^{-1}$ for every $\ell \geq \ell _0(\beta )$ , where the new constant $C=C(f,\beta )$ now depends on $\beta $ .

We claim that

$$ \begin{align*} |{\mathcal R}^c_{\tilde{x}^u}(\tilde{y}^u)|\asymp_C \frac{\unicode{x3bb}^c_{\tilde{x}_{-\ell}}(\ell)}{\unicode{x3bb}^u_{\tilde{x}_{-\ell}}(\ell)} \end{align*} $$

for some constant $C=C({\mathcal L},f,\beta )$ . To prove this claim, first notice that the segment of center manifold joining the points $\tilde {x}^u$ and $\tilde {y}^u$ is contained inside the local Pesin manifold $\widetilde W^c_{{\operatorname {loc}}}(\tilde {x}^u)$ . Indeed, from the distortion Lemma 2.28, we deduce that

$$ \begin{align*} \operatorname{length}(\tilde{f}^{-\ell}(\widetilde W^c_{{\operatorname{loc}}} (\tilde{x}^u)))\asymp_C\unicode{x3bb}^c_{\tilde{x}^u_{-\ell}}(\ell)^{-1}\asymp_C\unicode{x3bb}^c_{\tilde{x}_{-\ell}}(\ell)^{-1} \end{align*} $$

for some constant $C=C({\mathcal L},\tilde {f},\beta )$ , depending also on ${\mathcal L}$ . Since $d^c(\tilde {x}^u_{-\ell },\tilde {y}^u_{-\ell })\asymp _C(\unicode{x3bb} ^u_{\tilde {x}_{-\ell }}(\ell ))^{-1}$ , the domination of the partial hyperbolic splitting implies that $d^c(\tilde {x}^u_{-\ell },\tilde {y}^u_{-\ell })$ is exponentially smaller than the length of $\tilde {f}^{-\ell }(\widetilde W^c_{{\operatorname {loc}}}(\tilde {x}^u))$ . Therefore, for $\ell $ sufficiently large (depending on the constant $C({\mathcal L},\beta ,f))$ , we have that the entire center segment with endpoints $\tilde {x}^u$ and $\tilde {y}^u$ is contained in the local Pesin manifold $\widetilde W^c_{{\operatorname {loc}}}(\tilde {x}^u)$ . The distortion Lemma 2.28 then yields

$$ \begin{align*} d^c(\tilde{x}^u,\tilde{y}^u)\asymp_C\unicode{x3bb}^c_{\tilde{x}^u_{-\ell}} (\ell)d^c(\tilde{x}^u_{-\ell},\tilde{y}^u_{-\ell})\asymp_C\unicode{x3bb}^c_{\tilde{x}{-\ell}} (\ell) (\unicode{x3bb}^u_{\tilde{x}_{-\ell}}(\ell))^{-1}. \end{align*} $$

Now, the claim follows, for on the Lusin set ${\mathcal L}$ , the quantity ${\mathcal R}^c_{\tilde {x}^u}(\tilde {y}^u)$ is uniformly comparable to $d^c(\tilde {x}^u,\tilde {y}^u)$ .

Since the dynamics acts as a linear expansion on normal forms, this gives us that

$$ \begin{align*} |{\mathcal R}^c_{\tilde{x}^u_\tau}(\tilde{y}^u_\tau)|\asymp_C \frac{\unicode{x3bb}^c_{\tilde{x}^u}(\tau)\times\unicode{x3bb}^c_{\tilde{x}_{-\ell}}(\ell)}{\unicode{x3bb}^u_{\tilde{x}_{-\ell}}(\ell)}. \end{align*} $$

Since $\tilde {x}^u_\tau \in {\mathcal L}$ , we have that $\unicode{x3bb} ^c_{\tilde {x}^u}(\tau )$ and $\hat {\unicode{x3bb} }^c_{\tilde {x}^u}(\tau )$ are comparable by a constant depending on ${\mathcal L}$ . The lemma now follows from Remark 5.1.