2 Background and main technique

We state the preliminary definitions needed for the technique and introduce how to apply the technique to deduce the desired thermodynamic formalism.

2.1 Pressure

Let $f : X \to X$ be a continuous map on a compact metric space. We identify the set $X \times \mathbb {N}$ with the space of finite orbit segments by identifying $(x, n)$ with $(x, f(x), \ldots , f^{n-1}(x)).$

The Bowen metric of order n associated to f is defined by

$$ \begin{align*} d_n(x,y) = \max_{0 \le k < n} d(f^k(x), f^k(y)). \end{align*} $$

Given $x \in X$ , $\epsilon> 0$ , and $n \in \mathbb {N}$ , the Bowen ball of order n with center x and radius $\epsilon $ is

$$ \begin{align*} B_n(x, \epsilon) = \{y \in X : d_n(x, y) < \epsilon\}. \end{align*} $$

A set $E \subset X$ is called $(n, \epsilon )$ -separated if

$$ \begin{align*} d_n(x, y)\, \ge\, \epsilon \quad\text{for all } x, y \in E \text{ with } x \neq y. \end{align*} $$

Given $\mathcal {D} \subset X \times \mathbb {N}$ , we interpret $\mathcal {D}$ as a collection of finite orbit segments. Define

$$ \begin{align*} \mathcal{D}_n = \{ x \in X : (x, n) \in \mathcal{D} \}, \end{align*} $$

the set of initial points of all length-n orbit segments in $\mathcal {D}$ . We then consider the corresponding partition sum

$$ \begin{align*} \Lambda_n^{\mathrm{sep}} (\mathcal{D}, \varphi, \epsilon; f) = \sup \bigg\{\!\sum_{x \in E} e^{S_n \varphi(x)} : E \subset \mathcal{D}_n \text{ is } (n,\epsilon)\text{-separated}\bigg\}. \end{align*} $$

Definition 2.1. (Pressure)

The pressure of $\varphi $ on $\mathcal {D}$ at scale $\epsilon $ is given by

$$ \begin{align*} P(\mathcal{D}, \varphi, \epsilon; f) = \limsup_{n \to \infty} \dfrac{1}{n} \log \Lambda_n^{\mathrm{sep}}(\mathcal{D}, \varphi, \epsilon; f). \end{align*} $$

Then, the pressure of $\varphi $ on $\mathcal {D}$ is defined as

$$ \begin{align*} P(\mathcal{D}, \varphi; f) = \lim_{\epsilon \to 0}\, P(\mathcal{D}, \varphi, \epsilon; f). \end{align*} $$

Given $Z \subset X$ , define

$$ \begin{align*} P(Z, \varphi, \epsilon; f) := P(Z \times \mathbb{N}, \varphi, \epsilon; f). \end{align*} $$

Observe that $P(Z, \varphi; f)$ denotes the usual upper capacity pressure [Reference BowenBow74] and we often write $P(\varphi; f)$ in place of $P(X, \varphi; f)$ for the pressure on the entire space.

When $\varphi = 0$ , our definition yields the entropy of $\mathcal {D}$ :

(2.1) $$ \begin{align} h(\mathcal{D}, \epsilon; f) = h(\mathcal{D}, \epsilon) := P(\mathcal{D}, 0, \epsilon) \quad\text{and}\quad h(\mathcal{D}) = \lim_{\epsilon \to 0} h(\mathcal{D}, \epsilon). \end{align} $$

Definition 2.2. (Equilibrium state)

Let $\mathcal {M}(f)$ be the set of f-invariant Borel probability measures and $\mathcal {M}_e(f)$ the subset of ergodic measures in $\mathcal {M}(f)$ . By the variational principle for pressure [Reference WaltersWal82, Theorem 9.10],

$$ \begin{align*} P(\varphi; f) = \sup_{\mu \in \mathcal{M}(f)}\bigg(h_\mu(f) + \int \varphi \,d\mu \bigg) = \sup_{\mu \in \mathcal{M}_e(f)}\bigg(h_\mu(f) + \int \varphi \,d\mu\bigg). \end{align*} $$

A measure achieving this supremum is called an equilibrium state.

2.2 Obstructions to expansivity, specification, and regularity

Bowen showed in [Reference BowenBow74] that if $(X, f)$ is expansive and has the specification property, and if $\varphi $ satisfies certain regularity conditions, then there is a unique equilibrium state. We recall definitions and results from [Reference Climenhaga and ThompsonCT16], which demonstrate that non-uniform analogs of Bowen’s hypotheses are sufficient to guarantee uniqueness.

2.2.1 Expansivity

Given a continuous map $f : X \to X$ , the infinite Bowen ball of size $\epsilon> 0$ around a point $x \in X$ is defined by

$$ \begin{align*} \Gamma_\epsilon(x) := \{ y \in X : d(f^k x, f^k y) < \epsilon \text{ for all } k \in \mathbb{{\mathbb{N}}} \}. \end{align*} $$

If there exists $\epsilon> 0$ such that $\Gamma _\epsilon (x) = \{x\}$ for every $x \in X$ , then we say that $(X, f)$ is positively expansive. Throughout the rest of the paper, we drop ‘positively’ and simply write expansive, since all maps under consideration are non-invertible.

Definition 2.3. For $f : X \to X$ , the set of non-expansive points at scale $\epsilon $ is defined by

$$ \begin{align*} \mathrm{NE}(\epsilon) = \{ x \in X : \Gamma_\epsilon(x) \neq \{x\} \}. \end{align*} $$

An f-invariant measure $\mu $ is called almost expansive at scale $\epsilon $ if $\mu (\mathrm {NE}(\epsilon )) = 0$ .

Given a potential $\varphi $ , the pressure of obstructions to expansivity at scale $\epsilon $ is

$$ \begin{align*} P_{\mathrm{exp}}^{\perp}(\varphi,\epsilon) &= \sup_{\mu \in \mathcal{M}_e(f)} \bigg\{h_\mu(f) + \int \varphi \, d\mu: \mu(\mathrm{NE}(\epsilon))> 0\bigg\}. \end{align*} $$

This quantity is non-increasing in $\epsilon $ , so we define a scale-free version by

$$ \begin{align*} P_{\mathrm{exp}}^{\perp}(\varphi) = \lim_{\epsilon \to 0} P_{\mathrm{exp}}^{\perp}(\varphi,\epsilon). \end{align*} $$

In particular, the scale-free version $ P_{\exp }^{\perp }(\varphi ) $ satisfies

$$ \begin{align*} P_{\exp}^{\perp}(\varphi) \le P_{\exp}^{\perp}(\varphi,\epsilon) \end{align*} $$

for any fixed $\epsilon> 0$ .

Roughly speaking, $ P^\perp _{\exp }(f,\epsilon ) $ can be understood as follows. Recall that positive expansivity is equivalent to the condition

$$ \begin{align*} \bigcap_{n \ge 0} B_n(x,\epsilon) = \{x\}, \end{align*} $$

which holds for every $ x \in X $ when $\epsilon $ is sufficiently small. We define $ P^\perp _{\exp }(f,\epsilon ) $ to be the supremum of the pressures of all ergodic measures that assign positive weight to the set of points where this condition fails.

2.2.2 Specification

Definition 2.4. (W-specification at scale $\epsilon $ )

A collection of orbit segments

$$ \begin{align*} \mathcal D \subset X\times{\mathbb{N}} \quad (\text{with } (x,n)\text{ representing the segment } x,\,f(x),\ldots,f^{n-1}(x)) \end{align*} $$

has the (W)-specification property at scale $\epsilon>0$ if there exists $\tau =\tau (\epsilon )\in {\mathbb {N}}$ such that, for every finite set of segments

$$ \begin{align*} \{(x_j,n_j)\}_{j=1}^{k}\subset\mathcal D, \end{align*} $$

one can find ‘gluing times’ $\tau _1,\ldots ,\tau _{k-1}\in {\mathbb {N}}$ with $\tau _i\le \tau $ for every i and a point $y\in X$ satisfying

$$ \begin{align*} d_{n_j}\!(f^{t_{j}}(y),x_j)<\epsilon \quad\text{for each }1\le j\le k, \end{align*} $$

where

$$ \begin{align*} t_1:=0,\quad t_j:=\sum_{i=1}^{j-1}(n_i+\tau_i)\quad (2\le j\le k). \end{align*} $$

Here, $d_{n}(x,y)=\max _{0\le m<n}d(f^{m}x,f^{m}y)$ is the Bowen metric of length n.

We say that $\mathcal {D}$ has the (W)-specification property if the above condition holds for every $\epsilon> 0$ . We say that $(X, f)$ has the (W)-specification property if $X \times \mathbb {N}$ does.

As in [Reference Climenhaga and ThompsonCT16], we will sometimes say that $\mathcal {G}$ has (W)-specification, or simply specification. This version of the specification is weak in the sense that the ‘transition times’ $\tau _i$ are only assumed to be bounded by, rather than equal to, $\tau $ .

Definition 2.5. A triple $(\mathcal {P}, \mathcal {G}, \mathcal {S}) \subset (X \times \mathbb {N})^3$ is called a decomposition for $(X, f)$ if for every $x \in X$ and $n \in \mathbb {N}$ , there exist $p, g, s \in \mathbb {N}$ with $p + g + s = n$ such that

$$ \begin{align*} (x, p) \in \mathcal{P},\quad (f^p x, g) \in \mathcal{G},\quad (f^{p+g} x, s) \in \mathcal{S}. \end{align*} $$

In other words, every orbit segment can be decomposed into a ‘prefix’ from $\mathcal {P}$ , a ‘good’ core from $\mathcal {G}$ , and a ‘suffix’ from $\mathcal {S}$ .

Note that the symbol $(x, 0)$ denotes the empty set and the functions $p, g, s$ are permitted to take the value zero.

Definition 2.6. The pressure of obstructions to specification at scale $\epsilon $ is defined by

$$ \begin{align*} P_{\mathrm{spec}}^{\perp}(\varphi, \epsilon) := \inf \{P(\mathcal{P} \cup \mathcal{S},\varphi, 3\epsilon)\}, \end{align*} $$

where the infimum is taken over $\mathcal {P}, \mathcal {S} \subset X \times {\mathbb {N}}$ such that there exists a decomposition $(\mathcal {P},\mathcal {G},\mathcal {S}$ ) for $(X,f)$ for which every $\mathcal {G}$ has specification at scale $\epsilon $ . We then define the scale-free version:

$$ \begin{align*} P_{\mathrm{spec}}^{\perp}(\varphi) = \lim_{\epsilon \to 0} P_{\mathrm{spec}}^{\perp}(\varphi, \epsilon). \end{align*} $$

Again, since $P_{\mathrm {spec}}^{\perp }(\varphi , \epsilon )$ is monotonic as a function of $\epsilon $ , this limit exists. In contrast to $P_{\mathrm {exp}}^{\perp }(\varphi , \epsilon )$ , $P_{\mathrm {spec}}^{\perp }(\varphi , \epsilon )$ is non-decreasing in $\epsilon $ .

2.2.3 Regularity for potential

Definition 2.7. (Bowen property)

Given $\mathcal {D} \subset X \times \mathbb {N}$ , we say a function $\varphi : X \to \mathbb {R}$ has the Bowen property on $\mathcal {D}$ at scale $\epsilon $ if there is a constant $K = K(\varphi , \mathcal {D}, \epsilon )$ such that

$$ \begin{align*} | S_n \varphi(x) - S_n \varphi(y) | < K \end{align*} $$

for every $(x, n) \in \mathcal {D}$ and every $y \in B_n(x, \epsilon )$ . A function $\varphi $ has the Bowen property on $\mathcal {D}$ if it satisfies the Bowen property on $\mathcal {D}$ at some scale $\epsilon $ (and thus at all smaller scales as well).

2.3 General results on uniqueness of equilibrium states

Our main tool for existence and uniqueness of equilibrium state is [Reference Climenhaga and ThompsonCT16, Theorem 5.5].

Theorem 2.8. [Reference Climenhaga and ThompsonCT16, Theorems 5.5 and 5.7]

Let X be a compact metric space and $f \colon X \to X$ a continuous map. Let $\varphi \colon X \to \mathbb {R}$ be a continuous potential function. Suppose that

$$ \begin{align*} P_{\mathrm{exp}}^{\perp}(\varphi) < P_{\mathrm{top}}(\varphi) \end{align*} $$

and that $(X, f)$ admits a decomposition $(\mathcal {P}, \mathcal {G}, \mathcal {S})$ with the following properties:

1. (1) $\mathcal {G}$ has (W)-specification at any scale;

2. (2) $\varphi $ has the Bowen property on $\mathcal {G}$ ;

3. (3) $P(\mathcal {P} \cup \mathcal {S}, \varphi ) < P(\varphi; f )$ .

Then, there is a unique equilibrium state $\mu $ for $\varphi $ . Moreover, $\mu $ satisfies the following upper large deviations bound: if A is any weak*-closed and convex subset of the space of Borel probability measures on X, then

$$ \begin{align*} \limsup_{n\rightarrow\infty} \dfrac{1}{n} \log\mu(\mathcal{E}_n^{-1}(A)) \le \sup_{\nu\in A\cap \mathcal{M}(f)}\bigg(h_{\nu}(f)+ \int \varphi \,d\nu - P(\varphi; f)\bigg), \end{align*} $$

where $\mathcal {E}_n(x) = ({1}/{n})\sum _{k=0}^{n-1} \delta _{f^kx}$ is the nth empirical measure associated to x.

We note that the theorem we apply assumes that the pressures of obstructions to both expansivity and specification are defined in a scale-free manner.

3 Viana maps and their properties

We begin by gathering the necessary materials for the construction of Viana maps with the desired decomposition properties.

We consider from here on these maps f close to $f_{\alpha }$ restricted to $S^1 \times I$ . The main properties of f in a $C^3$ neighborhood of $f_{\alpha }$ are summarized below; see [Reference AlvesAlv00, Reference VianaVia97].

• • Non-uniformly expanding, that is, for $\alpha $ sufficiently small, there exists a positive constant $c_0> 0$ such that any map f sufficiently close to $f_{\alpha }$ in the $C^3$ topology has both Lyapunov exponents greater than $c_0$ at Lebesgue-almost every (a.e.) point, that is, for Lebesgue-a.e. $x=(\theta , t) \in S^1 \times I$ ,

  $$ \begin{align*}\liminf_{n\rightarrow \infty} \dfrac{1}{n} \log \|Df^n(x)\|> c_0.\end{align*} $$

• • Slow approximation to the critical set, that is, letting

  $$ \begin{align*} \mathcal C := \{x \in S^1 \times I : \det|Df_x|=0\} \end{align*} $$
  denote the critical set of f, for every $\gamma>0$ , there exists $\delta> 0$ such that for Lebesgue-a.e. $x \in S^1 \times I$ ,
  $$ \begin{align*} \limsup_{n\rightarrow\infty} \dfrac{1}{n}\sum_{j=0}^{n-1} -\log [\mathrm{dist}_{\delta}(f^j(x),\mathcal{C})]\leq \gamma. \end{align*} $$

  Here, $\mathrm {dist}_{\delta }$ is the $\delta $ -truncated distance defined by

  $$ \begin{align*}\mathrm{dist}_{\delta}(x,\mathcal{C}) = \begin{cases} 1 &\quad \text{if }\mathrm{dist}(x,\mathcal{C}) \geq \delta,\\ \mathrm{dist}(x,\mathcal{C}) &\quad \text{otherwise}. \end{cases}\end{align*} $$

• • Existence and uniqueness of an ergodic absolutely continuous invariant (SRB) measure $\mu _{\mathrm {SRB}}$ .

• • The attractor of f within the forward-invariant region $S^1 \times I$ is defined as the intersection

  $$ \begin{align*} \Lambda =\bigcap_{n\geq 0} f^n(S^1\times I) \end{align*} $$
  of all forward images of $S^1 \times I$ , which is just $\Lambda = f^2(S^1 \times I)$ , as long as the interval I is properly chosen.

• • Strongly topologically mixing: for any open set $ O \subset S^1 \times I $ , there exists $n= n(O) \in \mathbb {N} $ such that

  $$ \begin{align*} f^{n}(O) = \Lambda. \end{align*} $$

Next, we study the dynamics of the tangent bundle for Viana maps. Denote by $ f_0 $ the direct product map

$$ \begin{align*} f_0 = g \times h: S^1 \times I \to S^1 \times I, \quad (\theta, t) \mapsto (d\theta\, \mod 1, a_0 - t^2). \end{align*} $$

Since $ Df_0 \cdot ({\partial }/{\partial \theta }) = d \cdot ({\partial }/{\partial \theta }) $ and $ Df_0 \cdot ({\partial }/{\partial t}) = -2t \cdot ({\partial }/{\partial t}) $ , with $ d \geq 16> 4 $ and $ I \subset (-2, 2) $ , it is clear that $ f_0 $ is a partially hyperbolic map of the form $ T_{(\theta ,t)}(S^1 \times I) = E^u \oplus E^c $ , where $ E^u $ is uniformly expanding and $ E^c $ is dominated by $ E^u $ . From the theory of partial hyperbolicity, we know that if $ \alpha $ is sufficiently small, the map $ f_{\alpha } $ (like any $ C^1 $ -nearby map) also admits a partially hyperbolic splitting of the form $ E^u \oplus E^c $ , which varies continuously. As a consequence, we get the following proposition.

Proposition 3.1. Given $\epsilon> 0$ , if $\alpha $ is sufficiently small, the Lyapunov exponent

$$ \begin{align*}\unicode{x3bb}^u(x, f):= \lim_{n\rightarrow\infty}\frac{1}{n}\log \|Df^n|_{E^u(x)}\|\end{align*} $$

associated to $E^u$ at every point x satisfies

$$ \begin{align*}\log(d-\epsilon)\leq \unicode{x3bb}^u(x, f) \leq \log(d+\epsilon)\end{align*} $$

for any f close to $f_{\alpha }$ .

By applying the previous theorems, we derive a lower bound on the entropy of the SRB measure $ \mu _{\mathrm {SRB}}$ . By Pesin’s formula, the entropy $ h_{\mu _{\mathrm {SRB}}}(f) $ of $ \mu _{\mathrm {SRB}} $ is given by the integrated sum of the positive Lyapunov exponents of $ \mu _{\mathrm {SRB}} $ . Therefore, by Proposition 3.1, we obtain the following bound:

$$ \begin{align*} h_{\mu_{\mathrm{SRB}}}(f) \geq \log(d - \epsilon) + c_0. \end{align*} $$

4 Hyperbolic times and non-uniformly expanding maps

To advance the proof, we recall the notion of hyperbolic time for maps with a critical set, following the framework of Alves, Bonatti, and Viana [Reference Alves, Bonatti and VianaABV00], specialized to our partially hyperbolic setting.

Let $f\colon M \to M$ be a $C^2$ map that is a local diffeomorphism except on a set $\mathcal {C} \subset M$ of zero Lebesgue measure. We assume that there is a continuous $Df$ -invariant splitting $E^u \oplus E^c$ on M and a continuous cone field $\mathrm {Cone}^c = \{\mathrm {Cone}^c(x)\}_{x\in M}$ around $E^c$ such that $\mathrm {Cone}^c(x)$ contains the critical (center) direction at points of $\mathcal {C}$ .

We say that f is non-flat along the critical direction if there exist constants ${B>1}$ and $\ell>0$ such that, for all $x,y \notin \mathcal {C}$ with $2\,\mathrm {dist}(x,y) < \mathrm {dist}(x,\mathcal {C})$ and for every $v \in \mathrm {Cone}^c(x)\setminus \{0\}$ , we have:

• • $({1}/{B}) \mathrm {dist}(x, \mathcal {C})^l \leq {\|Df(x)v\|}/{\|v\|} \leq B \mathrm {dist}(x, \mathcal {C})^l;$

• • $|\!\log \!\|Df (x)^{-1}\|\!- \log \|Df (y)^{-1}\||\!\leq B ({\mathrm {dist}(x,y)}/{\mathrm {dist}(x, \mathcal {C})^l}) ;$

• • $|\!\log \!|\det Df (x)^{-1}|\!- \log \!|\det Df (y)^{-1}||\!\leq B ({\mathrm {dist}(x,y)}/{\mathrm {dist}(x, \mathcal {C})^l}).$

By dominated splitting and uniform control of the angle between $E^u$ and $E^c$ , these estimates along $\mathrm {Cone}^c$ are equivalent up to a uniform multiplicative constant to the corresponding estimates for the full operator norm $\|Df(x)^{-1}\|$ and the Jacobian $|\!\det Df(x)|$ , exactly as in [Reference Alves, Bonatti and VianaABV00].

In our application, we take $M = \Lambda $ , $\mathcal {C}$ equal to the critical set of the fiber map, $E^c$ the center bundle, and $\mathrm {Cone}^c$ a center cone field. As the reader can verify, the reference Viana map $f_{\alpha }$ satisfies these non-flatness assumptions along the center direction and since these conditions are $C^2$ -open, they hold for all Viana maps in a sufficiently small $C^2$ -neighborhood of $f_{\alpha }$ .

For the definition of hyperbolic times, we fix $0 < b < \min \{1/2, 1/(2\beta )\}$ .

Definition 4.1. Given $0 < \sigma < 1$ and $\delta> 0$ , we say that n is a $(\sigma ,\delta )$ -hyperbolic time for x if, for all $1 \leq k \leq n$ ,

$$ \begin{align*}\prod_{j=n-k}^{n-1}\|Df(f^j(x))^{-1}\| \leq \sigma^k\end{align*} $$

and

$$ \begin{align*}\mathrm{dist}_{\delta}(f^{n-k}(x),\mathcal{C}) \geq \sigma^{bk}.\end{align*} $$

Hyperbolic times are useful because they have the following key property.

Remark. In our setting, the non-flatness assumption is verified along the center direction, that is, on a $Df$ -invariant cone field $\mathrm {Cone}^c$ around $E^c$ and [Reference Alves, Bonatti and VianaABV00, Lemma 5.2] is applied to $\sup _{v\in \mathrm {Cone}^c(x), \|v\|=1}\|Df(x)^{-1}v\|$ . By dominated splitting and the uniform bound on the angle between $E^u$ and $E^c$ , there exists $C\ge 1$ such that

$$ \begin{align*} \sup_{v\in \mathrm{Cone}^c(x), \|v\|=1}\|Df(x)^{-1}v\| \kern1.5pt{\le}\kern1.5pt \|Df(x)^{-1}\| \kern1.5pt{\le}\kern1.5pt C\,\kern-0.9pt\sup_{v\in \mathrm{Cone}^c(x), \|v\|=1}\kern-1pt\|Df(x)^{-1}v\| \kern-1pt\quad\text{for all }x, \end{align*} $$

so the notion of $(\sigma ,\delta )$ -hyperbolic time used in Definition 4.1, which is formulated with the full operator norm $\|Df^{-1}\|$ , is equivalent to the one used in [Reference Alves, Bonatti and VianaABV00], up to adjusting $\sigma $ . Thus, Lemma 4.2 is precisely [Reference Alves, Bonatti and VianaABV00, Lemma 5.2] adapted to our center cone field setting.

Definition 4.3. For a fixed $\sigma <1$ , we define the expanding set $H(\sigma )$ to be the set of points $x\in M$ with the following two properties:

$$ \begin{align*}\limsup_{n\rightarrow \infty} \dfrac{1}{n}\sum_{j=0}^{n-1}\log\|Df(f^j(x)^{-1})\| \leq 3\log\sigma < 0\end{align*} $$

and, for any $\gamma> 0$ , there is some $\delta> 0$ satisfying

$$ \begin{align*}\limsup_{n\rightarrow \infty} \dfrac{1}{n}\sum_{j=0}^{n-1}-\log(\mathrm{dist}_{\delta}(f^j(x),\mathcal{C})) \leq \gamma.\end{align*} $$

5 Non-uniform expansivity

In this section, we show that Viana maps are systems for which expansivity fails, but this failure occurs in a lower entropy setting, giving rise to a form of non-uniform expansivity.

5.2 Slow recurrence

To overcome the loss of expansion caused by folding, we need to control the recurrence to the critical set $\mathcal {C}$ of generic points of invariant ergodic measures whose Lyapunov exponents are regular.

Lemma 5.1. Let $\eta $ be an ergodic measure such that $\unicode{x3bb} ^c(\eta )> -\infty $ , where $\unicode{x3bb} ^c$ is the Lyapunov exponent of $\eta $ associated to $E^c$ . Then,

$$ \begin{align*}\int |\!\log \mathrm{dist} (x, \mathcal{C})| \,d\eta < \infty.\end{align*} $$

Proof. Since $\eta $ is ergodic, $\unicode{x3bb} ^c(\eta ) = \int \log \|Df|_{E^c}\|\,d\eta> -\infty $ . However, the definition of $f_{\alpha }$ and $\alpha $ sufficiently small implies $\tfrac 13 \|Df(x)|_{E^c}\| \leq \mathrm {dist} (x, \mathcal {C}) \leq 3 \|Df(x)|_{E^c}\|$ . These two facts together finish the proof.

Lemma 5.2. Let $\eta $ be an ergodic measure with $\unicode{x3bb} ^c(\eta )> -\infty $ . Then, for any $\gamma> 0$ , there exists $\delta> 0$ such that for $\eta $ -a.e. point x,

$$ \begin{align*} \limsup_{n \to \infty} \dfrac{1}{n} \sum_{j=0}^{n-1}(-\!\log \mathrm{dist}_\delta(f^j(x), \mathcal{C})) \le \gamma. \end{align*} $$

Proof. By Lemma 5.1, the function $|\!\log \mathrm {dist}(x, \mathcal {C})|$ is $\eta $ -integrable and, in particular, $\eta (\mathcal {C}) = 0$ . Using these facts together with the definition of the $\delta $ -truncated distance $\mathrm {dist}_\delta $ , we deduce that for any $\gamma> 0$ , there exists $\delta> 0$ such that

$$ \begin{align*} \int -\log \mathrm{dist}_\delta(x,\mathcal{C}) \, d\eta \le \gamma. \end{align*} $$

Because $\eta $ is ergodic, an application of Birkhoff’s ergodic theorem to the function $-\log \mathrm {dist}_\delta (x, \mathcal {C})$ completes the proof.

5.3 Almost expansivity

Next, we prove that, for Viana maps, any ergodic f-invariant measure with large pressure must be positively expansive. We prove this using the strategy outlined below; see [Reference Arbieto, Matheus and SentiAMS05, Reference OliveiraOli03].

• • Select a certain set $\mathcal {K}$ of ergodic f-invariant measures, which are natural candidates for attaining the variational principle.

• • Prove that any measure outside $\mathcal {K}$ have strictly smaller pressure than the topological pressure hence cannot be an equilibrium state.

• • Show that any measure inside $\mathcal {K}$ is an expanding measure and any expanding measure is almost positively expansive.

We begin by describing $\mathcal {K}$ , the set of candidate measures for achieving the variational principle. Define

$$ \begin{align*} \mathcal{K} = \bigg\{\mu \in \mathcal{M}_e(f):\unicode{x3bb}^c(\mu)> \dfrac{c_0}{3}\bigg\}. \end{align*} $$

That is, $\mathcal {K}$ consists of those ergodic f-invariant measures whose central Lyapunov exponent is strictly bounded below by ${c_0}/{3}$ . The set $\mathcal {K}$ is non-empty, as it contains the unique SRB measure $\mu _{\mathrm {SRB}}$ , which has $\unicode{x3bb} ^c(\mu _{\mathrm {SRB}})> c_0$ by Pesin’s entropy formula.

5.3.1 Low pressure outside $\mathcal {K}$

The measures in $\mathcal {K}$ are natural candidates for realizing the variational principle because the measures outside of $\mathcal {K}$ have small pressure.

Lemma 5.3. For every $\eta \notin \mathcal K$ ,

$$ \begin{align*} h_{\eta}(f)+\int \varphi\,d\eta \le P(\varphi;f)-\frac{c_0}{6}. \end{align*} $$

In particular,

$$ \begin{align*} \sup_{\eta\in\mathcal M_e(f)\setminus\mathcal K} \bigg\{h_{\eta}(f)+\int \varphi\,d\eta\bigg\} \le P(\varphi;f)-\frac{c_0}{6}. \end{align*} $$

Proof. We compare $h_{\eta }(f) + \int \varphi \, d\eta $ with $h_{\mu _{\mathrm {SRB}}}(f) + \int \varphi \, d\mu _{\mathrm {SRB}}$ . Since $\eta \notin \mathcal {K}$ , it follows that $\unicode{x3bb} ^c(\eta ) \le ({c_0}/{3})$ . Therefore,

$$ \begin{align*} h_{\eta}(f) + \int \varphi \, d\eta &\le \log(d+\epsilon) +\frac{c_0}{3}+ \int \varphi \, d\eta \\ &\le \log(d+\epsilon) +\frac{c_0}{3} + \int \varphi \, d\mu_{\mathrm{SRB}} + (\sup\varphi-\inf\varphi) \\ &< \log(d+\epsilon) +\frac{c_0}{3} + \bigg(\frac{c_0}{2}-\log \frac{d+\epsilon}{d-\epsilon}\bigg)+ \int \varphi \, d\mu_{\mathrm{SRB}} \\ &= \log(d-\epsilon)+\frac{5c_0}{6}+\int \varphi \, d\mu_{\mathrm{SRB}} \\ &\le h_{\mu_{\mathrm{SRB}}}(f) + \int \varphi \, d\mu_{\mathrm{SRB}} -\frac{c_0}{6}. \end{align*} $$

Here, the first inequality follows from Ruelle’s inequality, with the unstable Lyapunov exponent at most $\log (d+\epsilon )$ . The second inequality uses $\int \varphi \,d\eta \le \int \varphi \,d\mu _{\mathrm {SRB}}+(\sup \varphi -\inf \varphi )$ . The strict inequality uses our assumption $\sup \varphi - \inf \varphi < ({c_0}/{2})-\log ({(d+\epsilon)}/{(d-\epsilon)})$ . The last inequality uses Pesin’s entropy formula so that $h_{\mu _{\mathrm {SRB}}}(f)\ge \log (d-\epsilon )+c_0$ . By the variational principle, this proves that $h_{\eta }(f) + \int \varphi \, d\eta \le P(\varphi; f )-({c_0}/{6})$ .

5.3.2 Expanding measures

We define the notion of expanding measures originally introduced in [Reference Alves, Bonatti and VianaABV00].

Definition 5.4. A measure $\mu $ is called a $\sigma ^{-1}$ -expanding measure if $x \in H(\sigma )$ for $\mu $ -a.e. point x. In other words, $\mu (H(\sigma )) = 1$ . Here, $H(\sigma )$ is the $\sigma ^{-1}$ -expanding set defined in Definition 4.3.

We now show that there exists a single $\sigma> 1$ such that every $\mu \in \mathcal {K}$ is $\sigma ^{-1}$ -expanding. As will become clear later, a uniform choice of $\sigma $ across all $\mu \in \mathcal {K}$ is necessary for establishing that every ergodic measure in $\mathcal {K}$ is almost expansive at the same scale.

Lemma 5.5. Any $\mu \in \mathcal {K}$ is $\sigma ^{-1}$ -expanding with $\sigma = \exp (-{c_0}/{9})$ .

Proof. Since $\mu $ is ergodic, we have

$$ \begin{align*} \unicode{x3bb}^c(x) = \unicode{x3bb}^c(\mu) = \int \log\|Df(x)|_{E^c}\| \, d\mu> \frac{c_0}{3} \end{align*} $$

for $\mu $ -a.e. x. By Birkhoff’s ergodic theorem, this is equivalent to

$$ \begin{align*} \lim_{n \to \infty} \dfrac{1}{n} \sum_{j=0}^{n-1} \log \|Df(f^j(x))^{-1}\| \leq 3 \cdot \bigg( -\frac{c_0}{9} \bigg) < 0, \end{align*} $$

because the central direction $E^c$ is one-dimensional.

This fact, combined with Lemma 5.2, gives us the assumptions needed for $H(\sigma )$ at $\mu $ -a.e. point.

5.3.3 Expansivity inside $\mathcal {K}$

Finally, we conclude the proof that any $\sigma ^{-1}$ -expanding measure $\mu $ is almost expansive at scale $\delta _1$ , which implies every $\mu \in \mathcal {K}$ is almost expansive.

Lemma 5.6. For any $\sigma ^{-1}$ -expanding measure $\mu $ , there exists $\delta _1> 0$ such that for any $\epsilon < \delta _1$ and for $\mu $ -a.e. x,

$$ \begin{align*} \Gamma_{\epsilon}(x) = \{x\}. \end{align*} $$

Proof. Proposition 4.4 guarantees that for $\mu $ -a.e. x, there are infinitely many hyperbolic times $n_i(x)$ , since $\mu $ is $\sigma ^{-1}$ -expanding. Therefore, applying Lemma 4.2, we conclude that there exists some $\delta _1> 0$ such that if $y \in \Gamma _{\epsilon }(x)$ with $\epsilon < \delta _1$ , then for any $n_i$ , we have

$$ \begin{align*} \text{dist}(x, y) \leq \sigma^{n_i/2} \, \text{dist}(f^{n_i}(x), f^{n_i}(y)) \leq \sigma^{n_i/2} \epsilon. \end{align*} $$

Since $n_i \to \infty $ as $i \to \infty $ , we deduce that $x = y$ .

5.4 Pressure gap

Theorem 5.7. We have the pressure gap

$$ \begin{align*} P_{\mathrm{exp}}^{\perp}(\varphi) < P(\varphi; f ). \end{align*} $$

Proof. By definition, for any $\epsilon < \delta _1$ ,

$$ \begin{align*} P_{\mathrm{exp}}^{\perp}(\varphi,\epsilon) = \sup_{\mu \in \mathcal{M}_e(f)} \bigg\{h_\mu(f) + \int \varphi \, d\mu : \mu(\mathrm{NE}(\epsilon))> 0\bigg\}. \end{align*} $$

Consider two cases for a measure $\mu $ .

Case 1: $\mu \notin \mathcal {K}$ . By Lemma 5.3, we have a uniform bound:

$$ \begin{align*} h_{\mu}(f)+\int \varphi\,d\mu < P(\varphi; f )- \frac{c_0}{6}. \end{align*} $$

In particular,

$$ \begin{align*} \sup_{\mu\in\mathcal M_e(f)\setminus\mathcal K} \bigg\{h_{\mu}(f)+\int \varphi\,d\mu\bigg\} \le P(\varphi;f)-\frac{c_0}{6}. \end{align*} $$

Hence, such measures cannot achieve a pressure at $P(\varphi; f )$ .

Case 2: $\mu \in \mathcal {K}$ . By Lemma 5.6, there exists $\delta _1> 0$ so that for all $\epsilon < \delta _1$ ,

$$ \begin{align*} \mu(\mathrm{NE}(\epsilon)) = 0. \end{align*} $$

Thus, for sufficiently small $\epsilon $ , these measures place no mass on $\mathrm {NE}(\epsilon )$ and do not contribute to $P_{\mathrm {exp}}^{\perp }(\varphi ,\epsilon )$ .

In summary, for small enough $\epsilon $ , every measure that contributes to $P_{\mathrm {exp}}^{\perp }(\varphi ,\epsilon )$ lies outside $\mathcal {K}$ , and each such measure has pressure below $P(\varphi; f )$ . Consequently,

$$ \begin{align*} P_{\mathrm{exp}}^{\perp}(\varphi) = \lim_{\epsilon\rightarrow 0} P_{\mathrm{exp}}^{\perp}(\varphi,\epsilon) < P(\varphi; f ). \end{align*} $$

This completes the proof.

6 Non-uniform specification and regularity

In this section, we first give an explicit description of how to decompose the collection of all orbit segments $(x,n)$ into $(\mathcal {P}, \mathcal {G},\mathcal {S})$ . We show that $\mathcal {G}$ has the specification property at any scale $\varepsilon $ and that orbits not in $\mathcal {G}$ carry small pressure. In addition, any Hölder continuous $\varphi $ has the Bowen property on $\mathcal {G}$ at some fixed scale $\varepsilon _0$ .

6.1 Construction of the orbit decomposition

Let $\psi ^c(x) = \log \|Df|_{E^c(x)}\|$ , so that $({1}/{n}) S_n \psi ^c(x)$ represents the average expansion (or contraction) undergone by an orbit segment $(x, n)$ in the center direction. Additionally, given $\mu \in \mathcal {M}_e(f)$ , $\unicode{x3bb} ^c(\mu ) =\int \psi ^c\, d\mu $ is the center Lyapunov exponent of $\mu $ .

To define the orbit decomposition (Figure 1), for some small $r>0$ , we consider the following collection of orbit segments:

(1) $$ \begin{align} \begin{aligned} \mathcal{P} &:=\emptyset, \\ \mathcal{G} &:=\{(x,n)\in \Lambda\times {\mathbb{N}}:\,S_{k}\psi^c(f^{n-k}x)> kr \text{ for all }\, 0< k\leq n\},\\ \mathcal{S} &:=\{(x,n)\in \Lambda\times {\mathbb{N}}:\, S_n\psi^c(x) \le nr\}. \end{aligned} \end{align} $$

Figure 1

Orbit decomposition (color online).

Figure 1 Long description

The top panel shows a horizontal line from x at the left to f to the n minus 1 x at the right, with multiple arcs above. Blue arcs span from x to intermediate points, while red arcs cluster near the right endpoint. The lower panel, below a downward double arrow, shows the same interval split at f to the n minus s minus 1 x, with blue arcs on the left labeled as belonging to script G and red arcs on the right labeled as script S. The right side contains a legend: a blue arc is labeled S sub k psi superscript c open parenthesis f to the n minus k x close parenthesis greater than k r, a red arc is labeled S sub k psi superscript c open parenthesis f to the n minus k x close parenthesis less than or equal to k r. Below, a yellow box shows two red arcs above a single red arc, with a downward double arrow indicating merging. All labels and mathematical expressions are rendered exactly as shown.

The collection $\mathcal {G}$ is chosen so that along orbit segments from $\mathcal {G}$ , the center-unstable manifolds are uniformly backward contracting. In addition, f is locally invertible along these ‘good’ orbits.

Together with the trivial collection $\{(x,0) : x \in \Lambda \}$ for $\mathcal {P}$ , these collections define a decomposition of any orbit $(x, n) \in \Lambda \times \mathbb {N}$ as follows: let s be the largest integer in $\{0, \ldots , n\}$ such that $ S_{s}\psi ^c(f^{n-s}x) \le sr$ , meaning $(f^{n-s}x, s) \in \mathcal {S}$ .

We must then have $(x, n-s) \in \mathcal {G}$ , since if $S_{k_0}\psi ^c(f^{n-s-k_0} x) \le k_0r$ for some $0 < k_0 \leq n-s$ , then

$$ \begin{align*} S_{s+k_0}\psi^c(f^{n-s-k_0}x) = S_{k_0}\psi^c(f^{n-s-k_0}x) + S_{s}\psi^c(f^{n-s}x) \le (s+k_0)r, \end{align*} $$

which would contradict the maximality of s.

6.2 Specification on $\mathcal {G}$

In the uniformly hyperbolic setting, a key step in establishing specification on a mixing, locally maximal hyperbolic set $\Lambda $ is that for any $\delta> 0$ , there exists $N \in \mathbb {N}$ such that for all $x, y \in \Lambda $ and $n \ge N$ ,

$$ \begin{align*} f^n(W_\delta^u(x)) \cap W_\delta^s(y) \neq \varnothing. \end{align*} $$

For Viana maps, we adapt this idea by replacing the local stable manifold with the point y itself and the local unstable manifold with the center-unstable manifold $W_\delta ^{cu}(x)$ . Indeed, Viana maps can be viewed as partially hyperbolic systems with one-dimensional unstable and center directions. Because each center-unstable manifold is merely a two-dimensional disk and every such disk is eventually mapped onto $\Lambda $ by the strong mixing property, the same construction effectively applies.

To prove specification on $\mathcal {G}$ , we begin by establishing that every ‘good’ orbit in $\mathcal {G}$ exhibits backward uniform contraction.

Lemma 6.1. There exist $\varepsilon _0>0$ and $C\ge 1$ such that for every $(x,n)\in \mathcal G$ and every $y\in B_n(x,\varepsilon _0)$ , we have

(2) $$ \begin{align} d(f^k x, f^k y) < C e^{-r(n-1-k)}\, d(f^{n-1}x, f^{n-1}y) < C e^{-r(n-1-k)}\,\varepsilon_0 \end{align} $$

for all $0\le k\le n-1$ .

Proof. From $(x,n)\in \mathcal G$ and the definition of $\psi ^c$ , we obtain a uniform backward contraction along the center:

$$ \begin{align*} \|Df^{-(n-1-k)}|_{E^c(f^{n-1} x)}\|< e^{-r(n-1-k)},\quad 0\le k\le n-1. \end{align*} $$

Uniform expansion along $E^u$ gives

$$ \begin{align*} \|Df^{-(n-1-k)}|_{E^u(f^{n-1} x)}\|< e^{-r(n-1-k)}. \end{align*} $$

Since the system is partially hyperbolic, the $Df$ -invariant splitting $TM = E^u \oplus E^c$ is continuous and dominated, with $\dim E^u = \dim E^c = 1$ . In particular, the angle between $E^u$ and $E^c$ is uniformly bounded away from zero, so there is a constant $C\ge 1$ such that

$$ \begin{align*} \|Df^{-(n-1-k)}(z)\| < C e^{-r(n-1-k)} \end{align*} $$

for every point z in a small neighborhood of $f^{n-1} x$ and every $0\le k\le n-1$ . Shrinking this neighborhood if needed and using compactness, we may choose $\varepsilon _0>0$ so that whenever $d(z,f^n x)<\varepsilon _0$ , the above estimate holds for all such k.

Now, take $y\in B_n(x,\varepsilon _0)$ . Then, $d(f^{n-1} x,f^{n-1} y)<\varepsilon _0$ . Let $\gamma $ be a minimizing geodesic from $f^{n-1} x$ to $f^{n-1} y$ ; it is contained in the $\varepsilon _0$ -ball around $f^{n-1} x$ . For each $0\le k\le n-1$ , we have

$$ \begin{align*} d(f^k x,f^k y)&= d(f^{-(n-1-k)} f^{n-1} x, f^{-(n-1-k)} f^{n-1} y) \\ &< \sup_{w\in\gamma} \|Df^{-(n-1-k)}(w)\|\cdot d(f^{n-1} x,f^{n-1} y), \end{align*} $$

and the supremum is bounded by $C e^{-r(n-1-k)}$ . This gives (2).

Theorem 6.2. The collection $\mathcal {G}$ has specification at any scale $\varepsilon>0$ .

Proof. Since having specification at some scale $\varepsilon _*>0$ implies specification at every larger scale $\varepsilon \ge \varepsilon _*$ , it is enough to prove the theorem for all sufficiently small $\varepsilon>0$ . We assume from now on that $0<\varepsilon \le \varepsilon _0$ , where $\varepsilon _0$ is given by Lemma 6.1.

Let $\{(x_i, n_i)\}_{i=1}^k$ be any finite collection of orbit segments from $\mathcal G$ . Since $(x_i,n_i)\in \mathcal G$ , Lemma 6.1 gives backward uniform contraction along this ‘good’ orbit: for all $y\in B(x_i,\varepsilon )$ and $0\le m\le n_i-1$ ,

$$ \begin{align*} d(f^m x_i, f^m y) < C e^{-r(n_i-1-m)}\,d(f^{n_i-1}x_i, f^{n_i-1}y) < C e^{-r(n_i-1-m)}\varepsilon. \end{align*} $$

Thus, the inverse branch of $f^{n_i-1}$ that maps back to $x_i$ is uniformly contracting on a neighborhood of $f^{n_i-1}x_i$ , with contraction rate $e^{-r}$ up to a uniform multiplicative constant. In particular, $f^{n_i-1}$ maps $B_{n_i}(x_i,\varepsilon )$ diffeomorphically onto an open neighborhood of $f^{n_i-1}x_i$ that contains the open disk $B(f^{n_i-1}x_i,\varepsilon /C)$ .

From the compactness and the strongly topologically mixing property of f, for any $\varepsilon>0 $ , there exists some $\tau = \tau (\varepsilon ) \in \mathbb {N}$ such that for any open disk $B(x,\varepsilon /C)$ , we have $f^j(B(x, \varepsilon /C)) = \Lambda $ for some $j \leq \tau $ . Hence, for any $i = 1, \ldots , k-1$ , we can select $\tau _i \leq \tau (\varepsilon )$ such that the image of $B(f^{n_i-1} x_i, \varepsilon /C)$ under $f^{\tau _i}$ covers $B_{n_{i+1}}(x_{i+1}, \varepsilon )$ .

As a consequence of these, we define

$$ \begin{align*} t_1=0 \quad \text{and} \quad t_j=\sum_{i=1}^{j-1} (n_i-1 + \tau_i) \quad \text{ for } j=2, \ldots, k \end{align*} $$

and set

$$ \begin{align*} U_1 := B_{n_1}(x_1, \varepsilon). \end{align*} $$

For $j=1,\ldots ,k-1$ , define

$$ \begin{align*} U_{j+1} := \{u\in U_j : f^{t_{j+1}}u \in B_{n_{j+1}}(x_{j+1},\varepsilon)\}. \end{align*} $$

By definition, each $U_j$ is open and the family is nested:

$$ \begin{align*} U_k \subset \cdots \subset U_{j+1} \subset U_j \subset \cdots \subset U_1. \end{align*} $$

We claim that for every $j=1,\ldots ,k$ :

1. (1) $U_j\neq \emptyset $ ;

2. (2) $f^{t_j}(U_j)=B_{n_j}(x_j,\varepsilon )$ .

For $j=1$ , this is clear, since $t_1=0$ and $U_1 = B_{n_1}(x_1,\varepsilon )$ .

Assume that items (1) and (2) hold for some $1\le j\le k-1$ . Let $z\in B_{n_{j+1}}(x_{j+1},\varepsilon )$ be arbitrary. By the choice of $\tau _j$ , there exists

$$ \begin{align*} w \in B (f^{n_j-1}x_j,\varepsilon/C) \subset W_j = f^{n_j-1}(B_{n_j}(x_j,\varepsilon)) \end{align*} $$

such that $f^{\tau _j}w = z$ . Since $w\in W_j$ , there is a unique $y\in B_{n_j}(x_j,\varepsilon )$ with $f^{n_j-1}y = w$ . By the induction hypothesis, $f^{t_j}(U_j)=B_{n_j}(x_j,\varepsilon )$ , so we can choose $u\in U_j$ with $f^{t_j}u = y$ . Then,

$$ \begin{align*} f^{t_{j+1}}u = f^{t_j+n_j-1+\tau_j}u = f^{\tau_j}w = z\in B_{n_{j+1}}(x_{j+1},\varepsilon), \end{align*} $$

so $u\in U_{j+1}$ . Since z was arbitrary, this shows that $f^{t_{j+1}}(U_{j+1}) = B_{n_{j+1}}(x_{j+1},\varepsilon )$ and, in particular, $U_{j+1}\neq \emptyset $ . This completes the induction.

Thus, all $U_j$ are non-empty and

$$ \begin{align*} U_k \subset \cdots \subset U_2 \subset U_1, \end{align*} $$

so

$$ \begin{align*} \bigcap_{j=1}^{k} U_j = U_k \neq \emptyset. \end{align*} $$

Choose any $y \in \bigcap _{j=1}^{k} U_j$ . Then, for every $1\le j\le k$ , we have $f^{t_j}(y) \in B_{n_j}(x_j,\varepsilon )$ , which is equivalent to

$$ \begin{align*} d_{n_j}(f^{t_{j}}(y),x_j)<\varepsilon \quad \text{for every }1\le j\le k. \end{align*} $$

Thus, the orbit of $y \varepsilon $ -shadows the collection of orbit segments $\{(x_j,n_j)\}_{j=1}^k$ with gaps bounded by $\tau (\varepsilon )$ , and $\mathcal G$ has specification at scale $\varepsilon $ .

6.3 Pressure gap

We prove that if $0<r\leq ({c_0}/{3})$ , then for any bounded potential $\varphi $ , the ‘bad orbits’ carry less topological pressure than the whole space. Recall we define $\psi ^c(x) = \log \|Df|_{E^c(x)}\|$ and

$$ \begin{align*} \mathcal{S} := \{ (x, n) \in \Lambda \times \mathbb{N} : S_n \psi^c(x) \leq nr \}. \end{align*} $$

We have the following pressure gap estimate.

Theorem 6.3. The collection of bad orbits from the decomposition carries pressure strictly smaller than the topological pressure:

$$ \begin{align*} P(\mathcal {P} \cup \mathcal {S}, \varphi )< P(\varphi; f ). \end{align*} $$

Proof. We start by taking $\mathcal {P} = \emptyset $ . To describe $P(\mathcal {S},\varphi )$ , recall we have proven in Lemma 5.3 that

$$ \begin{align*} \sup_{\mu \in \mathcal{M}_f} \bigg\{ h_{\mu}(f) +\int \varphi \,d\mu: \unicode{x3bb}^c(\mu) \leq \frac{c_0}{3} \bigg\} < P(\varphi; f ). \end{align*} $$

So, for $0<r\leq ({c_0}/{3})$ ,

(3) $$ \begin{align} \sup_{\mu \in \mathcal{M}(f)} \bigg\{ h_{\mu}(f) +\int \varphi \,d\mu: \unicode{x3bb}^c(\mu) \leq r \bigg\} < P(\varphi; f ). \end{align} $$

Since we define $\psi ^c(x) = \log \| Df|_{E^c(x)} \|$ , for an f-invariant measure $\mu $ , its center Lyapunov exponent satisfies $\unicode{x3bb} ^c(\mu ) = \int \psi ^c \, d\mu $ .

The left-hand side of (3) can be related to $P(\mathcal {S},\varphi )$ . Let $E_n \subset \mathcal {S}_n$ be any $(n,\varepsilon )$ -separated set. Consider the Borel probability measure

$$ \begin{align*} \nu_n = \frac{1}{\sum_{x\in E_n}e^{S_n\varphi(x)}}\, \sum_{x\in E_n}\delta_x e^{S_n\varphi(x)}, \quad \mu_n = \frac{1}{n}\sum_{k=0}^{n-1} f_*^k \nu_n. \end{align*} $$

Then, half of the proof of the variational principle (see [Reference WaltersWal82, Theorem 8.6]) shows that if $\mu $ is a limit point of $\mu _n$ , then $\mu $ is f-invariant and satisfies

$$ \begin{align*} P_{\mu}(\varphi) = h_{\mu}(f) +\int \varphi\, d\mu \geq P(\mathcal{S}, \varphi, \varepsilon). \end{align*} $$

Moreover, by weak $^*$ -convergence and the definition of $\mathcal {S}$ , we have

$$ \begin{align*} \unicode{x3bb}^c(\mu) = \int \psi^c \, d\mu \leq r. \end{align*} $$

Combining this with (3), we conclude that

$$ \begin{align*} P(\mathcal{S}, \varphi) < P(\varphi; f ). \end{align*} $$

Since $\mathcal {P} = \emptyset $ , this implies $P(\mathcal {P} \cup \mathcal {S}, \varphi ) < P(\varphi; f )$ . By definition, $P_{\mathrm {spec}}^{\perp }(\varphi )<P(\varphi; f )$ , which completes the proof.

6.4 Bowen property for potential

Theorem 6.4. Any Hölder continuous potential $\varphi $ satisfies the Bowen property on $\mathcal {G}$ at scale $\varepsilon _0$ .

Proof. Because $\varphi $ is Hölder, there exist constants $K>0$ and $\alpha \in (0,1)$ such that

$$ \begin{align*} |\varphi(x) - \varphi(y)| \le K\,(d(x,y))^\alpha\quad \text{for all } x,y \in \Lambda. \end{align*} $$

Take $(x,n)\in \mathcal {G}$ and $y \in B_n(x,\varepsilon )$ . By Lemma 6.1,

$$ \begin{align*} |S_n \varphi(x) - S_n \varphi(y)| \le K \sum_{k=0}^{n-1} (d(f^k x,\, f^k y))^\alpha < K \sum_{k=0}^{n-1} (Ce^{-r\,(n-1-k)}\,\varepsilon)^\alpha. \end{align*} $$

Re-indexing with $j = n-1-k$ and noting the resulting geometric series,

$$ \begin{align*} K\sum_{k=0}^{n-1}(Ce^{-r(n-1-k)}\,\varepsilon)^\alpha < KC^{\alpha}\,\varepsilon^\alpha\sum_{j=0}^{\infty} e^{-r\,\alpha\,j} < \infty. \end{align*} $$

Hence, $|S_n\varphi (x) - S_n\varphi (y)|$ is bounded uniformly in $x,n,$ and y, establishing the Bowen property at scale $\varepsilon $ .

7 Proof of main theorems

We now complete the proof that if f is a $C^3$ -perturbation of $f_{\alpha }$ and $\varphi :\Lambda \rightarrow {\mathbb {R}}$ , which satisfy the hypotheses of Theorem A, then the conditions of Theorem 2.8 are satisfied and hence there is a unique equilibrium state $\mu $ for $(\Lambda , f, \varphi )$ . Moreover, $\mu $ satisfies the upper level-2 large deviation principle.

By Theorem 5.7, $P_{\mathrm {exp}^+}^{\perp }(\varphi ) < P(\varphi; f )$ . We define the decomposition $(\mathcal {P}, \mathcal {G}, \mathcal {S})$ as in (1). In Theorem 6.2, we showed that $\mathcal {G}$ has specification at all scales. In Theorem 6.4, we showed that $\varphi $ has the Bowen property on $\mathcal {G}$ . By Theorem 6.3, $P(\mathcal {P} \cup \mathcal {S}, \varphi ) < P(\varphi; f )$ . Thus, we see that under the hypotheses of Theorem A, all the hypotheses of Theorem 2.8 are satisfied for the decomposition $(\mathcal {P}, \mathcal {G}, \mathcal {S})$ .