Proof. Let $\operatorname {vol}_f^u$ and $\operatorname {vol}_A^u$ denote Lebesgue measure along, respectively, strong-unstable leaves of f and unstable leaves of A. It is clear that we may find $C>1$ such that

(10) $$ \begin{align} \frac{1}{C} \operatorname{vol}^u_A(W^u_i(\pi(x))) \le \operatorname{vol}^u_f(\xi_i^u(x)) \le C \operatorname{vol}^u_A(W^u_i(\pi(x))) \end{align} $$

for every $i=1, \ldots , k$ and every $x\in {\mathcal M}_i$ . By construction, for every $n \ge 1$ , the image $A^n(W_i^u(\pi (x)))$ is an (essentially disjoint) union of unstable plaques $W_j^u(b)$ and, similarly, the image $f^n(\xi _i^u(x))$ is an (essentially disjoint) union of strong-unstable plaques $\xi _j^u(y)$ . Moreover, there is a one-to-one correspondence between the unstable plaques $W_j^u(b)$ inside $A^n(W_i^u(\pi (x)))$ and the strong-unstable plaques $\xi _j^u(y)$ inside $f^n(\xi _i^u(x))$ preserving the index j. Thus, equation (10) implies that

$$ \begin{align*} \frac{1}{C} \operatorname{vol}^u_A(A^n(W^u_i(\pi(x)))) \le \operatorname{vol}^u_f(f^n(\xi_i^u(x))) \le C \operatorname{vol}^u_A(A^n(W^u_i(\pi(x)))) \end{align*} $$

for every $n \ge 1$ , $i=1, \ldots , k$ , and $x\in {\mathcal M}_i$ . This implies that the two sequences of volumes grow at the same exponential rate, as claimed.

Proof. First, suppose that $h_w(A,{\mathcal W}^u)=0$ . By [Reference Viana and YangVY17, Proposition 2.5], this happens if and only if there is a full w-measure subset X that intersects every unstable leaf of A in at most one point. Then, $\pi ^{-1}(X)$ intersects every strong-unstable leaf of f in at most one point, since $\pi $ is a homeomorphism from strong-unstable leaves of f to unstable leaves of A. Using [Reference Viana and YangVY17, Proposition 2.5] once more, we conclude that $h_\mu (f,{\mathcal F}^{uu})=0$ .

From now on, suppose that $h_w(A,{\mathcal W}^u)>0$ . Initially, assume that $\mu $ is ergodic. Let $\eta ^u$ be any partition subordinate to the unstable foliation of A. Then, by definition,

(12) $$ \begin{align} h_{w}(A,{\mathcal W}^u)=H_{w}(A^{-1}\eta^{u} \mid \eta^{u}). \end{align} $$

Denote by $\xi ^{cu}$ the pull-back $\pi ^{-1}(\eta ^u)$ of $\eta ^u$ . Then, since $\pi _*(\mu )=w$ , we have

(13) $$ \begin{align} H_{\mu}(f^{-1}\xi^{cu} \mid \xi^{cu}) = H_{w}(A^{-1}\eta^u \mid \eta^u). \end{align} $$

Next, consider $\xi ^{u} = \xi ^{cu} \vee {\mathcal F}^{uu}$ , that is, the partition of M whose elements are the intersections of the elements of $\xi ^{cu}$ with the strong-unstable leaves of f. It is clear that $\xi ^{u}$ is finer than $\xi ^{cu}$ . Moreover, our assumptions on $\pi $ ensure that $\xi ^{u}$ is subordinate to ${\mathcal F}^{uu}$ . Thus,

(14) $$ \begin{align} h_\mu(f,{\mathcal F}^{uu})=H_{\mu}(f^{-1} \xi^{u} \mid \xi^{u}). \end{align} $$

We claim that

$$ \begin{align*} f^{-1} \xi^{u} = f^{-1}\xi^{cu} \bigvee \xi^{u}. \end{align*} $$

Indeed, given any $x\in M$ , both $f^{-1}\xi ^{u}(x)$ and $f^{-1}\xi ^{cu}(x) \cap \xi ^{u}(x)$ are contained in the strong-unstable leaf through x. Moreover, both are mapped to $A^{-1}\eta ^u (\pi (x))$ under $\pi $ . Since $\pi $ is a homeomorphism on strong-unstable leaves, it follows that the two sets coincide, as claimed. Now, using [Reference RokhlinRok67, §5.10], it follows that

(15) $$ \begin{align} \begin{aligned} H_{\mu}(f^{-1} \xi^{cu} \mid \xi^{cu}) & \geq H_{\mu}(f^{-1} \xi^{cu} \mid \xi^{u})\\ & = H_{\mu}\bigg(f^{-1} \xi^{cu} \bigvee \xi^{u} \mid \xi^{u}\bigg) = H_\mu(f^{-1} \xi^{u} \mid \xi^{u}). \end{aligned} \end{align} $$

Combining this with equations (12), (13), and (14), we get the statement of the proposition when $\mu $ is ergodic.

The general case now follows from the fact that the u-entropy is affine. Indeed, equation (11) means that

(16) $$ \begin{align} h_\mu(f,{\mathcal F}^{uu})= \int_{\mathcal P} h_{\mu_{P(x)}}(f,{\mathcal F}^{uu}) \, d\mu(x). \end{align} $$

Analogously,

(17) $$ \begin{align} h_\mu(A,{\mathcal W}^{u})= \int_{\mathcal Q} h_{w_{Q(a)}}(A,{\mathcal W}^{u}) \, dw(a). \end{align} $$

Here, ${\mathcal Q}$ is the partition of ${\mathbb T}^d$ defined by the condition that two points are in the same partition element if and only if they have the same Birkhoff average for A. It is clear that $\pi ^{-1}({\mathcal Q})$ is coarser than ${\mathcal P}$ . Thus, $\pi (P(x)))$ is contained in ${\mathcal Q}(\pi (x))$ and $\pi _*\mu _{P(x)} = w_{Q(\pi (x))}$ for almost every x. By the ergodic case treated previously, we also have that

$$ \begin{align*} h_{\mu_{P(x)}}(f,{\mathcal F}^{uu}) \le h_{w_{Q(\pi(x))}}(A,{\mathcal W}^{u}). \end{align*} $$

Now the claim follows by combining this with equations (16) and (17).

Proof. Let $\{\mu ^{cu}_y: y \in M\}$ denote a disintegration of $\mu $ relative to the partition $\xi ^{cu}=\pi ^{-1}(\eta ^u)$ . Since $w=\pi _*\mu $ , the fact that the disintegration is essentially unique implies that

(18) $$ \begin{align} \pi_*\mu_y^{cu} = w^u_{\pi(y)} \quad\text{for }\mu\text{-almost every }y. \end{align} $$

Since $\xi ^{u}$ is finer than $\xi ^{cu}$ , the conditional probabilities of each $\mu ^{cu}_y$ relative to the partition $\xi ^{u}$ coincide $\mu $ -almost everywhere with the conditional probabilities $\mu ^u_x$ of $\mu $ itself (transitivity of the disintegration, see [Reference RokhlinRok67, §1.7] or [Reference Viana and OliveiraVO16, Exercise 5.2.1]).

First, we prove the ‘only if’ claim. Assume that $h_\mu (f,{\mathcal F}^{uu}) = h_w(A,{\mathcal W}^u)$ . Then, in view of the relations in equations (12), (13), (14), and (15), the identity must hold in the latter:

(19) $$ \begin{align} H_{\mu}(f^{-1} \xi^{cu} \mid \xi^{cu}) = H_{\mu}(f^{-1} \xi^{cu} \mid \xi^{u}). \end{align} $$

Moreover, those arguments are not affected if we replace f with any forward iterate, just because ${\mathcal F}^{uu}$ remains the strong-unstable foliation. Thus, in this way, we get that

(20) $$ \begin{align} H_{\mu}(f^{-n} \xi^{cu} \mid \xi^{cu}) = H_{\mu}(f^{-n} \xi^{cu} \mid \xi^{u}) \end{align} $$

for every $n \ge 1$ . By [Reference RokhlinRok67, §5.10], this can only happen if $f^{-n} \xi ^{cu}$ and $\xi ^{u}$ are independent relative to $\xi ^{cu}$ , that is (see [Reference RokhlinRok67, §1.7]), if

$$ \begin{align*} \mu^{cu}_y(A \cap B) = \mu^{cu}_y(A) \mu^{cu}_y(B) \end{align*} $$

for any $f^{-n}\xi ^{cu}$ -measurable set A, any $\xi ^{u}$ -measurable set B, and $\mu $ -almost every y. See Figure 1.

Figure 1 Proving the ‘only if’ claim of Corollary 3.4.

By considering B arbitrarily small , we see that the conditional measure $\mu _x^u(A)$ is independent of x, and so

(21) $$ \begin{align} \mu_x^u(A) = \mu^{cu}_x(A) \end{align} $$

for $\mu $ -almost every y, $\mu ^{cu}_y$ -almost every x in $\xi ^{cu}(y)$ , and any $f^{-n}\xi ^{cu}$ -measurable set A. Moreover, equation (18) implies that $\mu ^{cu}_x(A)=w_{\pi (x)}^u(\pi (A))$ for every $f^{-n}\xi ^{cu}$ -measurable set A. This implies that

(22) $$ \begin{align}\pi_*\mu_x^u=\pi_*(\mu_x^{cu}) = w_{\pi(x)}^u \end{align} $$

restricted to the $\sigma $ -algebra of $f^{-n}\xi ^{cu}$ -measurable subsets of each $\xi ^{u}(x)$ , for any n. Now, since f is (uniformly) expanding along strong-unstable leaves, the family of partitions $\{f^{-n}\xi ^{cu}: n\ge 1\}\cap \xi ^{u}(x)$ generates the $\sigma $ -algebra of all measurable subsets of $\xi ^{u}(x)$ for every x. Thus, the previous identity yields the ‘only if’ statement: $\pi _*\mu _x^u=\pi _*(\mu _x^{cu}) = w_{\pi (x)}^u$ for $\mu $ -almost every x.

Next, assume that $\pi _*\mu ^u_x = w^u_{\pi (x)}$ for $\mu $ -almost every x. Using equation (18), it follows that $\pi _*\mu ^u_x = \pi _*\mu ^{cu}_x$ for $\mu $ -almost every x, which implies that

(23) $$ \begin{align} \mu_x^u(A) = \mu^{cu}_x(A) \end{align} $$

for $\mu $ -almost every x and any $f^{-n}\xi ^{cu}$ -measurable set A. Compare equation (21). Recalling that

$$ \begin{align*} \begin{aligned} H_{\mu}(f^{-1} \xi^{cu} \mid \xi^{cu}) & = \int H_{\mu^{cu}_x}(f^{-1}(\xi^{cu}){\mid}_{\xi^{cu}(x)}) \, d\mu(x) \quad\text{and}\\ H_{\mu}(f^{-1} \xi^{cu} \mid \xi^{u}) & = \int H_{\mu^{u}_x}(f^{-1}(\xi^{cu}){\mid}_{\xi^{u}(x)}) \, d\mu(x), \end{aligned} \end{align*} $$

we see that equation (23) implies equation (19). Using once more the relations in equations (12)–(15), we see that the later implies that $h_\mu (f,{\mathcal F}^{uu}) = h_w(A,{\mathcal W}^u)$ . This proves the ‘if’ statement.

Finally, recall that $\pi $ is a homeomorphism onto its image restricted to each strong-unstable leaf and, thus, to each element of $\xi ^{u}$ . Thus, the relation $\pi _*\mu ^u_x = w^u_{\pi (x)}$ means that each $\mu ^u_x$ is the push-forward of $w^u_{\pi (x)}$ under the inverse map $\pi ^{-1}$ . This implies the last claim in the statement.

Proof. By Proposition 3.3 and Lemma 3.1, every $\mu \in \operatorname {MM}^u(f)$ satisfies

$$ \begin{align*} h_w(A,{\mathcal W}^u)\geq h_\mu(f,{\mathcal F}^{uu})=h(f,{\mathcal F}^{uu})=h(A). \end{align*} $$

By equation (3), we also have that $h_w(A) \ge h_w(A,{\mathcal W}^u)$ . It follows that w is a measure of maximal entropy for A which, by uniqueness, implies that w is the Lebesgue measure of ${\mathbb T}^d$ . Since the total boundary $\partial {\mathcal R}$ of the Markov partition ${\mathcal R}$ has zero Lebesgue measure, as observed at the beginning of §2.2, it follows that the $\mu $ -measure of $\partial {\mathcal M}$ is equal to zero. Let $\xi ^{u}$ be the family of all $\xi ^{u}_i(x)$ with x in the interior of some ${\mathcal M}_i$ , and, similarly, $\eta ^u$ be the family of all $W^u_i(a)$ with a in the interior of some ${\mathcal R}_i$ . By the previous observation, these are partitions of (full measure subsets of) M and ${\mathbb T}^d$ , respectively. Moreover, $\eta ^u$ is subordinate to the unstable foliation of A, and $\xi ^{u}$ is its pull-back under the map $\pi $ to each strong-unstable leaf of f, which means that these two partitions verify the assumptions of Proposition 3.3 and Corollary 3.4. This means that we are in a position to apply the corollary, and conclude that

$$ \begin{align*} \pi_*\mu_x^u = \operatorname{vol}^u_{\pi(x)} = \text{normalized Lebesgue measure on }\eta^u_{\pi(x)} \end{align*} $$

for $\mu $ -almost every x. This proves that $\mu $ is a c-Gibbs u-state.

Proof. Let $\{\mu _P: P \in {\mathcal P}\}$ be the ergodic decomposition of a c-Gibbs u-state $\mu $ , and $\hat \mu $ be the corresponding quotient measure in ${\mathcal P}$ . This means that every $\mu _P$ is an ergodic mesure and

(24) $$ \begin{align} \mu = \int_{\mathcal P} \mu_P \, d\hat\mu(P). \end{align} $$

For each i, let $\mu _i = \mu {\mid} _{{\mathcal M}_i}$ and ${\mathcal P}_i$ be the restriction of ${\mathcal P}$ to the domain ${\mathcal M}_i$ . For each $P\in {\mathcal P}_i$ , denote by $\mu _{P,i}$ the restriction of $\mu _P$ to ${\mathcal M}_i$ . Restricting equation (24) to the measurable subsets of ${\mathcal M}_i$ , we see that

(25) $$ \begin{align} \mu_i = \int_{{\mathcal P}_i} \mu_{P,i} \, d\hat\mu(P). \end{align} $$

As before, we denote by $\{\mu ^u_{i,x}: x \in {\mathcal M}_i\}$ the disintegration of $\mu _i$ with respect to the partition $\xi _i^u$ of ${\mathcal M}_i$ into strong-unstable plaques. The fact that Birkhoff averages are constant on strong-unstable leaves ensures that ${\mathcal P}_i$ is a coarser partition than $\xi ^{u}_i$ (see Remark 3.2). Moreover, for each $P\in {\mathcal P}_i$ , let $\{\mu ^u_{P,i,x}: x\in P\}$ be the disintegration of $\mu _{P,i}$ with respect to the partition $\xi _i^u$ . By the transitivity of the disintegration (see Remark 3.2) and the definition of c-Gibbs u-state,

$$ \begin{align*} \mu^u_{P(x),i,x} = \mu^u_{i,x} = \nu^u_{i,x} \quad\text{for }\mu_i\text{-almost every }x\in{\mathcal M}_i. \end{align*} $$

By equation (25), this implies that

$$ \begin{align*} \mu^u_{P,i,x} = \nu^u_{i,x} \quad\text{for }\mu_{P,i}\text{-almost every }x\text{ and }\hat\mu\text{-almost every }P\in{\mathcal P}_i. \end{align*} $$

Since i is arbitrary, this just means that $\mu _P$ is a c-Gibbs u-state for $\hat \mu $ -almost every $P\in {\mathcal P}$ .

Proof. We begin by proving that $\mu (\partial {\mathcal M})=0$ . In view of Lemma 3.6, for this purpose, it is no restriction to suppose that $\mu $ is ergodic. Then,

(26) $$ \begin{align} \frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^j(y)} \to \mu \end{align} $$

for $\mu $ -almost every y. Consider any $i \in \{1, \ldots , k\}$ such that the Markov element ${\mathcal M}_{i}$ has positive $\mu $ -measure. By the definition of c-Gibbs u-state, it follows that equation (26) holds for $\nu ^u_{i,x}$ -almost every y and $\mu $ -almost every $x\in {\mathcal M}_{i}$ . In particular, using the dominated convergence theorem,

$$ \begin{align*} \frac 1n \sum_{j=0}^{n-1} f_*^j\nu^u_{i,x} \to \mu \end{align*} $$

for $\mu $ -almost every $x\in {\mathcal M}_{i}$ . By the definition of c-Gibbs u-state, the image of each $\nu _{i,x}$ under the projection $\pi $ is the normalized Lebesgue measure on $W^u_i(\pi (x))$ , which we denote as $\operatorname {vol}_{i,\pi (x)}$ . Thus, the previous relation implies that

$$ \begin{align*} \frac 1n \sum_{j=0}^{n-1} A_*^j \operatorname{vol}_{i,\pi(x)} \to \pi_*\mu. \end{align*} $$

Now, it is well known and easy to check that the left-hand side converges to the Lebesgue measure $\operatorname {vol}$ on the torus ${\mathbb T}^d$ . So, we just proved that $\pi _*\mu =\operatorname {vol}$ . As observed previously, the total boundary $\partial {\mathcal R}$ has zero Lebesgue measure. It follows that $\mu (\partial {\mathcal M})=0$ , as claimed.

Let $\xi ^{u}$ be the family of all $\xi ^{u}_i(x)$ with x in the interior of some ${\mathcal M}_i$ , and, similarly, $\eta ^u$ be the family of all $W^u_i(a)$ with a in the interior of some ${\mathcal R}_i$ . By the previous observation, these are partitions of (full measure subsets of) M and ${\mathbb T}^d$ , respectively. Moreover, $\eta ^u$ is subordinate to the unstable foliation of A, and $\xi ^{u}$ is its pull-back under the map $\pi $ to each strong-unstable leaf of f. By the definition of c-Gibbs u-state,

$$ \begin{align*} \pi_*\mu_x^u = \operatorname{vol}^u_{\pi(x)} = \text{normalized Lebesgue measure on }\eta^u_{\pi(x)} \end{align*} $$

for $\mu $ -almost every x. Applying Corollary 3.4, we conclude that $h_\mu (f,{\mathcal F}^{uu})=h_{\operatorname {vol}}(A,{\mathcal W}^u)$ . Since A is linear, we also have that $h_{\operatorname {vol}}(A,{\mathcal W}^u)$ coincides with the topological u-entropy $h(A,{\mathcal W}^u)$ . Moreover, by Lemma 3.1, the two topological entropies $h(A,{\mathcal W}^u)$ and $h(f,{\mathcal F}^{uu})$ are the same. Combining these identities, we find that $h_\mu (f,{\mathcal F}^{uu})=h(f,{\mathcal F}^{uu})$ . This means that $\mu $ is indeed a measure of maximal u-entropy.

Proof. Let $(f_n)_n$ be a sequence of diffeomorphisms converging to f in the $C^1$ topology. We want to prove that $f_n$ has a c-mostly contracting center for every large n. For this, by Proposition 4.2, we only need to show that the center Lyapunov exponents for any ergodic c-Gibbs u-state of $f_n$ are all negative, if n is large enough.

Consider any $g\in {\mathcal U}$ . By Lemma 3.1, its topological u-entropy $h(g, {\mathcal F}^{uu})$ is equal to the topological entropy $h(A)$ of A. Recall that ([Reference Hu, Wu and ZhuHWZ21]) the set $\operatorname {MM}^u(g)$ of measures of maximal u-entropy is non-empty, convex, and compact, and its extreme points are ergodic measures. By Theorem B, $\operatorname {MM}^u(g)$ coincides with the space of c-Gibbs u-states, for any $g\in {\mathcal U}$ . Thus, denoting by ${\mathcal F}^{uu}_g$ the strong-unstable foliation of g,

$$ \begin{align*} h_\mu(g,{\mathcal F}^{uu}_g) \leq h(A) \end{align*} $$

for any invariant probability $\mu $ of g, and the identity holds if and only if $\mu $ is a c-Gibbs u-state.

Let $(\mu _n)_n$ be a sequence of probability measures such that each $\mu _n$ is $f_n$ -invariant and the weak $^*$ limit $\mu =\lim _n \mu _n$ exists. Then, $\mu $ is f-invariant and, by Yang [Reference YangYan17],

$$ \begin{align*} \limsup_n h_{\mu_n}(f_n,{\mathcal F}^{uu}_{f_n}) \leq h_\mu(f,{\mathcal F}^{uu}_f). \end{align*} $$

Together with the previous paragraph, this implies that

(28) $$ \begin{align} \limsup_n \operatorname{Gibbs}^u_c(f_n) \subset \operatorname{Gibbs}^u_c(f). \end{align} $$

Lemma 5.2. A diffeomorphism $g\in {\mathcal U}$ has a c-mostly contracting center if and only if there are $m \ge 1$ and $a <0$ such that

$$ \begin{align*} \frac 1m \int_M \log \|Dg^m{\mid}_{E_g^{cs}}\| \, d\mu < a \quad\text{for any ergodic }c\text{-Gibbs }u\text{-state }\mu\text{ of }g. \end{align*} $$

Let us assume this fact for a while, and use it deduce the proof of the proposition from it. Since $\operatorname {Gibbs}^u_c(f)$ is weak $^*$ -compact, Lemma 5.2, together with the assumption that f has a c-mostly contracting center, implies that there are $m \ge 1$ and $a<0$ such that

$$ \begin{align*} \frac 1m \int_M \log \|Df^m{\mid}_{E_f^{cs}}\| \, d\mu < a \quad\text{for every } \mu \in \operatorname{Gibbs}^u_c(f). \end{align*} $$

Then, using equation (28) and the fact that the center-stable bundle $E^{cs}$ depends continuously on the diffeomorphism,

$$ \begin{align*} \frac 1m \int_M \log \|Df_n^m{\mid}_{E_{f_n}^{cs}}\| \, d\mu_n < \frac{a}{2} \quad\text{for any }\mu_n \in \operatorname{Gibbs}^u_c(f)\text{ and any large }n\ge 1. \end{align*} $$

Invoking Lemma 5.2 once more, we see $f_n$ does have a c-mostly contracting center for every large n. This reduces the proof of Proposition 5.1 to proving the previous lemma.

Proof of Lemma 5.2

Let g have a c-mostly contracting center. By Proposition 4.2, the center Lyapunov exponents of any ergodic c-Gibbs u-state $\mu $ of g are all negative. So, for any $\mu $ , there are $m_\mu \ge 1$ and $a_\mu <0$ such that

$$ \begin{align*} \frac 1{m_\mu} \int_M \log \|Dg^{m_\mu}{\mid}_{E^{cs}}\| \, d\mu < a_\mu. \end{align*} $$

Observe that the inequality remains true for any probability measure in an open neighborhood of $\mu $ . Thus, by compactness, we may take $m=m_\mu $ and $a=a_\mu $ independent of $\mu $ . This proves the ‘only if’ part of the statement.

To prove the converse, assume there are $m \ge 1$ and $a<0$ such that

$$ \begin{align*} \frac 1m \int_M \log \|Dg^{m} {\mid}_{E^{cs}}\| \, d\mu < a \end{align*} $$

for any ergodic c-Gibbs u state $\mu $ of g. According to the theorem of Oseledets, the largest center-stable Lyapunov exponent coincides with

$$ \begin{align*} \lim_n \frac{1}{nm} \int_M \log \|Dg^{nm}{\mid}_{E^{cs}}\| \, d\mu \end{align*} $$

which, by subadditivity, is bounded above by

$$ \begin{align*} \frac{1}{m} \int_M \log \|Dg^{m}{\mid}_{E^{cs}}\| \, d\mu < a <0. \end{align*} $$

This shows that the center Lyapunov exponents of any c-Gibbs u-state are negative. By Proposition 4.2, it follows that g has a c-mostly contracting center.

The proof of Proposition 5.1 is complete.

Proof. Dynamical coherence was proven by Potrie [Reference PotriePot15, Theorem A.1]. Moreover, Franks [Reference FranksFra70] proved that there exists a continuous surjective map $\pi : {\mathbb T}^3\to {\mathbb T}^3$ such that $\pi \circ f=A\circ \pi $ . This gives condition (H1). Potrie [Reference PotriePot15, Corollary 7.7 and Remark 7.8] give that the semiconjugacy $\pi $ maps each strong-unstable leaf of f to an unstable leaf of A, as required in condition (H2). Finally, Potrie [Reference PotriePot15, Theorem 7.10] also proved the corresponding statement for center-stable leaves, as in condition (H3).

At this point f is known to satisfy all the assumptions of Theorem B and thus of Lemma 3.1. According to that lemma, for any c-Gibbs u-state $\mu $ ,

$$ \begin{align*} h_\mu(f) \ge h_\mu(f,{\mathcal F}^{uu}) = h_{\mathrm{top}}(A)=\log k_3. \end{align*} $$

We also know, from [Reference Viana and YangVY17, Theorem 3.6], that the push-forward map $\pi _*$ preserves the entropy, and is a bijection restricted to the subsets of invariant ergodic probability measures with entropy larger than $|\!\log k_1|$ . These facts imply that $h_{\pi _*\mu }(A) = h_{\mathrm{top}}(A)$ , and so $\pi _*\mu $ coincides with the (unique) measure of maximal entropy of A, namely, the Lebesgue measure $\nu $ on ${\mathbb T}^3$ . By the injectivity of $\pi _*$ , it also follows that $\mu $ is the unique measure of maximal entropy of f, and the unique c-Gibbs u-state. We denote this measure by $\mu _f$ to highlight its dependence on the diffeomorphism.

We need to show that the center Lyapunov exponent of $\mu _f$ is negative. Indeed, we claim that

(29) $$ \begin{align} \unicode{x3bb}^c(\mu_f) \leq \log k_2 <0. \end{align} $$

This was proved by Ures [Reference UresUre12, Theorem 5.1] in the special case of $C^2$ partially hyperbolic diffeomorphisms. In fact, he used the stronger, so-called absolute, version of partially hyperbolicity, whereas here, we always refer to the more general pointwise version. However, the only step where absolute partial hyperbolicity is used in his argument is for proving that the unstable foliation is quasi-isometric, and that has been proved to hold in the general pointwise case, by Hammerlindl and Potrie [Reference Hammerlindl and PotrieHP14, §3]. Thus, to fully prove (29) we only have to remove the $C^2$ restriction.

For any $C^1$ element $f\in {\mathcal D}(A)$ , consider a sequence of $C^2$ diffeomorphisms $f_n\in {\mathcal D}(A)$ converging to f in the $C^1$ topology. Let $\mu _n$ denote the measure of maximal entropy of each $f_n$ . We claim that $(\mu _n)_n$ converges to $\mu _f$ . To prove this, let $\tilde \mu $ be any accumulation point. Since these diffeomorphisms are away from homoclinic tangencies, it follows from [Reference Liao, Viana and YangLVY13] that the entropy varies upper semi-continuously:

$$ \begin{align*} h_{\tilde\mu}(f) \geq \limsup_n h_{\mu_n}(f_n) = h_{\mathrm{top}}(A). \end{align*} $$

Thus, by our previous arguments, $(\pi _f)_*(\tilde \mu )$ is the Lebesgue measure on ${\mathbb T}^3$ , and $\tilde \mu $ is the unique maximal measure of f. This proves the claim. Now, as the center bundle is one-dimensional (1D), it follows that $\unicode{x3bb} ^c(\mu _n) \to \unicode{x3bb} ^c(\mu _f)$ . Since we already know that $\unicode{x3bb} ^c(\mu _n) \le \log k_2$ for every n, this completes the proof of equation (29).

We are left to prove that the support of $\mu _f$ is connected. Let $l \ge 1$ be the number of connected components. Since $\mu _f$ is ergodic for f, these connected components are cyclically permuted by f. Also, cf. equation (27), the ergodic components of $\mu _f$ for the iterate $f^l$ are the normalized restrictions of $\mu _f$ to each connected component of $\operatorname {supp}\mu _f$ . Now, $\mu _f$ is still a measure of maximal u-entropy for $f^l$ and thus, so are its ergodic components. This shows that $f^l$ has l distinct measures of maximal u-entropy. However, it is clear that $f^l \in {\mathcal D}(A^l)$ , and so we may apply the previous arguments to it. In particular, we get that $f^l$ has a unique measure of maximal u-entropy. Thus, $l=1$ .

Proof. Let $f_c$ denote the map induced by any given $f\in \operatorname {SPH}_1(M)$ on the quotient space $M_c=M/{\mathcal F}^c$ , and let $\pi : M \to M_c$ be the canonical quotient map. Then, $\pi $ is a semi-conjugacy from f to $f_c$ . Moreover, $f_c$ is a topological Anosov homeomorphism, that is, a globally hyperbolic homeomorphism, in the sense of [Reference VianaVia08, §1.3] or [Reference Viana and YangVY13, §2.2]. By a result of Hiraide [Reference HiraideHir89], the map $f_c$ is conjugate to a linear Anosov torus diffeomorphism A. Up to replacing $\pi $ with its composition with this conjugacy, if necessary, it is no restriction to suppose that $f_c=A$ . We do so from now on. This gives condition (H1).

As before, let ${\mathcal F}^c$ , ${\mathcal F}^{ss}$ , ${\mathcal F}^{uu}$ , and ${\mathcal F}^{cs}$ denote respectively the center, strong-stable, strong-unstable, and center-stable foliations of f. We claim that every center leaf ${\mathcal F}^c(x)$ intersects the strong-stable leaf ${\mathcal F}^{ss}(x)$ only once. Indeed, suppose that there exists another point y in the intersection. The distance between $f^n(x)$ and $f^n(y)$ along the strong-stable leaf decreases exponentially. Then, clearly, the same is true for the distance in the ambient manifold. Since $f^n(x)$ and $f^n(y)$ belong to the same center leaf, and the center leaves are a continuous family of $C^1$ embedded circles, this can only happen if the distance between the points along the center leaf also decreases at the same rate. That is impossible because, by domination, the contraction rates along center leaves are strictly weaker than along strong-stable leaves. This contradiction proves the claim. It follows that, under the map $\pi :M\to {\mathbb T}^d$ , the center-stable leaf ${\mathcal F}^{cs}(x)$ projects down to the stable manifold of $\pi (x)$ for the linear automorphism A. Analogously, the strong-unstable leaf ${\mathcal F}^{uu}(x)$ projects down to the unstable manifold of $\pi (x)$ for A. Thus, conditions (H2) and (H3) are also proved.

Proof. This now follows directly from [Reference Ures, Viana and YangUVY21, Lemma 5.1], where it was proved that the space of c-Gibbs u-states (called $\nu $ -Gibbs u-states and denoted as $\mathrm {Gibb}^u_\nu (f)$ in that paper) coincides with the the finite-dimensional simplex (denoted as $\operatorname {MM}^-(f)$ in that paper) generated by the ergodic measures of maximal entropy and negative center Lyapunov exponent.

Proof. First, we deal with the case $f=f_0$ and then we explain how the arguments can be adapted to any $C^1$ -small perturbation.

Let $\pi ^{cs}:M \to S^1$ be the canonical projection $\pi ^{cs}(\theta ,x)=\theta $ . For each $(\theta ,x)\in \Lambda (f_0)$ and $n \in {\mathbb Z}$ , define $\theta _n=\pi ^{cs}(f_0^n(\theta ,x))$ . The sequence $(\theta _n)_{n\in {\mathbb Z}}$ is an orbit for the circle map $x \to kx \ \ \mod 1$ . Moreover, since $g_0: \Lambda (g_0) \to \Lambda (g_0)$ is the natural extension of that map, there exists exactly one point $y \in D$ such that $(\theta ,y) \in \Lambda (g_0)$ and $\theta _n=\pi ^{cs}(g_0^n(\theta , y))$ for every $n \in {\mathbb Z}$ .

Let $\pi :\Lambda (f_0) \to \Lambda (g_0)$ be the map $\pi : (\theta ,x) = (\theta ,y)$ defined in this way. This map $\pi $ is continuous and surjective, and it is a semiconjugacy between $f_0$ and $g_0$ . Moreover, by construction, it maps each local center-stable ‘leaf’ $\Lambda (f_0) \cap (\{\theta \} \times D)$ of $f_0$ to the center-stable ‘leaf’ $\Lambda (g_0) \cap (\{\theta \} \times D)$ of $g_0$ . Thus, we proved conditions (H1) and (H3) for $f=f_0$ , in versions suitable for the present setting.

We are left to verify condition (H2). Let $(\theta ,x)$ and $(\theta ^\prime ,x^\prime )$ be any two points of $\Lambda (f_0)$ in the same local unstable manifold. Let $y^\prime $ be the point of D such that $\pi (\theta ^\prime ,x^\prime )=(\theta ^\prime ,y^\prime )$ . For every $n<0$ , the points $f_0^n(\theta ,x)$ and $f_0^n(\theta ^\prime ,x^\prime )$ are contained in a cylinder $I_n \times D$ , where $I_n$ is an interval whose length goes to zero when $n\to -\infty $ . Since $f_0^n(\theta ,x)$ and $g_0^n(\theta ,y)$ belong to the same vertical disk $\{\theta _n\} \times D$ for every n, and the same holds for $f_0^n(\theta ^\prime ,x^\prime )$ and $g_0^n(\theta ^\prime ,y^\prime )$ , we conclude that $g_0^n(\theta ,y)$ and $g_0^n(\theta ^\prime ,y^\prime )$ are also contained in the cylinder $I_n \times D$ . This can only occur if $(\theta ,y)$ and $(\theta ^\prime ,y^\prime )$ belong to the same local strong-unstable leaf of $g_0$ . Thus, we proved that $\pi $ maps every strong-unstable leaf of $f_0$ inside an strong-unstable leaf of $g_0$ , as we wanted to prove.

Now let us extend this construction to a $C^1$ -neighborhood of f. The stability theorem of [Reference Hirsch, Pugh and ShubHPS77] gives that if f is $C^1$ -close to $f_0$ then its center foliation ${\mathcal F}^c_f$ is conjugate to the center foliation of $f_0$ (that is, the vertical lamination $\{\{\theta \}\times D: \theta \in S^1\}$ ) by a homeomorphism close to the identity. Let $\pi ^{cs}_f: M \to M/{\mathcal F}^c_f$ be the quotient map: the quotient space is a topological circle, and the induced quotient map $f_{cs}$ is conjugate to the circle map $x \mapsto kx \ \ \mod 1$ by some homeomorphism $\psi :M/{\mathcal F}^c_f \to S^1$ .

For every $(\theta ,x)\in \Lambda (f)$ , consider the sequence $(\theta ^f_n)_n$ on $S^1$ defined by

$$ \begin{align*} \theta_n^f = \psi(\pi^{cs}_f(f^n(\theta,x))). \end{align*} $$

As previously, this sequence identifies a unique point $(\theta _0,y) \in \Lambda (g_0)$ such that

$$ \begin{align*} \theta_n=\pi^{cs}(g_0^n(\theta_0,y)) \quad\text{for any } n\in {\mathbb Z}. \end{align*} $$

Let $\pi _f:\Lambda (f) \to \Lambda (g_0)$ be the map given by $\pi _f(\theta ,x)=(\theta _0,y)$ . By the same arguments as before, $\pi _f$ is a semiconjugacy between $f {\mid} _{\Lambda (f)}$ and $g_0 {\mid} _{\Lambda (g_0)}$ , verifying the conditions (H1), (H2), and (H3).

Proof. Observe that the unstable Lyapunov exponent of $f_0$ relative to any invariant measure is equal to $\log k$ . Since $E^u$ is 1D, this exponent varies continuously with the diffeomorphism. In particular, there exists a neighborhood of $f_0$ such that $\unicode{x3bb} ^u(\cdot ,f)> \log k - c$ for any f in that neighborhood. Let $\mu $ be any ergodic c-Gibbs u-state $\mu $ of f. Lemma 3.1 gives that

(33) $$ \begin{align} h_\mu(f{\mid}_{\Lambda(f)}) \geq h_\mu(f{\mid}_{\Lambda(f)},{\mathcal F}^{uu})=h(g_0)=\log k. \end{align} $$

Combining this with equation (32), we find that

$$ \begin{align*} \unicode{x3bb}^c_1(\mu,f)> c-\unicode{x3bb}^u(\mu,f) > -\log k. \end{align*} $$

If $\unicode{x3bb} ^c_2(\mu ,f)$ were non-negative, then the Ruelle inequality (Ruelle [Reference RuelleRue78]) for $f^{-1}$ would yield

$$ \begin{align*} h_\mu(f{\mid}_{\Lambda(f)}) = h_\mu(f^{-1}{\mid}_{\Lambda(f)}) \leq -\unicode{x3bb}^c_1(\mu,f) < \log k, \end{align*} $$

and that would contradict equation (33). Thus, $\unicode{x3bb} ^c_2(\mu ,f) < 0$ for any ergodic c-Gibbs u-state $\mu $ , which means that f has a c-mostly contracting center.

Let $\mu _{\mathrm{max}}$ be any ergodic measure of maximal entropy and $\mu $ be any ergodic c-Gibbs u-state $\mu $ of f. Then,

(34) $$ \begin{align} h_{\mu_{\mathrm{max}}}(f{\mid}_{\Lambda(f)})\geq h_\mu(f{\mid}_{\Lambda(f)})\geq \log k. \end{align} $$

By the previous argument, starting from equation (34) rather than equation (33), we get that both center Lyapunov exponents of $\mu _{\mathrm{max}}$ are negative, and so $\mu _{\mathrm{max}}$ is a hyperbolic measure. By Proposition A.4 and Lemma 3.1, it follows that

$$ \begin{align*} h_{\mu_{\mathrm{max}}}(f{\mid}_{\Lambda(f)}) = h_{\mu_{\mathrm{max}}}(f{\mid}_{\Lambda(f)}, {\mathcal F}^{uu}) \leq \log k. \end{align*} $$

This has several consequences. To begin with, we get that $\mu _{\mathrm{max}}$ is a measure of maximal u-entropy. In other words, we have shown that every ergodic measure of maximal entropy is also a measure of maximal u-entropy. Since these are convex spaces whose extreme points are ergodic measures, it follows that the space of measures of maximal entropy is contained in the space of measures of maximal u-entropy, which (by Theorem B) is known to coincide with the space of c-Gibbs u-states. Another consequence of the previous argument is that the topological entropy $h(f {\mid} _{\Lambda (f)}) = h_{\mu _{\mathrm{max}}}(f {\mid} _{\Lambda (f)})$ is equal to $\log k$ . Hence, using Lemma 3.1, we also get that every c-Gibbs u-state is a measure of maximal entropy.