2.1 Gibbs u-states

Following Pesin and Sinai [Reference Pesin and Sinai43] and Bonatti and Viana [Reference Bonatti and Viana11] (see also [Reference Bonatti, Díaz and Viana10, Chapter 11]), we call Gibbs u-state any invariant probability measure whose conditional probabilities (Rokhlin [Reference Rokhlin45]) along strongly unstable leaves are absolutely continuous with respect to the Lebesgue measure on the leaves. In fact, assuming that the derivative $Df$ is Hölder continuous, the Gibbs-u state always exists, and the densities with respect to Lebesgue measures along unstable plaques are continuous. Moreover, the densities vary continuously with respect to the strongly unstable leaves. As a consequence, the space of Gibbs u-states of f, denoted by $\mbox {Gibbs}^u(\cdot )$ , is compact relative to the weak-* topology in the probability space.

The set of Gibbs u-states plays important roles in the study of physical measures for partially hyperbolic diffeomorphisms. The proofs for the following basic properties of Gibbs u-states can be found in the book of Bonatti, Díaz and Viana [Reference Bonatti, Díaz and Viana10, Subsection 11.2] (see also Dolgopyat [Reference Dolgopyat24]):

The semi-continuity of Gibbs u-states with respect to $C^{1+}$ diffeomorphisms under $C^1$ topology was recently proved by the author of this article in [Reference Yang52].

Proposition 2.2. Suppose that $f_n$ ( $n=1,\cdots , \infty $ ) and f are $C^{1+}$ partially hyperbolic diffeomorphisms such that $f_n\overset {C^1}{\to } f$ . Then

$$ \begin{align*}\limsup \mbox{Gibbs}^u(f_n)\subset \mbox{Gibbs}^u(f),\end{align*} $$

where the convergence is in the Hausdorff topology of the probability space.

The following lemma shows the relation between the Gibbs u-states of a diffeomorphism and its iterations.

2.2 Partial entropy along unstable foliation

In this section, we give the precise definition of the partial metric entropy of $\mu $ along the unstable foliation ${\mathcal F}^u$ of f, which depends on a special class of measurable partitions. The partial entropy has been proven to be a powerful tool in the study of partially hyperbolic diffeomorphisms, thanks to its semi-continuity in the $C^1$ topology.

Ledrappier and Strelcyn [Reference Ledrappier and Strelcyn34] proved that the Pesin unstable lamination admits some $\mu $ -subordinate measurable partition. The following result is contained in Lemma 3.1.2 of Ledrappier and Young [Reference Ledrappier and Young35].

This allows us to define the partial entropy of $\mu $ using any $\mu $ -subordinate partition.

Although partially entropies are well defined for $C^1$ diffeomorphisms and behave well under $C^1$ topology, one still need higher regularity such as $C^2$ or at least $C^{1+}$ in order to relate it with other quantities such as Lyapunov exponents or Gibbs u-states. The following upper bound for the partial entropy along the unstable foliation ${\mathcal F}^u$ follows [Reference Ledrappier and Young35, Reference Ledrappier and Young36].

Proposition 2.8. Let f be $C^{1+}$ and $\mu $ be an invariant probability measure of f; then

$$ \begin{align*}h_\mu(f,{\mathcal F}^u)\leq\int \log \operatorname{Jac}^u(x) d\mu(x).\end{align*} $$

Moreover,

(2) $$ \begin{align} h_\mu(f,{\mathcal F}^u)=\int \log \operatorname{Jac}^u(x) d\mu(x) \end{align} $$

if and only if $\mu $ is a Gibbs u-state of f.

The following equality was built in [Reference Ledrappier and Young36, Proposition 5.1] when f is $\operatorname {C}^2$ . As explained above, it also holds under the general situation assuming only $C^{1+}$ .

Proposition 2.9. Let $\mu $ be a probability measure of f such that all of the center exponents of $\mu $ are nonpositive. Then

$$ \begin{align*}h_\mu(f,{\mathcal F}^u)=h_\mu(f).\end{align*} $$

2.3 Other invariant measure subspaces

Proposition 2.1 (4) states that when f is $C^{1+}$ , Gibbs u-states are the natural candidates of the physical measures of f. However, this statement falls apart when f is only $C^1$ . Here let us recall that the main result of [Reference Hua, Yang and Yang32] (see also [Reference Crovisier, Yang and Zhang23]) shows that for $C^1$ diffeomorphisms, every limit point of the empirical measures at Lebesgue almost every point must satisfy ((2), Proposition 2.8). However, such measures may not be Gibbs u-states due to the lack of Pesin’s formula for $C^1$ diffeomorphisms. To this end, we introduce two candidate spaces of physical measures for such f. See [Reference Hua, Yang and Yang32], [Reference Crovisier, Yang and Zhang23] and [Reference Catsigeras, Cerminara and Enrich Pesin21] for their properties.

We write

$$ \begin{align*}\operatorname{G}(f)=\operatorname{G}^{u}(f)\cap \operatorname{G}^{cu}(f).\end{align*} $$

We first observe that the spaces above are nonempty; moreover, the space $\operatorname {G}(f)$ contains all of the candidates of physical measures.

The following property shows that $\operatorname {G}^u(\cdot )$ shares similar properties with $\mbox {Gibbs}^u(\cdot )$ (Proposition 2.1).

We need to observe that, in general, the space $\operatorname {G}(f)$ may not have such properties (especially when it comes to the ergodic components). Indeed, in Proposition 5.17, we will show that the above properties hold for $\operatorname {G}(g)$ when g is $C^1$ close to f which is $C^{1+}$ with a mostly expanding center.

3.1 Derived from Anosov diffeomorphisms

We assume A to be a linear Anosov diffeomorphism over ${\mathbb T}^3$ with three positive simple real eigenvalues $0<k_1<1<k_2<k_3$ .

3.1.1 Local derived from Anosov diffeomorphisms

Let us begin by recalling the construction of Mañé’s example, which is a local $C^0$ perturbation of A. The statement below is a little different from the original construction in history.

Example 3.1. Let p be a fixed point of A and U a small neighbourhood of p. There is a partially hyperbolic diffeomorphism $f_0$ that coincides with A on ${\mathbb T}^3\setminus U$ . $f_0$ is topological Anosov, and

(3) $$ \begin{align} \mid Df_0\mid_{E^c(x)} \mid \geq 1, \end{align} $$

where the equality holds if and only if $x=p$ .

Because $Df_0\mid _{E^c(\cdot )}$ is expanding everywhere except at the point p, it is clear that $f_0$ has a mostly expanding center. Thus, by [Reference Yang52], $f_0$ admits a $C^1$ neighbourhood ${\mathcal U}$ such that every $C^{1+}$ diffeomorphism belonging to ${\mathcal U}$ has a mostly expanding center.

3.2 Perturbation of volume-preserving partially hyperbolic diffeomorphisms

In [Reference Dolgopyat25], Dolgopyat showed the following.

The following result in [Reference Yang52] allows us to obtain more examples using $C^1$ perturbation.

Proposition 3.4 contains an abundance of systems: by Avila [Reference Avila5], $\operatorname {C}^\infty $ volume-preserving diffeomorphisms are $\operatorname {C}^1$ dense. And by Baraviera and Bonatti [Reference Baraviera and Bonatti6], the volume-preserving partially hyperbolic diffeomorphisms with a 1-dimensional center and nonvanishing center exponent are $\operatorname {C}^1$ open and dense. Moreover, the subset of accessible systems is $\operatorname {C}^1$ open and $\operatorname {C}^k$ dense for any $k\geq 1$ among all partially hyperbolic diffeomorphisms with a 1-dimensional center direction, due to the work of Burns et al. [Reference Burns, Rodriguez Hertz, Rodriguez Hertz, Talitskaya and Ures17]; see also Theorem 1.5 in Niţică and Török [Reference Niţică and Török40].

Indeed, the accessibility assumption in the above proposition can be replaced by another hypothesis.

The diffeomorphisms with minimal strongly stable and unstable foliations are also quite common; they fill an open and dense subset of volume-preserving partially hyperbolic diffeomorphisms with a 1-dimensional center and have compact center leaves. This follows from a conservative version of the results in [Reference Bonatti, Díaz and Ures9].

3.3 Product of diffeomorphisms with a mostly expanding center

The following was shown by Ures, Viana and the author of this article in [Reference Ures, Viana and Yang49].

5.1 Hyperbolic times

Definition 5.4. Given $b>0$ , we say that n is a b- hyperbolic time for a point x if

$$ \begin{align*}\frac{1}{k}\sum_{j=n-k+1}^n\log\|Df^{-1}\mid_{E^{cu}(f^j(x))}\|\leq -b \text{ for any }0<k \leq n.\end{align*} $$

Let D be any $\operatorname {C}^1$ disk; we use $d_D(\cdot ,\cdot )$ to denote the distance between two points in the disk. Recall that for the dominated splitting $E^s\oplus E^{cu}$ , one can define the center unstable cone field, which is invariant under forward iteration.

The next lemma states that if n is a hyperbolic time for x, then on the disk $f^n(D)$ , one picks up a contraction by $e^{-b}$ for each backward iteration.

Lemma 5.5 ([Reference Alves, Bonatti and Viana1] Lemma 2.7)

For any $b>0$ , there is $r>0$ such that, given any $\operatorname {C}^1$ disk D tangent to the center-unstable cone field, $x\in D$ and $n\geq 1$ a $b/2$ -hyperbolic time for x, we have

$$ \begin{align*}d_{f^{n-k}(D)}(f^{n-k}(y),f^{n-k}(x))\leq e^{-kb/2} d_{f^n(D)}(f^n(x),f^n(y)),\end{align*} $$

for any point $y\in D$ with $d_{f^n(D)}(f^n(x),f^n(y))\leq r$ and any $1\leq k \leq n$ .

By Lemma 5.3 and Proposition 2.12, for any $g\in {\mathcal U}$ , there is a full-volume subset $\Gamma _g$ such that for any $x\in \Gamma _g$ , any limit of the sequence $\frac {1}{n}\sum _{i=0}^{n-1}\delta _{g^i(x)}$ belongs to $\operatorname {G}(g)$ . Thus, for any $x\in \Gamma _g$ ,

(8) $$ \begin{align} \begin{aligned} \limsup_{n\to \infty}\frac{1}{n}\sum_{i=0}^{n-1}\log \|Dg^{-1}\mid_{E^{cu}_g(g^i(x))}\|&=\limsup \int \log \|Dg^{-1}\mid_{E^{cu}_g(x)}\| d\frac{1}{n}\sum_{i=0}^{n-1}\delta_{g^i(x)}\\ &<-b_0<0. \end{aligned} \end{align} $$

Define $H(b_0/2,x,g)$ to be the set of $b_0/2$ -hyperbolic times for $x\in \Gamma _g$ ; that is, the set of times $m\geq 1$ such that

(9) $$ \begin{align} \frac{1}{k}\sum_{i=m-k+1}^{m} \log \|Dg^{-1}\mid_{E^{cu}_g(g^i(x))}\|\leq -b_0/2 \text{ for all }1\leq k \leq m. \end{align} $$

By the Pliss lemma (see [Reference Alves, Bonatti and Viana1]), such hyperbolic times have positive density on the orbit segment from $0$ to n: there exists $n_x\geq 1$ and $\delta _1>0$ such that

(10) $$ \begin{align} \#(H(b_0/2,x,g)\cap [1,n))\geq n\delta_1 \text{ for all }n\geq n_x. \end{align} $$

By Lemma 5.5 and Remark 5.6, there is $r_1>0$ that only depends on ${\mathcal U}$ and $b_0/2$ , such that for any $x\in \Gamma _g$ and any disk D tangent to the center-unstable cone field, $x\in D$ , $n\in H(b_0/2,x,g)$ , we have

(11) $$ \begin{align} d_{D}(x,y)\leq e^{-nb_0/2}d_{g^n(D)}(g^n(x),f^n(y)), \end{align} $$

for any $y\in D$ with $d_{g^n(D)}(g^n(x),g^n(y))\leq r_1$ . (We also assume that $r_1$ satisfies the condition (15) below, which depends only on the neighbourhood ${\mathcal U}$ .) In particular, for $x\in \Gamma _g$ and disk D tangent to the center-unstable cone field containing x, $g^n(D)$ contains a smaller disk $D_n$ with diameter $r_1$ for $n\in H(b_0/2,x,g)$ sufficiently large. Then $\cup _{z\in D_n}{\mathcal F}^s(z)$ contains an open ball with radius $r_1$ .

Definition 5.7. Denote by ${\mathcal H}(b_0/2,g)$ the set of points x such that for any $k\geq 1$ ,

(12) $$ \begin{align} \frac{1}{k}\sum_{i=0}^{k-1} \log \|Dg^{-1}\mid_{E^{cu}_g(g^{-i}(x))}\|\leq -b_0/2 \text{ for all }k\geq 0. \end{align} $$

In other words, for every $n>0$ , n is a hyperbolic time for the point $f^{-n}(x)$ .

The next lemma shows that there are plenty of hyperbolic times on the forward orbit of x, every $\mu \in \operatorname {G}^u(g)$ and $\mu $ almost every x.

Lemma 5.8. For any $g\in {\mathcal U}$ and any $\mu \in \operatorname {G}^u(g)$ , we have

(13) $$ \begin{align} \mu({\mathcal H}(b_0/2,g))\geq \delta_1, \end{align} $$

where $\delta _1$ is given in (10).

Proof. By Proposition 2.13, we may assume $\mu $ to be ergodic. By Birkhoff’s theorem, we only need to show that for $\mu $ almost every x, $\liminf \frac {1}{n}\#\{1\leq k \leq n; f^k(x)\in {\mathcal H}(b_0/2,g)\}\geq \delta _1$ . It is equivalent to show that for some fixed $m_x$ ,

(14) $$ \begin{align} \liminf\frac{1}{n}\#\{1\leq k \leq n; f^{k+m_x}(x)\in {\mathcal H}(b_0/2,g)\}\geq \delta_1. \end{align} $$

By Lemma 5.3, take x be a typical point of $\mu $ , such that

$$ \begin{align*}\lim\frac{1}{n}\sum_{i=0}^{n-1} \log \|Dg^{-1}\mid_{E^{cu}_g(g^{-i}(x))}\|\leq -b_0.\end{align*} $$

We claim that there is $m>0$ such that $g^{-m}(x)\in {\mathcal H}(b_0/2,g)$ . Otherwise, for any $g^{-n}(x)$ , there is $i_n>0$ such that $\frac {1}{i_n}\sum _{i=0}^{i_n-1} \log \|Dg^{-1}\mid _{E^{cu}_g(g^{-i-n}(x))}\|\geq -b_0/2$ . Recursively, we obtain a sequence of points: $n_1=i_0$ , $n_2=n_1+i_{n_1}, \cdots $ ; by induction, we have

$$ \begin{align*}\frac{1}{n_k}\sum_{i=0}^{n_k-1} \log \|Dg^{-1}\mid_{E^{cu}_g(g^{-i}(x))}\|\geq -b_0/2.\end{align*} $$

This contradicts the choice of x.

Moreover, it is easy to see that for any $k\in H(b_0/2,g^{-m}(x),g)$ , $g^{k-m}(x)\in {\mathcal H}(b_0/2,g)$ . Then by (10) and taking $m_x=m$ in (14), we conclude the proof. □

5.2 Fake foliations

In order to avoid assuming dynamical coherence of f, we use locally invariant (fake) foliations, a construction that follows Burns and Wilkinson [Reference Burns and Wilkinson18] and goes back to Hirsch, Pugh and Shub [Reference Hirsch, Pugh and Shub31]. We fix ${\mathcal U}$ a small $C^1$ neighbourhood of f provided by Lemma 5.3.

For $g\in {\mathcal U}$ and any $x\in M$ , we consider the following three types of Bowen balls:

• • finite Bowen ball: $B_n(g,x,\varepsilon )=\{y\in M: d(g^i(x),g^i(y))<\varepsilon , |i|<n\},$

• • negative Bowen ball: $B^-_\infty (g,x,\varepsilon )=\{y\in M: d(g^i(x),g^i(y))<\varepsilon , i<0\},$

• • (two sided) infinite Bowen ball:

  $$ \begin{align*}B_\infty(g,x,\varepsilon)=\{y\in M: d(g^i(x),g^i(y))<\varepsilon, i\in \mathbb{Z}\}.\end{align*} $$

The following was shown in the proof of [Reference Liao, Viana and Yang37], [Theorem 3.1].

We may take $r_1$ in the previous section to satisfy that

(15) $$ \begin{align} r_1<r_0/2. \end{align} $$

Then as a consequence of Lemma 5.5, we show that for every point in ${\mathcal H}(b_0/2,g)$ , the unstable manifold has uniform size.

Lemma 5.11. For any $x\in {\mathcal H}(b_0/2,g)$ , $\hat {{\mathcal F}}^{cu}_{g,x}(x,r_1)\subset W_{\operatorname {loc}}^u(x)$ . More precisely, for any $y\in \hat {{\mathcal F}}^{cu}_{g,x}(x,r_1)$ ,

$$ \begin{align*}d_{\hat{{\mathcal F}}^{cu}_{g,g^{-n}(x)}(g^{-n}(x))}(g^{-n}(x),g^{-n}(y))\leq e^{-nb_0/2}d_{\hat{{\mathcal F}}^{cu}_{g,x}(x)}(x,y).\end{align*} $$

The goal of this subsection is to show that the measures in $\operatorname {G}^u(g)$ for $g\in {\mathcal U}$ are uniformly entropy expansiveness.

Lemma 5.12. For any $g\in {\mathcal U}$ and any measure $\mu \in \operatorname {G}^u(g)$ , for $\mu $ almost every point x,

$$ \begin{align*}B_\infty(g,x,r_1)=x.\end{align*} $$

Proof. By Lemma 5.10 and the choice of $r_1\leq r_0/2$ , we have

$$ \begin{align*}B_\infty(g,x,r_1)\subset B^-_\infty(g,x,r_1)\subset \hat{{\mathcal F}}^{cu}_{g,x}(x,r_1).\end{align*} $$

Let x be a $\mu $ typical point; by Lemma 5.8, we may assume that the forward orbit of x enters ${\mathcal H}(b_0/2,g)$ infinitely many times. Suppose that there is a distinct point $y\in B_\infty (g,x,r_1/2)\subset \hat {{\mathcal F}}^{cu}_{g,x}(x,r_1)$ , we are going to prove by contradiction that $x$ and $y$ coincide with each other. Suppose that $f^n(x)\in {\mathcal H}(b_0/2,g)$ , then $f^n(y)\in B_\infty (g,g^n(x),r_1/2)\in \hat {{\mathcal F}}^{cu}_{g,x}(x,r_1)$ . By Lemma 5.11,

$$ \begin{align*}d_{\hat{{\mathcal F}}^{cu}_{g,x}(x)}(x,y)\leq e^{-nb_0/2}d_{\hat{{\mathcal F}}^{cu}_{g,g^n(x)}(g^n(x))}(g^n(x),g^n(y))\leq e^{-nb_0/2}r_1.\end{align*} $$

Taking $n\to \infty $ , we have $d_{\hat {{\mathcal F}}^{cu}_{g,x}(x)}(x,y)=0$ . Hence, $x=y$ , a contradiction with the hypothesis that x and y are distinct. The proof is complete. □

5.3 Upper semi-continuity of metric entropy

In this section, we are going to show that the metric entropy for measures in $\operatorname {G}^u(\cdot )$ is upper semi-continuous, which is a consequence of the uniform entropy expansiveness for measures among $\operatorname {G}^u(\cdot )$ .

Define the $\varepsilon $ -tail entropy at x by

$$ \begin{align*}h^*(g,x,\varepsilon)=h_{top}(g,B_\infty(g,x,\varepsilon)),\end{align*} $$

where we use Bowen’s definition of the topological entropy [Reference Bowen12]:

$$ \begin{align*}h_{top}(g,K) = \lim_{\varepsilon\to 0}\limsup_{n\to\infty} \frac1n\log r_n(\varepsilon,K); \end{align*} $$

here $r_n(\varepsilon ,K)$ is the minimum cardinality of $(n,\varepsilon )$ -spanning sets of K. For any probability measure $\mu $ of g, let $h^*(g,\mu ,\varepsilon )=\int h^*(g,x,\varepsilon ) d\mu (x)$ .

As a direct consequence of Lemma 5.12, we get the following.

We also need the following lemma of [Reference Cao, Liao and You20], [Theorem 1.2].

By Lemma 5.14 and Lemma 5.15, we conclude that $h_\mu (g)=h_\mu (g,{\mathcal P})$ for any finite measurable partition ${\mathcal P}$ with $diam({\mathcal P})\leq r_1$ . In particular, by a standard argument for the upper semi-continuity of metric entropy (see, for instance, [Reference Liao, Viana and Yang37], [Lemma 2.3]), we have the following.

Corollary 5.16. Let $g_n$ ( $n\geq 0$ ) be a sequence of $C^1$ partially hyperbolic diffeomorphisms inside ${\mathcal U}$ and $\mu _n\in \operatorname {G}^u(g_n)$ . Suppose yjsy $g_n\to g_0$ in $C^1$ topology and $\mu _n \to \mu _0\in \operatorname {G}^u(g_0)$ in weak-* topology; then

$$ \begin{align*}\limsup_{n\to \infty}h_{\mu_n}(g_n)\leq h_{\mu_0}(g_0).\end{align*} $$

5.4 Physical measures

In this section, we will provide a uniform treatment on the existence of physical measures for all $C^{1+}$ diffeomorphisms in ${\mathcal U}$ . For this purpose, let $r_1>0$ be given by Lemma 5.5 and (15).

Proof. Recall that

$$ \begin{align*}\operatorname{G}^{cu}(g)=\left\{\mu\in {\mathcal M}_{\mbox{inv}}(f): h_\mu(g)\geq \int \log(\det(Dg\mid_{E^{cu}(x)}))d\mu(x)\right\}.\end{align*} $$

Because the metric entropy function is affine, it follows that $\operatorname {G}^{cu}(f)$ is convex. By Proposition 2.13, $\operatorname {G}^u(g)$ is convex, and so is $\operatorname {G}(g)=\operatorname {G}^u(g)\bigcap \operatorname {G}^{cu}(g)$ .

The compactness of $\operatorname {G}(g)$ follows from Corollary 5.16. More precisely, suppose that there is a sequence of invariant probabilities $\{\mu _n\}_{n=0}^\infty $ of g such that $\mu _n\in \operatorname {G}(g)$ and assume $\lim _{n\to \infty } \mu _n=\mu $ . Because $\mu _n\in \operatorname {G}^{cu}(g)$ , we have

$$ \begin{align*}h_{\mu_n}(g)\geq \int \log(\det(Dg\mid_{E^{cu}(x)}))d\mu_n(x).\end{align*} $$

Note that $\mu _n\in \operatorname {G}^{u}(g)$ , and by Proposition 2.13, $\operatorname {G}^u(g)$ is compact; thus, we have $\mu \in \operatorname {G}^u(g)$ . It then follows from Corollary 5.16 that $\limsup _{n\to \infty } h_{\mu _n}(g)\leq h_\mu (g)$ , which implies

$$ \begin{align*}h_{\mu}(g)\geq \int \log(\det(Dg\mid_{E^{cu}(x)}))d\mu(x).\end{align*} $$

This means that $\mu \in \operatorname {G}^{cu}(g)$ ; thus, $\mu \in \operatorname {G}^u(g)\cap \operatorname {G}^{cu}(g)=\operatorname {G}(g)$ .

Indeed, by Corollary 5.16 and a similar proof as above, for a sequence of $C^1$ maps $g_n\in {\mathcal U},g_n\to g\in {\mathcal U}$ in $C^1$ topology and $\mu _n\in \operatorname {G}(g_n)$ converging to $\mu $ in weak-* topology, we have $\mu \in \operatorname {G}(g).$ Then the map ${\mathcal G}(\cdot )$ is upper semi-continuous, as claimed.

Suppose that $\mu $ is any extreme element of $\operatorname {G}(g)$ ; then it is contained in $\operatorname {G}^u(g)$ . We claim the following.

Lemma 5.18. $\mu $ is ergodic.

Proof. Let $\tilde {\mu }$ be a typical ergodic component in the ergodic decomposition of $\mu $ . We are going to show that $\tilde {\mu }\in \operatorname {G}(g)$ ; this implies that $\tilde {\mu }$ is also an extreme element of $\operatorname {G}(g)$ and thus it coincides with $\mu $ .

By Proposition 2.13, $\tilde {\mu }\in \operatorname {G}^u(g)$ . Thus, it suffices to show that $\tilde {\mu }\in \operatorname {G}^{cu}(g)$ .

Because $g\in {\mathcal U}$ , by Lemma 5.3, any measure $\nu \in \operatorname {G}^u(g)$ has positive center exponent. By Ruelle’s inequality,

$$ \begin{align*}h_{\tilde{\mu}}(g)\leq \int \log(\det(Dg\mid_{E^{cu}(x)}))d\tilde{\mu}(x).\end{align*} $$

Because $\mu \in \operatorname {G}^{cu}(g)$ ,

$$ \begin{align*}h_\mu(g)\geq \int \log(\det(Dg\mid_{E^{cu}(x)}))d\mu(x).\end{align*} $$

Because entropy function is an affine functional with respect to invariant measures, we must have $h_{\tilde {\mu }}(g)= \int \log (\det (Dg\mid _{E^{cu}(x)}))d\tilde {\mu }(x)$ for typical ergodic components $\tilde {\mu }$ of $\mu $ . Thus, $\tilde {\mu }\in \operatorname {G}^{cu}(g)$ . The proof is complete. □

We continue the proof of Proposition 5.17. Assume that $g\in {\mathcal U}$ is a $C^{1+}$ partially hyperbolic diffeomorphism. First we suppose that $\mu $ is an extreme element of $\operatorname {G}(g)$ . Then by the discussion above, $\mu $ is ergodic with positive center exponents. Moreover, by Ruelle’s inequality, we get

$$ \begin{align*}h_\mu(g)= \int \log(\det(Dg\mid_{E^{cu}(x)}))d\mu(x).\end{align*} $$

By the entropy formula of Ledrappier-Young [Reference Ledrappier and Young35], the disintegration of $\mu $ along the Pesin unstable manifold is equivalent to the Lebesgue measure on the leaves. This means, for $\mu $ almost every x, Lebesgue almost every point on the Pesin unstable manifold of x is a typical point of $\mu $ . Because the basin of $\mu $ is saturated by stable leaves (we use the fact that $E^s$ is uniformly contracting) and the stable foliation is absolutely continuous, the union of the stable leaves of the previous full Lebesgue measure subset of $W^u(x)$ is contained in the basin of $\mu $ and has full volume inside a ball with center at x. Note, however, that such a ball may not have uniform radius $r_1$ .

To obtain a ball with radius $r_1$ in the basin of $\mu $ , we apply Lemma 5.8 to obtain an $n>0$ such that $g^n(x)\in {\mathcal H}(b_0/2,g)$ . Then by Lemma 5.11, $W^u(g^n(x),g)$ contains a disk with radius $r_1$ , where Lebesgue typical points in this disk are typical points of $\mu $ . By the uniform transversality between the bundles $E^s$ and $E^{cu}$ , the basin of $\mu $ contains Lebesgue almost every point of a ball at $g^n(x)$ with radius $r_1$ , which we denote by $B_{g^n(x)}(r_1)$ . It then follows that

(16) $$ \begin{align} \mu(B_{g^n(x)}(r_1))>0. \end{align} $$

To simplify notation, we write any ball obtained in the above way by $B_{\mu }$ .

Because the basins of different physical measures are disjoint, $\operatorname {G}(g)$ has only finitely many extreme elements. We denote them by $\mu _1,\cdots , \mu _k$ .

Now we prove that every physical measure $\mu $ of g is an extreme element of $\operatorname {G}(g)$ . Because the basin of $\mu $ , ${\mathcal B}(\mu )$ , has positive volume, by Proposition 2.12, Lebesgue almost every point $x\in {\mathcal B}(\mu )$ must satisfy $\mu =\lim \frac {1}{n}\sum _{i=0}^{n-1}\delta _{g^i(x)}\in \operatorname {G}(g)$ . Thus, it only remains to show that $\mu $ is ergodic.

Because $\operatorname {G}(g)$ is convex with finitely many extreme elements, $\mu $ can be written as a combination:

$$ \begin{align*}\mu=a_1 \mu_1+\cdots +a_k \mu_k,\end{align*} $$

where $0\leq a_1,\cdots , a_k \leq 1$ and $\sum _{i=1}^k a_k=1$ . There is $1\leq j\leq k$ such that $a_j>0$ . Then by (16), $\mu (B_{\mu _j})=a_j \mu _j(B_{\mu _j})>0$ .

Thus, for every point $x\in {\mathcal B}(\mu )$ and n sufficiently large, $\frac {1}{n}\sum _{i=0}^{n-1} \delta _{g^i(x)} (B_{\mu _j})>0$ . In particular, there is $n_j>0$ such that $g^{n_j}(x)\in B_{\mu _j}$ .

Because $\operatorname {vol}({\mathcal B}(\mu ))>0$ , take $x\in {\mathcal B}(\mu )$ a Lebesgue density point of ${\mathcal B}(\mu )$ ; that is, it satisfies

$$ \begin{align*}\lim_{r\to 0^+} \frac{\operatorname{vol}(B_x(r)\cap {\mathcal B}(\mu))}{\operatorname{vol}(B_x(r))}\to 1.\end{align*} $$

Then the above argument shows that ${\mathcal B}(\mu )\bigcap B_{\mu _j}=g^{n_j}({\mathcal B}(\mu ))\bigcap B_{\mu _j}$ has positive Lebesgue measure.

Recall that $B_{\mu _j}$ is in the basin of $\mu _j$ . Therefore, the basin of $\mu $ and $\mu _j$ has nontrivial intersection. This implies $\mu = \mu _j$ , and $\mu $ is ergodic. The proof of Proposition 5.17 is complete. □

6.1 Liao’s shadowing lemma

Definition 6.1. An orbit segment $(x,f(x),\cdots ,f^n(x))$ is called $\lambda $ -quasi hyperbolic if there exists $0< \lambda <1$ such that for $1 \leq k \leq n$ ,

(18) $$ \begin{align} \prod_{i=0}^{k-1} \|Df^{-1}\mid_{E^{cu}(f^{n-i}(x))}\|< \lambda^k, \end{align} $$

(19) $$ \begin{align} \prod_{i=0}^{k-1}\|Df|_{E^s(f^i(x))}\| \le \lambda^k \end{align} $$

and

(20) $$ \begin{align} \frac{\|Df|_{E^s(f^i(x))}\|}{m(Df|_{E^{cu}(f^i(x))} )}\le \lambda^2, \end{align} $$

where $m(\cdot )$ is the minimum norm.

Note that (19) and (20) are automatically satisfied due to the uniform contraction of $Df$ on $E^s$ and the domination between $E^s$ and $E^{cu}$ . In other words, $(x,f(x),\cdots ,f^n(x))$ is $\lambda $ -quasi-hyperbolic if n is a $(-\log \lambda )$ -hyperbolic time for x. In this subsection we need the following shadowing lemma by Liao, which allows a quasi-hyperbolic, periodic pseudo-orbit to be shadowed by a periodic orbit with large unstable manifold.

Remark 6.3. The parameters in the previous lemma can be made uniform for diffeomorphisms in a $C^1$ neighbourhood ${\mathcal U}$ of f. Moreover, one can take $\delta $ sufficiently small; then $d(f^i(x),f^i(p))\leq L d(x,f^n(x))$ is sufficiently small for any $0\leq i \leq n-1$ , and then

$$ \begin{align*}\prod_{i=0}^{k-1} \|Df^{-1}\mid_{E^{cu}(f^{n-i}(p))}\|\leq \lambda^k \end{align*} $$

for any $1 \leq k \leq n$ . In particular, if one takes $\lambda = e^{-b_0/2}$ , then the size of the unstable manifold of p can be chosen to be $r_1>0$ , which is the constant given by Lemma 5.11.

Definition 6.4. A periodic point p of $g\in {\mathcal U}$ is called a $\lambda $ -hyperbolic periodic point if it satisfies

(21) $$ \begin{align} \prod_{i=0}^{k-1} \|Df^{-1}\mid_{E^{cu}(f^{n-i}(p))}\|\leq \lambda^k \end{align} $$

for any $1 \leq k \leq \pi (p)$ .

By the previous discussion and Remark 5.6, we have shown the following.

6.2 Existence of skeleton

In this section, we will show that any $C^1$ diffeomorphism $g\in {\mathcal U}$ admits a skeleton. The main result of this section is Proposition 6.8.

In order to apply Liao’s shadowing lemma, we need to establish the existence of orbit segments that are quasi-hyperbolic. This follows from Proposition 2.12 and (17).

Proposition 6.6. Suppose $g\in {\mathcal U}$ . There is a full-volume subset $\Gamma _g$ such that for Lebesgue almost every point $x\in \Gamma _g$ ,

$$ \begin{align*}\limsup_{n\to \infty}\frac{1}{n}\sum_{i=0}^{n-1} \log\|Dg^{-1}\mid_{E^{cu}(g^{n-i}(x))}\|\leq -b_0.\end{align*} $$

By the Pliss lemma (see [Reference Alves, Bonatti and Viana1]), there exists $n_x\geq 1$ and $\delta _1>0$ such that

(22) $$ \begin{align} \#(H(b_03/4,x,g)\cap [1,n))\geq n\delta_1 \text{ for all }n\geq n_x, \end{align} $$

where $H(b_03/4,x,g)$ is the collection of $b_03/4$ -hyperbolic times along the forward orbit of x.

Taking a sequence of integers $n_x\leq n_1< n_2 <\cdots $ such that $n_i\in H(b_03/4,x,g))$ , we may assume that $x_{n_i} = f^{n_i}(x)$ converges to a point $x_0$ . For $\lambda =e^{-\frac {3b_0}{4}}$ , $\rho $ and L are obtained by Lemma 6.2. We may further assume that $\sup _{i,j}\{d(x_{n_i},x_{n_j})\}\leq \rho _0 \leq \rho $ , where $\rho _0$ satisfies that for any two points $y,z\in M$ with $d(y,z)\leq L \rho _0$ , we have

(23) $$ \begin{align} \mid \log\|Dg^{-1}\mid_{E^{cu}(y)}\|-\log\|Dg^{-1}\mid_{E^{cu}(z)}\| \mid \leq b_0/4. \end{align} $$

Because for any $i<j$ , the pseudo-orbit $\{x_{n_i},x_{n_i+1},\cdots ,x_{n_j-1}\}$ is $b_03/4$ -quasi-hyperbolic, by Lemma 6.2, this pseudo-orbit is $L d(x_{n_i},x_{n_j})\leq L \rho _0$ shadowed by a periodic orbit $p_{x,i,j}$ . Because $x_{n_i}\to x_0$ as $i\to \infty $ , all of the periodic points $p_{x,i,i+1}$ converge to $x_0$ .

Moreover, by the choice of $\rho _0$ in (23), $p_{x,i,j}$ is a $e^{-b_0/2}$ -quasi-hyperbolic periodic point. By Lemma 6.5, each periodic point $p_{x,i,j}$ has unstable manifold with size at least $r_1$ . Their stable manifolds already have uniform size due to $E^s$ being uniformly contracting (note that all $p_{x,i,j}$ s have stable index $i_s$ ). Thus, there is $m_x$ such that for any $i,j>m_x$ , $p_{x,i,i+1}$ and $p_{x,j,j+1}$ are homoclinic related to each other and

$$ \begin{align*} {\mathcal F}^s_{\operatorname{loc}}(x_{i})\pitchfork W^u_{r_1}(p_{j,j+1})\neq \emptyset. \end{align*} $$

Furthermore, ${\mathcal F}^s_{\operatorname {loc}}(p_i)$ will intersect transversally with any disk center at $x_j$ , tangent to the $cu$ cone with radius at least $r_1$ .

To simplify notation, we will write $p_{x,i,i+1}=p_{x,i}$ .

Proof. Let U be any small neighbourhood of x. Because for any $i>m_x$ , all of the hyperbolic periodic points $p_{i}$ are homoclinic related to each other, we only need to show that there is $i>m_x$ such that ${\mathcal F}^s(\mbox {Orb}(p_i))\cap U\neq \emptyset $ ; then the lemma will follow from the inclination lemma.

We take $\varepsilon>0$ small enough such that $\hat {{\mathcal F}}^{cu}_{g,x}(x,\varepsilon )\subset U$ . By Lemma 5.5, for $i>m_x$ sufficiently large, $g^{i}(\hat {{\mathcal F}^{cu}}_{g,x}(x,\varepsilon ))\supset \hat {{\mathcal F}^{cu}}_{g,x_{n_i}}(x_{n_i},r_1)$ , where the latter is a disk tangent to a $cu$ cone with uniform diameter. This means that when i is sufficiently large,

$$ \begin{align*}g^{n_i}(\hat{{\mathcal F}^u}_{g,x}(x,\varepsilon))\pitchfork {\mathcal F}^s_{\operatorname{loc}}(p_{i})\neq \emptyset.\end{align*} $$

By the invariance of the stable manifold under the iteration of g, we have $U\cap {\mathcal F}^s(\mbox {Orb}(p_i))\neq \emptyset $ .

The proof is complete. □

Now we are ready to construct the skeleton for $g\in {\mathcal U}$ . By Proposition 6.6, for each $x\in \Gamma _g$ , we fix any one of $p_{x,i}$ for $i>m_x$ and denote it by $p_x$ . Then by the previous lemma, the union $\bigcup _{x\in \Gamma }{\mathcal F}^s(\mbox {Orb}(p_x))$ is dense in the manifold M.

Moreover, because each periodic point $p_x$ has stable and unstable manifold with size at least $r_1$ , there are only finitely many of them that are not homoclinically related to each other, with number uniformly bounded from above. Take $\{p_1,\cdots ,p_k\}$ a subset of $\{p_x\}_{x\in \Gamma }$ that are not homoclinic related and have maximal cardinality.

We claim that $\bigcup _{i=1,\cdots ,k}{\mathcal F}^s(\mbox {Orb}(p_i))$ is dense in the manifold M. Assume that this is not the case; then we can take $p_x$ for $x\in M\setminus \bigcup _{i=1,\cdots ,k}\mbox {Cl}({\mathcal F}^s(\mbox {Orb}(p_i)))$ . By the choice of $\{p_1,\cdots ,p_k\}$ , $p_x$ must be homoclinically related to some $p_i$ . However, this means that $\mbox {Cl}({\mathcal F}^s(\mbox {Orb}(p_i)))=\mbox {Cl}({\mathcal F}^s(\mbox {Orb}(p_x)))$ by the inclination lemma. Lemma 6.7 then shows that $x\in \mbox {Cl}({\mathcal F}^s(\mbox {Orb}(p_i))) $ , which is a contradiction.

Thus, $\{p_1,\cdots ,p_k\}$ forms a pre-skeleton. By Lemma 4.5, we have shown the following.

From now on, we fix ${\mathcal S}(g)=\{p_1,\cdots ,p_k\}$ a skeleton obtained as above.

6.3 Skeleton and measures

In this section we assume $g\in {\mathcal U}$ to be $C^{1+}$ ; then by Lemma 5.3, g has a mostly expanding center. We will establish a one-to-one correspondence between elements of ${\mathcal S}(g)$ and the physical measures of g.

By Proposition 5.17, g has only finitely many physical measures $\{\mu _1,\cdots ,\mu _l\}$ . Moreover, from Lemma 5.8 and Proposition 5.17, there is $r_1>0$ only depending on ${\mathcal U}$ and $b_0$ such that, for any physical measure $\mu _j$ of g, there is a $\mu _j$ regular point $x_j$ such that

1. (a) $x_j\in {\mathcal H}(b_0/2,g)$ and thus has Pesin unstable manifold with size larger than $r_1$ ;

2. (b) $\mu $ regular points consist of Lebesgue almost every point on the Pesin unstable manifold of $x_j$ .

The main result of this section is the following.

Proof. Fix any physical measure $\mu _j$ of g. By (a) above, there is $p_i\in {\mathcal S}(g)$ such that ${\mathcal F}^s(\mbox {Orb}(p_i))\pitchfork W^u_{r_1}(x_j,g)\neq \emptyset $ . By the inclination lemma, $g^n(W^u_{r_1}(x_j,g))$ converges to $W^u(\mbox {Orb}(p_i),g)$ . Because $W^u_{r_1}(x_j,g)\subset \operatorname {supp}(\mu _j)$ by (b) above, we have $\mbox {Cl}(W^u(\mbox {Orb}(p_i),g))\subset \operatorname {supp}(\mu _j)$ .

To show the reversed inclusion, note that for n large enough, by the inclination lemma, $g^n(W^u_{r_1}(x,g))$ approaches $W^u_{\operatorname {loc}}(p)$ in the following sense: there is a stable holonomy map from $W^u_{\operatorname {loc}}(p)$ to $g^n(W^u_{r_1}(x,g))$ induced by the stable foliation. Because the set of $\mu _j$ typical points is invariant under iteration, Lebesgue almost every point of $g^n(W^u_{r_1}(x,g))$ is also typical for $\mu _j$ . Because stable foliation is absolutely continuous and the basin of $\mu _j$ is s-saturated, it follows that Lebesgue almost every point of $W^u(\mbox {Orb}(p_i),g)$ belongs to the basin of $\mu _j$ .

Take any point $y\in W^u(p_i)\cap {\mathcal B}(\mu _j)$ . Because $g^n(y)\in W^u(\mbox { Orb}(p_i))$ for any $n\geq 0$ , $\mu _j=\lim \frac {1}{n}\sum \delta _{g^i(y)}$ is supported on $\mbox {Cl}(W^u(\mbox {Orb}(p_i),g))$ . As a conclusion,

(24) $$ \begin{align} \operatorname{supp}(\mu_j)=\mbox{Cl}(W^u(\mbox{Orb}(p_i),g)). \end{align} $$

Because Lebesgue almost every point on $W^u(\mbox {Orb}(p_i),g)$ belongs to the basin of $\mu _j$ , the map $j\to i(j)$ is injective; in particular, we have $k\geq l$ . After reordering the periodic points of ${\mathcal S}(g)$ , we may assume that $i(j)=j$ for $j=1,\cdots , l$ .

In order to prove $k=l$ , we only need to show that $\{p_1,\cdots ,p_l\}$ is a pre-skeleton; that is, $\bigcup _{i=1}^l {\mathcal F}^s(\mbox {Orb}(p_i))$ is dense in the manifold M. By Proposition 5.17, the union of basins of physical measures has full volume; thus, it suffices to prove that for each $1\leq i \leq l$ , the closure of ${\mathcal F}^s(\mbox {Orb}(p_i))$ coincides with the closure of ${\mathcal B}(\mu _i)$ .

By (24) we have $p_i\in \operatorname {supp}(\mu _i)$ . Take $r>0$ sufficiently small such that $\mu _i(B_r(p_i))>0$ and $B_r(p_i)\subset {\mathcal O}_i=\bigcup _{y\in W^u(\mbox {Orb}(p_i),g)}{\mathcal F}^s(y)$ . For any $x\in {\mathcal B}(\mu _i)$ , because we have $\frac {1}{n}\sum _{i=0}^{n-1}\delta _{f^i(x)}\to \mu _i$ , there is n sufficiently large such that $\frac {1}{n}\sum _{i=0}^{n-1}\delta _{f^i(x)}(B_r(p_i))>0$ . This shows that there is $m>0$ such that $f^m(x)\in B_r(p_i)\subset {\mathcal O}_i$ . By (ii) of Proposition 4.2, $f^m(x)$ is accumulated by ${\mathcal F}^s(\mbox { Orb}(p_i))$ and so is x. Thus, we have shown that the basin of $\mu _i$ is contained in the closure of ${\mathcal F}^s(\mbox {Orb}(p_i))$ , and the reversed inclusion follows immediately from the u-saturation of $\operatorname {supp}(\mu _i)$ . We now conclude that $k=l$ .

The proof is complete. □

Proof. Proof of Theorem B

By Proposition 6.8, f admits an index $i_s$ skeleton. Let ${\mathcal S}=\{p_1,\cdots ,p_k\}$ be any index $i_s$ skeleton of f. By Proposition 6.9, the number of physical measures is precisely $k=\#{\mathcal S}$ , and for each $p_i\in {\mathcal S}$ there exists a distinct physical measure $\mu _i$ such that

1. (1) the closure of $W^u(Orb(p_i))$ coincides with $\operatorname {supp}(\mu _i)$ and by (ii) of Proposition 4.2, they also coincide with the homoclinic class of the orbit $\mbox {Orb}(p_i)$ .

2. (2) the closure of ${\mathcal F}^s(\mbox {Orb}(p_i))$ coincides with the closure of the basin of the measure $\mu _i$ .

Moreover, by (ii) of Proposition 4.2,

$$ \begin{align*}\operatorname{Int}(\mbox{Cl}({\mathcal B}(\mu_i)))\cap \operatorname{Int}(\mbox{Cl}({\mathcal B}(\mu_j)))= \emptyset\end{align*} $$

for $1 \leq i \neq j \leq k$ . The proof is finished. □

6.4 Proof of Corollary C

We finish this section with the proof of Corollary C.