2.1 Skew products

We denote by $\operatorname {Diff}^\infty (N)$ the space of $C^{\infty }$ diffeomorphisms of N, equipped with the Whitney $C^{\infty }$ topology.

Let $f: M \rightarrow M$ be a homeomorphism and $g_x\in \operatorname {Diff}^\infty (N)$ for every $x\in M$ .

Definition 2.1. (Skew product)

By a skew product F on $M\times N$ , we mean a homeomorphism $F: M\times N \rightarrow M\times N$ defined by

$$ \begin{align*} F:M\times N &\longrightarrow M\times N\\ (x, y) &\mapsto (f(x), g_x(y)), \end{align*} $$

such that the following maps are continuous:

$$ \begin{align*} M &\longrightarrow \operatorname{Diff}^{\infty}(N) \\ x &\longmapsto g_x \end{align*} $$

and

$$ \begin{align*} M &\longrightarrow \operatorname{Diff}^{\infty}(N) \\ x &\longmapsto g^{-1}_x. \end{align*} $$

2.2 Lyapunov exponents

Assume that the skew product map F is equipped with an F-invariant measure $\mu $ . According to Furstenberg and Kesten’s theorem [Reference Furstenberg and KestenFK], the following limits,

$$ \begin{align*} \begin{aligned} \lambda_1^c(F,x,y)&=\lim_{n \rightarrow \infty} \dfrac{1}{n} \log \|D_y g_x^n \|, \\ \lambda_2^c(F,x,y)&=\lim_{n \rightarrow \infty} \dfrac{1}{n} \log \|D_y (g_x^{n})^{-1}\| ^{-1}, \end{aligned} \end{align*} $$

exist for $\mu $ -almost every (a.e.) $(x,y)$ , where

$$ \begin{align*} g_x^n= \begin{cases} \mathrm{Id} & \text{ if } n=0, \\ g_{f^{n-1}(x)} \circ \cdots \circ g_{f(x)} \circ g_{x} & \text{ if } n>0, \\ (g_{f^n(x)})^{-1} \circ \cdots \circ(g_{f^{-1}(x)})^{-1}& \text{ if } n<0, \end{cases} \end{align*} $$

and

$$ \begin{align*} \lambda_1^c(F,x,y)\geq \lambda_2^c(F,x,y). \end{align*} $$

The numbers $\lambda _j^c(F,x,y)$ are called fiber Lyapunov exponents of F with respect the measure $\mu $ . For $j=1,2$ , define

$$ \begin{align*} \lambda_j^c(F,\mu)=\int \lambda_j^c(F,x,y)\, d\mu. \end{align*} $$

If $\lambda _1^c(F,x,y)\neq \lambda _2^c(F,x,y)$ , then, by Oseledets theorem [Reference OseledetsO], there is a measurable splitting

$$ \begin{align*} \{x\} \times T_y N = E_1(x, y) \oplus E_2(x, y) \end{align*} $$

such that

$$ \begin{align*} E_j{}(x,y):=\bigg\{v \in T_y N\setminus \{0\} : \lim_{n \rightarrow \pm \infty} \dfrac{1}{n} \log \|D_y g_x^n v\| =\lambda_j^c(F, x,y)\bigg\} \cup\{0\}. \end{align*} $$

The subspaces $E_j(x, y)$ are invariant under F.

If $\mu $ is ergodic, then for $\mu $ -a.e. $(x,y)$ , $\lambda _j^c(F,x,y)=\lambda _j^c(F, \mu )$ and the dimensions of the subspaces $E_j(x, y)$ are constant.

2.3 Fiber entropy

Let $\pi $ denote the canonical projection $\pi : M \times N \rightarrow M$ . For each $x \in M$ , we define the fiber of the product manifold $M \times N$ over x as

$$ \begin{align*} N_x := \{x\} \times N = \pi^{-1}(x). \end{align*} $$

Let $\mu $ be an F-invariant probability measure and $\nu = \pi _*\mu $ . Consider the family $\{\mu _x\}_{x \in M}$ of conditional measures along the fibers $N_x$ (see [Reference RokhlinRo]). This family is uniquely determined up to a $\nu $ -null set.

Given a measurable partition $\mathcal {P}$ of N, define

$$ \begin{align*} \mathcal{P}_x^n=\bigvee_{j=1}^n g_x^{-1} \circ g_{f(x)}^{-1} \circ \cdots \circ g_{f^{j-1}(x)}^{-1} \mathcal{P}. \end{align*} $$

Suppose that

(1) $$ \begin{align} \int_{M} H_x(\mathcal{P})\,d \nu<\infty, \end{align} $$

where $H_x(\mathcal {P})=-\sum _{P \in \mathcal {P}} \mu _x(P) \log \mu _x(P)$ .

Theorem 2.4. [Reference Abramov and RokhlinAR]

Let F be a skew product, $\mu $ an F-invariant probability measure, and $\nu = \pi _*\mu $ . For every $\mathcal {P}$ satisfying (1),

$$ \begin{align*} h_\mu(F \mid \pi, \mathcal{P}) := \lim_{n \rightarrow \infty} \dfrac{1}{n} \int_M H_x(\mathcal{P}_x^n)\, d\nu. \end{align*} $$

This limit exists and is finite. Moreover, if we define

$$ \begin{align*} h_\mu(F \mid \pi) = \sup_{\mathcal{P}} h_\mu(F \mid \pi, \mathcal{P}), \end{align*} $$

then

(2) $$ \begin{align} h_\mu(F) = h_{\nu}(f) + h_\mu(F \mid \pi). \end{align} $$

The quantity $h_\mu (F \mid \pi )$ is called the fiber entropy of the skew product F.

Theorem 2.5. [Reference Bahnmüller and BogenschützBB]

For a skew product F, and an F-invariant and ergodic measure $\mu $ , the following Margulis–Ruelle inequality type holds:

$$ \begin{align*} h_\mu(F \mid \pi) \leq \sum_{\lambda_j^c(F,\mu)> 0} \lambda_j^c(F,\mu) \operatorname{\mathrm{dim}} E_j. \end{align*} $$

Since the fiber entropy coincides with the conditional entropy of F with respect to $\pi ^{-1} \mathcal {B}$ , where $\mathcal {B}$ is the Borel $\sigma $ -algebra of f (see [Reference KiferK1]), and the $\sigma $ -algebra generated by $\pi ^{-1}$ is F-invariant, then it follows that the fiber entropy satisfies the equality

(3) $$ \begin{align} h_\mu(F \mid \pi) = h_\mu(F^{-1} \mid \pi). \end{align} $$

In particular, we obtain the reverse Margulis–Ruelle inequality,

$$ \begin{align*} h_\mu(F \mid \pi) \leq \sum_{\lambda_j^c(F,\mu)< 0} -\lambda_j^c(F,\mu) \operatorname{dim} E_j. \end{align*} $$

2.4 Main results

Theorem C. Let M be a compact metric space and N a closed smooth surface. Let $F_k$ , $k\in \mathbb {N}$ , be a skew product, and let $\vartheta _k$ be an $F_k$ -invariant and ergodic measure. Suppose that:

• • the limits $\lim _k \lambda _1^c(F_k, \vartheta _k)$ and $\lim _k h_{\vartheta _k}(F_k\mid \pi )$ exist and are positive;

• • $F_k$ converges to a skew product F;

• • $\vartheta _k \xrightarrow {w^*} {\mu }$ for some ${F}$ -invariant measure ${\mu }$ .

Then, there exist $\beta \in (0,1]$ and two F-invariant measures $\mu _0$ and $\mu _1$ such that $\mu =(1-\beta )\mu _0+\beta \mu _1$ , $h_{\mu _1} (F\mid \pi )> 0$ , and

$$ \begin{align*} \lim _{k \rightarrow \infty} h_{\vartheta_k}(F_k\mid\pi)\leq \beta h_{\mu_1}(F\mid \pi). \end{align*} $$

Moreover, if $\lambda _2(F,x,y)\leq 0$ for $\mu $ -a.e. $(x,y)$ ,

$$ \begin{align*} \lim _{k \rightarrow \infty} \lambda_1^c(F_k,\vartheta_{k})=\beta \lambda_1^c(F, \mu_1). \end{align*} $$

Corollary 2.6. Let $F_k$ be a skew product, and let $\vartheta _k$ be an $F_k$ -invariant and ergodic measure. Suppose that $F_k$ converges to a skew product F, and $\vartheta _k\to {\mu }$ in the weak* topology for some ${F}$ -invariant and ergodic probability measure ${\mu }$ . If

$$ \begin{align*} h_{\vartheta_k}(F_k\mid\pi)\to h_{\mu}(F\mid \pi)>0, \end{align*} $$

then

$$ \begin{align*} \lambda_1^c(F_k,\vartheta_k) \to \lambda_1^c(F, \mu)\quad \text{and}\quad \lambda_2^c(F_k,\vartheta_k) \to \lambda_2^c(F, \mu). \end{align*} $$

5.1 Ledrappier–Young type formulas

Let F be a skew product and let $\mu $ be an F-invariant ergodic measure of saddle type. Let $\mathcal {R}_{\mu }$ denote the set of points where the Lyapunov exponents of $\mu $ are well defined.

Definition 5.1. (Fiber-wise unstable manifolds)

The set

$$ \begin{align*} \widetilde{W}^u(x, y):=\bigg\{z \in N: \limsup_{n \rightarrow+\infty}\dfrac{1}{n} \log d(g_x^{-n} (y), g_x^{-n} (z)) \leq-\lambda_1^c(F,\mu)\bigg\} \end{align*} $$

is called the fiber-wise unstable manifold of F at $(x, y)$ , where $(x, y)\in \mathcal {R}_{\mu }$ .

Each $ \widetilde {W}^u(x, y) $ is a $ C^{\infty } $ immersed submanifold of $ N $ and $\{x\}\times \widetilde {W}^u(x, y)$ is tangent at $ (x, y) $ to $E_1(x, y)$ .

In this section, we define quantities referred to as local entropy along fiber-wise unstable manifolds. In the ergodic case, the concept of local entropy along the $\widetilde {W}^u$ -manifolds is represented by a number $h^u$ , which measures the degree of randomness along the leaves of the $\widetilde {W}^u$ -manifolds. This notion of entropy originates from Ledrappier and Young [Reference Ledrappier and YoungLY]. Here, we present a version for skew products as described in [Reference Liu and XieLX].

Let $\eta $ be a partition of $M\times N$ that is subordinate to the $\widetilde {W}^u$ -manifolds. For each $x \in M$ , we define $\eta _x$ as the partition $\{\eta _x(y) : y \in N\}$ of N. Recall that $\{\mu _x\}_{x\in M}$ is the disintegration of $\mu $ along the fibers $\{x\}\times N$ . For each $x\in M$ , consider

$$ \begin{align*}\{(\mu_x)_y^{\eta_x}\}_{y \in N},\end{align*} $$

as the canonical family of conditional measures of $\mu _x$ associated with $\eta _x$ .

For $\varepsilon> 0$ , $(x, y) \in M\times N$ , and $n \in \mathbb {Z}^{+}$ , define

$$ \begin{align*} B^F_x(y; n, \varepsilon) := \{ z \in N : d(g_x^j(y), g_x^j(z)) < \varepsilon \text{ for all } 0 \leq j < n \}. \end{align*} $$

In a manner similar to the approach taken by Ledrappier and Young [Reference Ledrappier and YoungLY], the following quantities are introduced:

$$ \begin{align*} \begin{aligned} & \underline{h}^u(x, y; \varepsilon, \eta):=\liminf _{n \rightarrow+\infty}-\dfrac{1}{n} \log (\mu_x)_y^{\eta_x}(B^F_x(y; n, \varepsilon)), \\ & \overline{h}^u(x, y; \varepsilon, \eta):=\limsup _{n \rightarrow+\infty}-\dfrac{1}{n} \log (\mu_x)_y^{\eta_x}(B^F_x(y; n, \varepsilon)). \end{aligned} \end{align*} $$

These functions are clearly measurable. In addition, it is important to observe that the functions increase as $\varepsilon \to 0$ .

Finally, the concepts of lower and upper local entropy along the $\widetilde {W}^u$ -manifolds at $(x, y)$ , with respect to $\eta $ , can be defined. We remark that in [Reference Liu and XieLX], the authors use the intrinsic metric on the unstable sub-manifold to define these quantities. We remark that although we use the global metric, both definitions coincide because $\varepsilon \to 0$ .

$$ \begin{align*} \begin{aligned} \underline{h}^u(x, y; \eta)&:=\lim_{\varepsilon \rightarrow 0} \underline{h}^u(x, y; \varepsilon, \eta) = \lim_{\varepsilon \rightarrow 0} \liminf _{n \rightarrow+\infty}-\dfrac{1}{n} \log (\mu_x)_y^{\eta_x}(B^F_x(y; n, \varepsilon)), \\ \overline{h}^u(x, y; \eta)&:=\lim_{\varepsilon \rightarrow 0} \overline{h}^u(x, y; \varepsilon, \eta)=\lim_{\varepsilon \rightarrow 0}\limsup _{n \rightarrow+\infty}-\dfrac{1}{n} \log (\mu_x)_y^{\eta_x}(B^F_x(y; n, \varepsilon)). \end{aligned} \end{align*} $$

We denote by $\sigma $ the partition $\pi ^{-1}(\mathcal {E})$ , where $\mathcal {E}$ represents the partition of M into points. Let $\eta $ be a partition subordinate to the $\widetilde {W}^u$ -manifolds such that $\eta>\sigma $ , $F^{-1} \eta> \eta $ , $F^{-n}\eta $ tends to the partition into single points, and $h_\mu (F\mid \pi )=H_{\mu }(\eta \mid F(\eta ) \vee \sigma )$ . Such a partition always exists (see [Reference Bahnmüller and LiuBL, Proposition 3.7]).

Proposition 5.3. [Reference Liu and XieLX, Proposition 8.6]

Let $\eta $ be a partition satisfying the properties above, then for $\mu $ -a.e. $(x, y)$ ,

$$ \begin{align*} \underline{h}^u(x, y; \eta)=\overline{h}^u(x, y; \eta)=H_{\mu}(\eta \mid F(\eta) \vee \sigma). \end{align*} $$

Using this result, we conclude that the fiber entropy of a skew product F can be expressed for $\mu $ -a.e. $(x,y)$ as the local entropy along the $\widetilde {W}^{u}$ -manifolds, that is,

(6) $$ \begin{align} h_\mu(F\mid \pi)=\lim_{\varepsilon \rightarrow 0}\liminf _{n \rightarrow+\infty}-\dfrac{1}{n} \log (\mu_x)_y^{\eta_x}(B^{F}_x( y; n, \varepsilon)). \end{align} $$

Observe that, in particular, the proposition implies that this limit does not depend on the partition $\eta $ .

Definition 5.4. We define the following quantity:

$$ \begin{align*} r_F(n, \varepsilon,(\mu_x)_{y}^{\eta_x}, \gamma):=\inf\{r^F_x(n, \varepsilon, Z): Z \subset N \ \text{and}\ (\mu_x)_{y}^{\eta_x}(Z)>\gamma\}, \end{align*} $$

where

$$ \begin{align*} r^F_x(n, \varepsilon, Z) = \inf \bigg\{\#A: Z \subset \bigcup_{y\in A} B^F_x(y; n, \varepsilon)\bigg\}. \end{align*} $$

Proposition 5.5. Let F be a skew product and suppose that the measure $\mu $ is an F-invariant ergodic measure of saddle type. Then, for $\mu $ -a.e. point $(x, y) \in M \times N$ , we have

(7) $$ \begin{align} h_\mu(F\mid \pi)=\inf_{\gamma>0}\lim_{\varepsilon \rightarrow 0}\liminf_{n \rightarrow+\infty}\dfrac{1}{n} \log r_F(n, \varepsilon, (\mu_x)_y^{\eta_x}, \gamma). \end{align} $$

The proof is an adaptation of the arguments in [Reference ZangZ].

Proof. For brevity, we denote the fiber entropy $h_{\mu }(F \mid \pi )$ simply by $h_F$ throughout this proof.

Let $\Gamma _\mu $ denote the full measure set of points satisfying Proposition 5.3. Then, we can assume that, for every $(x,y) \in \Gamma _\mu $ , the local unstable entropy satisfies

$$ \begin{align*}\lim _{\varepsilon \rightarrow 0} \liminf_{n \rightarrow+\infty}-\dfrac{1}{n} \log (\mu_x)_y^{\eta_x}(B^F_x(y;n,\varepsilon))=h_{F}. \end{align*} $$

Let us fix an arbitrary $\tau> 0$ . We define a family of sets $\Lambda _\tau ^{\varepsilon }$ for each $\varepsilon> 0$ as follows:

$$ \begin{align*} \Lambda_\tau^{\varepsilon} :=\bigg\{(x,y) \in \Gamma_\mu: \liminf _{n \rightarrow+\infty}-\dfrac{1}{n} \log (\mu_x)_y^{\eta_x}(B^F_x(y; n, 2 \varepsilon))>h_F-\tau\bigg\}. \end{align*} $$

It follows that the union of these sets covers $\Gamma _\mu $ , that is, $\bigcup _{\varepsilon>0} \Lambda _\tau ^{\varepsilon }\supset \Gamma _\mu $ .

Now, fix $\varepsilon>0$ and a point $p=(x_p, y_p) \in \Lambda _\tau ^{\varepsilon }$ . Define

$$ \begin{align*} \Lambda_{\tau}^{\varepsilon}(x_p):=\{y\in N : (x_p,y) \in \Lambda_\tau^{\varepsilon}\}. \end{align*} $$

For any $j \in \mathbb {N}$ , we further refine this set by defining

$$ \begin{align*} \Lambda_{\tau}^{\varepsilon}(p,j)=\{y\in \Lambda_\tau^{\varepsilon}(x_p): (\mu_{x_p})_{y_p}^{\eta_{x_p}}(B^F_{x_p}(y; n, 2 \varepsilon)) \leq e^{-n(h_F-\tau)} \text{ for all } n \geq j\}. \end{align*} $$

By construction, these sets are nested, $\Lambda _\tau ^{\varepsilon }(p, j) \subset \Lambda _\tau ^{\varepsilon }(p, j+1)$ , and their union covers $\Lambda _\tau ^{\varepsilon }(x_p)$ in terms of the measure $(\mu _{x_p})_{y_p}^{\eta _{x_p}}$ :

$$ \begin{align*} (\mu_{x_p})_{y_p}^{\eta_{x_p}}\bigg(\!\bigcup_j \Lambda_\tau^{\varepsilon}(p, j)\bigg)=(\mu_{x_p})_{y_p}^{\eta_{x_p}}(\Lambda_\tau^{\varepsilon}(x_p)). \end{align*} $$

Let $\gamma> 0$ and $p=(x_p, y_p)\in \Gamma _\mu $ be arbitrary. Consider

$$ \begin{align*} \Gamma_{\mu}(x_p):=\{y\in N: (x_p,y)\in \Gamma_{\mu}\}. \end{align*} $$

Since the union $\bigcup _j \Lambda _\tau ^{\varepsilon }(p, j)$ covers $\Lambda _\tau ^{\varepsilon }(x_p)$ in measure, and $\bigcup _{\varepsilon>0} \Lambda _\tau ^{\varepsilon }(x_p)\supset \Gamma _\mu (x_p)$ , we can choose $\varepsilon> 0$ sufficiently small and $L \in \mathbb {N}$ sufficiently large such that

$$ \begin{align*} (\mu_{x_p})_{y_p}^{\eta_{x_p}}(\Lambda_\tau^{\varepsilon}(p, j)) \geq 1-\dfrac{\gamma}{2} \quad \text{for all } j \geq L. \end{align*} $$

For a given $n \in \mathbb {N}$ , let $C_n \subset \Gamma _\mu (x_p)$ be a set such that $|C_n|$ represents the minimum number of $(n, \varepsilon )$ -Bowen balls required to cover a subset of $\Gamma _\mu (x_p)$ with measure at least $\gamma $ under the measure $(\mu _{x_p})_{y_p}^{\eta _{x_p}}$ . Moreover, assume that $U_n = \bigcup _{\bar {y} \in C_n} B^F_{x_p}(\bar {y}; n, \varepsilon )$ satisfies $(\mu _{x_p})_{y_p}^{\eta _{x_p}}(U_n) \geq \gamma $ .

Therefore, we conclude

$$ \begin{align*} (\mu_{x_p})_{y_p}^{\eta_{x_p}}(\Lambda_\tau^{\varepsilon}(p, n) \cap U_n) \geq \dfrac{\gamma}{2} \quad \text{for all } n \geq L. \end{align*} $$

Let $A_n$ be the points $\bar {y} \in C_n$ for which the corresponding ball $B^F_{x_p}(\bar {y}; n, \varepsilon )$ has a non-empty intersection with $\Lambda _\tau ^{\varepsilon }(p, n)$ . That is, $A_n = \{ \bar {y} \in C_n : B^F_{x_p}(\bar {y}; n, \varepsilon ) \cap \Lambda _\tau ^{\varepsilon }(p, n) \neq \emptyset \}$ .

The union of balls centered in $A_n$ covers the intersection $\Lambda _\tau ^{\varepsilon }(p, n) \cap U_n$ . For each $\bar {y} \in A_n$ , choose an arbitrary point $\widetilde {y} \in B^F_{x_p}(\bar {y}; n, \varepsilon ) \cap \Lambda _\tau ^{\varepsilon }(p, n)$ . Then,

$$ \begin{align*} B^F_{x_p}(\bar{y}; n, \varepsilon) \subset B^F_{x_p}(\widetilde{y}; n, 2 \varepsilon). \end{align*} $$

Hence, for $n \geq L$ ,

$$ \begin{align*} \dfrac{\gamma}{2} &\leq (\mu_{x_p})_{y_p}^{\eta_{x_p}}\bigg(\!\bigcup_{\bar{y} \in A_n} B^F_{x_p}(\widetilde{y}; n, 2 \varepsilon)\bigg) \\ &\leq r_F(n, \varepsilon,(\mu_{x_p})_{y_p}^{\eta_{x_p}},\gamma) \times \sup _{y \in \Lambda_\tau^{\varepsilon}(p, n)} (\mu_{x_p})_{y_p}^{\eta_{x_p}}(B^F_{x_p}(y; n, 2 \varepsilon)) \\ &\leq r_F(n, \varepsilon,(\mu_{x_p})_{y_p}^{\eta_{x_p}},\gamma) \times e^{-n(h_F-\tau)}. \end{align*} $$

Therefore, for any $\varepsilon = \varepsilon (\gamma )$ small enough and n large enough,

$$ \begin{align*} r_F(n, \varepsilon,(\mu_{x_p})_{y_p}^{\eta_{x_p}},\gamma) \geq \dfrac{\gamma}{2} \cdot e^{n(h_F-\tau)}. \end{align*} $$

Since $\gamma>0$ and $\tau>0$ were arbitrary, it follows that

$$ \begin{align*} \begin{aligned} &h_{F} \leq \inf _\gamma \lim _{\varepsilon \rightarrow 0} \liminf _{n \rightarrow+\infty} \dfrac{1}{n} \log r_F(n, \varepsilon,(\mu_{x_p})_{y_p}^{\eta_{x_p}},\gamma). \end{aligned} \end{align*} $$

The proof of the other inequality proceeds analogously using the identity with $\limsup $ .

5.2 Entropy formulas on the projective bundle

To analyze entropy in a way that is more compatible with the projective bundle structure, [Reference Buzzi, Crovisier and SarigBCS1] introduces a two-scale framework. First, let us recall some definitions.

Let F be a skew product and recall Definition 3.1 of the lift $\widehat {F}$ acting on $M\times \widehat {N}$ . That is,

$$ \begin{align*} \widehat{F} (x,(y,E)) = (f(x),\widehat{g}_x(y,E)), \end{align*} $$

where

$$ \begin{align*} \widehat{g}_x(y,E) = (g_x(y), D_yg_x (E)). \end{align*} $$

Fix $x\in M$ and a point $\widehat {y} = (y, E) \in \widehat {N}$ . Given two positive scales $\varepsilon , \widehat {\varepsilon }> 0$ , for each $n \in \mathbb {N}$ , we define the x-fibered $(n, \varepsilon , \widehat {\varepsilon })$ -Bowen ball centered at $\widehat {y} \in \widehat {N}$ as the set

$$ \begin{align*} \begin{aligned} &B^{\widehat{F}}_x(\widehat{y}, n, \varepsilon, \widehat{\varepsilon})\\ &\quad := \{ \widehat{z} \in \widehat{N} : d (\widehat{g}^j_x(\widehat{z}), \widehat{g}^j_x(\widehat{y})) < \widehat{\varepsilon} \ \text{and} \ d(g^j_x(z), g^j_x(y)) < \varepsilon \ \text{for all } 0 \leq j < n \}. \end{aligned} \end{align*} $$

Given a subset $\widehat {Z} \subset \widehat {N}$ , we denote by

$$ \begin{align*} r^{\widehat{F}}_x(n, \varepsilon, \widehat{\varepsilon}, \widehat{Z}) \end{align*} $$

the $(n, \varepsilon , \widehat {\varepsilon })$ -covering number, which is the minimal number of x-fibered $(n, \varepsilon , \widehat {\varepsilon })$ -Bowen balls of order n needed to cover $\widehat {Z}$ .

Let $\widehat {\mu }$ be an $\widehat {F}$ -invariant measure and fix $0 < \gamma < 1$ . Consider the family $\{\widehat {\mu }_x\}_{x \in M}$ of conditional measures defined on the fibers $\{x\} \times \widehat {N}$ . We then define the $(n, \varepsilon , \widehat {\varepsilon }, \gamma )$ - covering number of $\widehat {\mu }_x$ as the minimal number of x-fibered $(n, \varepsilon , \widehat {\varepsilon })$ -Bowen balls needed to cover a set of $\widehat {\mu }_x$ -measure at least $\gamma $ . This quantity is denoted by

$$ \begin{align*} r^{\widehat{F}}_x(n, \varepsilon, \widehat{\varepsilon}, \widehat{\mu}_x, \gamma). \end{align*} $$

For an F-invariant measure $\mu $ , we can define, analogously, $r^{F}_x(n, \varepsilon , \mu _x, \gamma ).$ If $\widehat {\mu }$ is a lift of $\mu $ , [Reference Buzzi, Crovisier and SarigBCS1, Claim 4.4] extends to the present context to obtain the following claim.

Claim 5.6. Fix $x\in M$ . For any $\widehat {\varepsilon },\alpha>0$ , there exist $C, \varepsilon _{*}>0$ such that, for any $0<\varepsilon \leq \varepsilon _{*}$ and any $\gamma>0$ ,

$$ \begin{align*} r^{\widehat{F}}_x(n, \varepsilon, \widehat{\varepsilon}, \widehat{\mu}_x, \gamma)\leq C e^{\alpha n}r^{F}_x(n, \varepsilon, \mu_x, \gamma). \end{align*} $$

Given an F-invariant measure $\mu $ , we define the essential fiber entropy by

$$ \begin{align*} \bar{h}_\mu(F\mid \pi):=\text{ess-sup}\, h_{\mu_e}(F\mid \pi), \end{align*} $$

where the supremum is taken over the ergodic components of $\mu $ .

Proposition 5.7. Let F be a skew product and $\mu $ an F-invariant measure such that $\pi _*\mu =\nu $ . Given $\varepsilon>0$ and $\gamma>0$ , for $\nu $ -a.e $x\in M$ ,

$$ \begin{align*} \limsup_{n\to \infty} \dfrac{1}{n} \log r^{F}_x(n, \varepsilon, \mu_x, \gamma)\leq \bar{h}_\mu(F\mid \pi). \end{align*} $$

Proof. Let $\mathcal {P}$ be a partition of N. For $x\in M$ and $n\in \mathbb {N}$ , consider the partition $\mathcal {P}^n_x$ as in Theorem 2.4. Given $y\in N$ , denote $\mathcal {P}^n_x(y)$ the element of $\mathcal {P}^n_x$ that contains y.

When the measure $\mu $ is ergodic, Belinskaja [Reference BelinskajaB] established a fibered version of the Shannon–McMillan–Breiman theorem which states that for $\mu $ -a.e. $(x,y)$ ,

$$ \begin{align*} \lim_{n\to \infty}-\dfrac{1}{n} \log \mu_{x}(\mathcal{P}^n_x(y))=h_\mu(F \mid \pi, \mathcal{P}). \end{align*} $$

Therefore, in this case, the proposition follows analogously to the proof of the Katok entropy formula [Reference KatokKa].

For the general case, it is possible to obtain a non-ergodic version of Belinskaja’s theorem using the same reasoning than in [Reference Denker, Grillenberger and SigmundDGS] to prove the non-ergodic version of Shannon–McMillan–Breiman. Then, the conclusion follows again by Katok’s arguments.

Proposition 5.7 and Claim 5.6 imply the following result.

Corollary 5.8. Let F be a skew product, $\mu $ an F-invariant measure such that $\pi _*\mu =\nu $ , and $\widehat {\mu }$ is a lift of $\mu $ . Given $\widehat {\varepsilon }>0$ and $\gamma>0$ , for $\nu $ -a.e $x\in M$ ,

$$ \begin{align*} \lim_{\varepsilon\to 0}\limsup_{n\to \infty} \dfrac{1}{n} \log r^{\widehat{F}}_x(n, \varepsilon, \widehat{\varepsilon}, \widehat{\mu}_x, \gamma)\leq \bar{h}_\mu(F\mid \pi). \end{align*} $$

6.1 Reparameterization results

As in the work of [Reference Buzzi, Crovisier and SarigBCS1], one of the main tools we employ is the Yomdin theory. We present here the statement of a version that closely follows theirs. In particular, it is possible to prove the existence of admissible reparameterizations whose cardinality can be controlled by the minimal number of Bowen balls needed to cover the curve. The proof of the proposition below is an easy adaptation of [Reference Buzzi, Crovisier and SarigBCS1, Corollary 4.14].

Let $r\geq 2$ and consider the following quantities:

$$ \begin{align*} Q_r(F):=\max \Big(\sup_{x \in M} \|g_x\|_{C^{r}},\; \sup_{x \in M} \|\widehat{g}_x\|_{C^{r}}\Big),\quad Q_{r,L}(F):=\max _{j=1,2, \ldots, L} Q_r(F^j). \end{align*} $$

Recall that $\|D \widehat {F}\|_{\mathrm {sup}}=\sup _{x \in M} \sup _{\widehat {y} \in \widehat {N}} \|d_{\widehat {y}}\widehat {g}_x\|$ .

Proposition 6.7. Let F be a skew product. For every $2\leq r<\infty $ and $Q> 0$ , there exist constants $\Upsilon =\Upsilon (r)> 0$ and $\varepsilon _Y = \varepsilon _Y(r,Q)> 0$ with the following properties. Given $x \in M$ , suppose:

1. (1) $Q_{r, L}(F) < Q$ ;

2. (2) the curve $\sigma _x : [0,1] \to M \times N$ is regular and has $C^{r}$ -size bounded by $(\varepsilon , \widehat {\varepsilon })$ , where $0 < \varepsilon , \widehat {\varepsilon } < \varepsilon _Y$ ;

3. (3) $L, n \geq 1$ , $J \subset [0,1]$ .

Under these assumptions, there exists a collection $\mathcal {R}$ of reparameterizations of $\sigma $ over J, which is $(C^{r}, F, L, \varepsilon , \widehat {\varepsilon })$ -admissible up to time n, and whose cardinality satisfies the estimate

$$ \begin{align*} |\mathcal{R}| \leq C(r,F) \cdot r_{f(x)}^{\widehat{F}}(n, \varepsilon, \widehat{\varepsilon}, \pi_2(\widehat{F} \circ \widehat{\sigma}_x(J))), \end{align*} $$

where

$$ \begin{align*} C(r,F)=\Upsilon(r)^{\lceil {n}/{L}\rceil}\|D\widehat{F}^L\|^{{\lfloor {n}/{L}\rfloor}/{(r-1)}}_{\mathrm{sup}}\|D\widehat{F}^{n-L\lfloor {n}/{L}\rfloor}\|^{{1}/{(r-1)}}_{\mathrm{sup}} \end{align*} $$

and $\pi _2\colon M\times \widehat {N}\to \widehat {N}$ is the projection onto the second coordinate.

Observe that the constant $C(r,F)$ depends on the regularity r that has been fixed. Although the map is acting $C^{\infty }$ in the fibers, since we do not know how the constant $\Upsilon (r)$ behaves with r, we are not able at this stage to simplify this quantity considering $r\to \infty $ .

In the following, we present a consequence of Proposition 6.7. The next result generalizes [Reference Buzzi, Crovisier and SarigBCS1, Proposition 5.1] to the present setting. We remark that the main difficulty appears when we apply Corollary 5.8 since now all the parameters depend on the point $x\in M$ .

Define $\lambda (\widehat {F}):=\lim _{n\to \infty } ({1}/{n})\log \|D\widehat {F}^n\|_{\mathrm {sup}}$ .

Proof. We consider a compact set $K_{\gamma }\subset M$ such that the disintegration $x\mapsto \widehat {\mu }_x$ is continuous on $K_{\gamma }$ , $\nu (K_{\gamma })>1-\gamma ^2$ , and Corollary 5.8 holds for every $x\in K_{\gamma }$ .

Let $\Upsilon (r)$ and $\varepsilon _Y$ be given by Proposition 6.7 and $L_1>0$ such that for every $L\geq L_1$ ,

(8) $$ \begin{align} \Upsilon(r)<\exp \bigg(\frac{\rho L}{10}\bigg) \end{align} $$

and

$$ \begin{align*} \dfrac{1}{L}\log\|D\widehat{F}^L\|_{\mathrm{sup}}< \lambda(\widehat{F})+ \frac{\rho}{10}. \end{align*} $$

Fix $\widehat {\varepsilon }$ in $(0, \varepsilon _Y)$ and $L\geq L_1$ .

By Corollary 5.8, for every $x\in K_{\gamma }$ , there exists $0<\bar {\varepsilon }(x)<\varepsilon _Y$ and $\bar {n}(x)>L$ such that for any $\varepsilon < \bar {\varepsilon }(x)$ and $n \geq \bar {n}(x)$ ,

(9) $$ \begin{align} \dfrac{1}{n} \log r^{\widehat{F}}_x\bigg(n, \frac{\varepsilon}{4}, \frac{\widehat{\varepsilon}}{4}, \widehat{\mu}_x, 1-\gamma^2\bigg)<\bar{h}_{\mu}(F\mid \pi)+\frac{\rho}{4}. \end{align} $$

Moreover, there exist $0<\varepsilon _1<\varepsilon _Y$ , $n_1> L$ , and a compact set $K^{\prime }_{\gamma }\subset K_{\gamma }$ with $\nu (K^{\prime }_{\gamma })>1-\gamma ^2$ such that for every $x\in K^{\prime }_{\gamma }$ , $\bar {\varepsilon }(x)\geq \varepsilon _1$ and $\bar {n}_1(x)\leq n_1$ . We can assume that $n_1$ was chosen such that for every $n\geq n_1$ ,

(10) $$ \begin{align} \log \|D\widehat{F}^L\|_{\mathrm{sup}}<\frac{n\rho}{10}. \end{align} $$

Fix $\varepsilon <\varepsilon _1$ and $n\geq n_1$ . By (9), for every $x\in K^{\prime }_{\gamma }$ , there exists a set $\widehat {Z}_x \subset \widehat {N}$ with $\widehat {\mu }_x(\widehat {Z}_x)> 1 - \gamma ^2$ such that

(11) $$ \begin{align} r^{\widehat{F}}_x\bigg(n, \frac{\varepsilon}{4}, \frac{\widehat{\varepsilon}}{4}, \widehat{Z}_x\bigg) < \exp\bigg(n \bar{h}(F \mid \pi) + \frac{\rho}{4} \bigg). \end{align} $$

By the regularity of $\widehat {\mu }_x$ , there exists a compact set $\widehat {K}_x\subset \widehat {Z}_x$ such that $\widehat {\mu }_x(\widehat {K}_x)> 1 - \gamma ^2$ and $\widehat {K}_x$ also satisfies (11).

Therefore, for every $x\in K^{\prime }_{\gamma }$ , there is a neighborhood $\widehat {U}_{1,x}$ of $\widehat {K}_x$ such that $\widehat {U}_{1,x}$ is contained in the union of a collection $\mathcal {C}_x$ of x-fibered $(n, {\varepsilon }/{4}, {\widehat {\varepsilon }}/{4})$ -Bowen balls for $\widehat {F}$ with cardinality at $\operatorname {most} \exp (n(\bar {h}(F\mid \pi )+{\rho }/{4}))$ .

Since $x\mapsto \widehat {\mu }_x$ is continuous on $K^{\prime }_{\gamma }$ and $K^{\prime }_{\gamma }$ is compact, there exists $x_1, \ldots , x_l\in K^{\prime }_{\gamma }$ and $\delta _i>0$ such that $K^{\prime }_{\gamma }\subset \bigcup _{i=1}^l B(x_i, \delta _i)$ and if $y\in B(x_i,\delta _i)$ , then $\widehat {\mu }_y(\widehat {U}_{1,x_i})> 1 - \gamma ^2$ and $\widehat {U}_{1,x_i}$ is contained in the union of a collection $\mathcal {C}_y$ of y-fibered $(n, {\varepsilon }/{2}, {\widehat {\varepsilon }}/{2})$ -Bowen balls for $\widehat {F}$ with cardinality at $\operatorname {most} \exp (n(\bar {h}(F\mid \pi )+{\rho }/{2}))$ .

We define the following subset:

$$ \begin{align*} \begin{aligned} \widehat{U}_1=B(x_1,\delta_1)\times \widehat{U}_{1,x_1}&\cup (B(x_2,\delta_2)\setminus \overline{B(x_1, \delta_1)})\times \widehat{U}_{1,x_2}\cdots\\ &\cup (B(x_l,\delta_l) \setminus \bigcap_{i=1}^{l-1}\overline{B(x_i, \delta_i)})\times \widehat{U}_{1,x_l}. \end{aligned} \end{align*} $$

Observe that $\widehat {U}_1$ is an open subset of $M\times \widehat {N}$ , and we can suppose that $\widehat {\mu }(\partial \widehat {U}_{1})=0$ and $\widehat {\mu }(\widehat {U}_1)>1-\gamma ^2$ .

For every $x\in M$ and $\sigma _x$ regular curve, we know that

$$ \begin{align*}\widehat{F}(\sigma_x([0,1])\cap \widehat{U}_1)\subset \{f(x)\}\times g_x(\widehat{U}_{1,x_i}) \quad \text{for some } i=1,\ldots,l.\end{align*} $$

This implies that there exists $\mathcal {U}_1$ , a neighborhood of F in the space of skew products, such that for every $G\in \mathcal {U}_1$ , $\widehat {G}(\sigma _x([0,1])\cap \widehat {U}_1)$ can be covered using fibered $(n, \varepsilon ,\widehat {\varepsilon })$ -Bowen balls for $\widehat {G}$ with cardinality at $\operatorname {most} \exp (n(\bar {h}(F\mid \pi )+{\rho }/{4}))$ .

Moreover, we suppose that for every $G\in \mathcal {U}_1$ ,

(12) $$ \begin{align} \|D\widehat{G}^L\|_{\mathrm{sup}}\leq e^{{nL}/{10}}\|D\widehat{F}^L\|_{\mathrm{sup}}\quad \text{and} \quad \|D\widehat{G}\|_{\mathrm{ sup}}\leq e^{{\rho}/{10}}\|D\widehat{F}\|_{\mathrm{sup}}. \end{align} $$

If $G\in \mathcal {U}_1$ , we conclude

(13) $$ \begin{align} r_{g(x)}^{\widehat{G}}(n, \varepsilon, \widehat{\varepsilon}, \widehat{g}_x^G(\widehat{\sigma}[0,1] \cap \widehat{U}_{1})) \leq \exp \bigg(n\bigg(\bar{h}(F\mid \pi)+\frac{\rho}{4}\bigg)\bigg). \end{align} $$

Since $\varepsilon , \widehat {\varepsilon }<\varepsilon _Y$ , if $Q_L(G)<Q$ , we can apply Corollary 6.7 to obtain a family of reparameterizations which are admissible up to time n and satisfy

$$ \begin{align*} |\mathcal{R}| \leq C(r,G)\; r_{g(x)}^{\widehat{G}}(n, \varepsilon, \widehat{\varepsilon}, \widehat{g}_x^G (\widehat{\sigma}[0,1] \cap \widehat{U}_{1})). \end{align*} $$

Using (8), (10), (12), and (13),

$$ \begin{align*} \begin{aligned} |\mathcal{R}|\leq\exp \bigg[n\bigg(\bar{h}(F\mid \pi)+\frac{\lambda(\widehat{F})}{r-1}+\rho\bigg)\bigg]. \end{aligned} \\[-34pt] \end{align*} $$

7.1 Proof of Theorem C

Let $(F_k, \vartheta _k)$ and $(F,\mu )$ satisfy the hypotheses of the theorem. That is:

• • the limits $\lim _k \lambda _1^c(F_k, \vartheta _k)$ and $\lim _k h_{\vartheta _k}(F_k\mid \pi )$ exist and are positive;

• • $F_k$ converges to a skew product F;

• • $\vartheta _k \xrightarrow {w^*} {\mu }$ for some ${F}$ -invariant measure ${\mu }$ .

For k big enough, $h_{\vartheta _k}(F_k\mid \pi )>0$ , then applying Theorem 2.5, we can suppose that $\vartheta _k$ are of saddle type.

Recall that when the measure $\vartheta _k$ is of saddle type, we can consider a partition $\eta ^k$ subordinated to the fiber-wise unstable manifolds and a system of conditional measures $(\vartheta _{k,x})_{y}^{\eta ^k_{x}}$ as in §5.1. By Proposition 4.4, there exists $\beta \in (0,1]$ , and two F-invariant measures $\mu _0$ and $\mu _1$ such that $\mu =(1-\beta )\mu _0+\beta \mu _1$ .

Proposition 7.1. Let $r\leq 2 <\infty $ . For any $\gamma> 0$ , $\rho> 0$ , $\varepsilon>0$ , and $\widehat {\varepsilon }>0$ sufficiently small and L big enough, there exists $k_*> 0$ such that for all $k \geq k_*$ , there exist:

• • a set $A_k\subset M\times N$ such that $\vartheta _k(A_k)>0$ ; and

• • an integer $N_k \in \mathbb {N}$ with the following property.

For every $(x_k,y_k) \in A_k$ , the set $\widetilde {W}_k(x_k, y_k)$ and the measure $(\vartheta _{k,x_k})_{y_k}^{\eta ^k_{x_k}}$ are well defined. Moreover, for any $C^{\infty }$ regular curve $\sigma _k: [0,1] \to \widetilde {W}_k^u(x_k, y_k)$ satisfying $\|\sigma _k\|_{C^{\infty }} < (\varepsilon , \widehat {\varepsilon })$ , any measurable set $J_k \subset [0,1]$ for which

$$ \begin{align*} (\vartheta_{k,x_k})_{y_k}^{\eta^k_{x_k}}(\sigma_k(J_k) \cap A_k)> 0, \end{align*} $$

and for all $n \geq N_k$ , there exists a family of reparameterizations $\mathcal {R}_n^k$ of $\sigma _k$ over $J_k$ which is $(C^{r}, F_k, L, \varepsilon , \widehat {\varepsilon })$ -admissible up to time n and whose cardinality satisfies

$$ \begin{align*} |\mathcal{R}_n^k| \leq \exp \bigg[n\bigg(\beta h_{\mu_1}(F \mid \pi) + \frac{\lambda(\widehat{F})}{r-1}+ O(\rho) + O(\gamma) \bigg) \bigg]. \end{align*} $$

For k big enough, let $A_k$ be given by Proposition 7.1. Then, there exists $(x_k, y_k)\in A_k$ such that:

• • $(x_k,y_k)$ satisfies Theorem 6.6;

• • $y_k$ belongs to the support of the restriction of $(\vartheta _{k,x_k})_{y_k}^{\eta ^k_{x_k}}$ to $A_k$ .

Let $\sigma _k\colon [0,1]\to \widetilde {W}_k(x_k, y_k)$ be a $C^{\infty }$ regular curve that parameterizes a neighborhood of $y_k$ and has $C^{\infty }$ size smaller than $(\varepsilon , \widehat {\varepsilon })$ . Considering $J_k=\sigma _k^{-1}(A_k)$ , and applying Theorem 6.6 and Proposition 7.1, we conclude

$$ \begin{align*} h_{\vartheta_k}(F_k \mid \pi)\leq \beta h_{\mu_1}(F \mid \pi) + \frac{\lambda(\widehat{F})}{r-1}+O(\rho) + O(\gamma). \end{align*} $$

Passing to the limits $k\to \infty $ , and then $\gamma \to 0$ and $\rho \to 0$ , we conclude

$$ \begin{align*} \lim_{k\to \infty} h_{\vartheta_k}(F_k \mid \pi)\leq \beta h_{\mu_1}(F \mid \pi) + \frac{\lambda(\widehat{F})}{r-1}. \end{align*} $$

Observe that since the map F acts $C^{\infty }$ on the fibers, the inequality holds for arbitrary large r. Therefore, the second term goes to zero and the first part of the theorem follows.

Now, let us assume that $\lambda _2(F,x,y)\leq 0$ for $\mu $ -a.e. $(x,y)$ . As $\beta>0$ , $\mu _1$ -a.e. $(x,y)$ has $\lambda _2(F,x,y)\leq 0$ , so by Proposition 4.4,

$$ \begin{align*} \begin{aligned} \lim _{k \rightarrow \infty} \lambda_1^c(F_k, \vartheta_k)=\beta\, \lambda_1^c(F, \mu_1). \end{aligned} \end{align*} $$