2.1. Lyapunov exponents of partially hyperbolic diffeomorphisms

In our first main theorem, we prove that linear Anosov diffeomorphisms maximize (respectively minimize) the sum of unstable (respectively stable) Lyapunov exponents, in any homotopy path totally inside partially hyperbolic diffeomorphisms.

The notion of metric entropy appears naturally when dealing with Lyapunov exponents. For a partially hyperbolic diffeomorphism, it is possible to define entropy of an invariant measure along unstable foliation and there is variational principle results for such entropy. A u-maximal entropy measure is a measure which attains the supremum of unstable entropy among all invariant measures. See §3.2.1 and references for more details.

Theorem A. Let $f\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ be a $C^2$ volume-preserving DA diffeomorphism in $Ph_{d_s, d_u}$ with linearization $A\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ such that f and A are homotopic by a path fully contained in $Ph_{d_s, d_u}({\mathbb {T}}^d)$ , then

$$ \begin{align*} \sum_{i = 1}^{d_u} \unicode{x3bb}^u_i(f, x) \leq \sum_{i = 1}^{d_u} \unicode{x3bb}^u_i(A) \,\,\,\, \text{and} \,\,\,\, \sum_{i = 1}^{d_s} \unicode{x3bb}^s_i(f, x) \geq \sum_{i = 1}^{d_s} \unicode{x3bb}^s_i(A), \end{align*} $$

for m-almost every (a.e.) $x \in {\mathbb {T}}^d$ and, the first (respectively second) inequality is strict unless m is a measure of u-maximal entropy (respectively u-maximal for $f^{-1}$ ).

This is related to the result of [Reference Micena and Tahzibi16]: let $f\colon \mathbb {T}^3 \rightarrow \mathbb {T}^3$ be a $C^2$ conservative derived from Anosov partially hyperbolic diffeomorphism with linearization A. Then, $\unicode{x3bb} ^u(x) \leq \unicode{x3bb} ^u(A)$ for almost every $x.$ We recall that the authors used quasi-isometric property of unstable foliation for the three-dimensional derived from Anosov diffeomorphisms. They do not assume that f is in the same connected component of A.

Theorem B. Let $f\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ be a $C^2$ volume-preserving DA diffeomorphism in $Ph_{d_s, d_u}$ with linearization $A\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ belonging to $Ph_{d_s, d_u}.$ Suppose that for $\sigma \in \{s, u\}$ , there is a $(d - d_{\sigma })$ -dimensional subspace $P_{\sigma } \subset \mathbb {R}^d, $ such that $\angle (E^{\sigma }_f(x), P_{\sigma })> \alpha > 0$ for all $x \in \mathbb {R}^d$ ( $E^{\sigma }_f$ stands for the lift of the bundles), then

$$ \begin{align*} \sum_{i = 1}^{d_u} \unicode{x3bb}^u_i(f, x) \leq \sum_{i = 1}^{d_u} \unicode{x3bb}^u_i(A) \quad \text{and} \quad \sum_{i = 1}^{d_s} \unicode{x3bb}^s_i(f, x) \geq \sum_{i = 1}^{d_s} \unicode{x3bb}^s_i(A), \end{align*} $$

for m-a.e. $x \in \mathbb {T}^d.$

Observe that the assumption on the existence of $P_{\sigma }$ in the above theorem is a mild condition and is satisfied if the stable and unstable bundles of f do not vary too much. Indeed, we require that $\{ E^{\sigma }_f(x), x \in \mathbb {R}^d\}$ is not dense in the Grassmannian of $d_{\sigma }$ -dimensional subspaces.

2.2. Lyapunov exponents of non-uniformly Anosov systems

In this section, we deal with dynamics that are not necessarily partially hyperbolic. However, they enjoy some dominated splitting property: Oseledets’ splitting (stable/unstable) is dominated.

In the next theorem, we compare Lyapunov exponents of a non-uniformly Anosov diffeomorphism with those of linear Anosov automorphism if they are homotopic. Observe that by definition, the number of negative (positive) Lyapunov exponents of a non-uniformly Anosov diffeomorphism is constant almost everywhere and coincides with the dimension of the bundles in the dominated splitting. However, the Lyapunov exponents may depend on the orbit, as we do not assume ergodicity. In fact, it is interesting to know whether, in general, all topologically transitive non-uniformly Anosov diffeomorphisms are ergodic or not. We conjecture that all topologically transitive non-uniformly Anosov diffeomorphisms are ergodic.

Let $A\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ be a linear Anosov diffeomorphism such that $T \mathbb {T}^d = E^s_{A} \oplus E^u_{A}.$ Denote by $d_s =\mathrm {dim} (E^s_A)$ and $d_u =\mathrm {dim}(E^u_A).$ After changing to an equivalent Riemannian norm, if necessary, there are $0<\unicode{x3bb} <1<\gamma $ such that $\|A|_{E^s}\| \leq \unicode{x3bb} $ , $m(A|_{E^u}) \geq \gamma $ and $E^s_A$ is orthogonal to $E^u_A.$ Indeed, to use fewer constants in the proofs, we assume that $E^u_i$ and $E^u_j, i \neq j$ are orthogonal, where $E^u_i$ terms are generalized eigenspaces of A which coincide with the Oseledets decomposition. Recall that for a linear transformation T, $m(T):= \|T^{-1}\|^{-1}.$ We refer to $\gamma , \unicode{x3bb} $ as rates of hyperbolicity of A. Observe that Lyapunov exponents of any diffeomorphism are independent of the choice of the equivalent norm.

Let us comment on the hypotheses: the first and second one ask for some compatibility of invariant bundles and the third one asks that any possible expansion in the dominated bundle E is less than the expansion rate of A and similarly any possible contraction in the dominating bundle F is weaker than the contraction rate of $A.$

Bonatti and Viana [Reference Bonatti and Viana6] constructed the first examples of robustly transitive diffeomorphisms that are not partially hyperbolic, which was later generalized in higher dimensions by [Reference Tahzibi23]. Those classes of examples satisfy the hypotheses of the above theorem.

In the above theorem, the integrability of invariant subbundles E and F is proved in [Reference Buzzi and Fisher8].

2.3. Regularity of foliations

Any partially hyperbolic diffeomorphism admits invariant stable and unstable foliations. As the proofs of our main results are based on the regularity of such foliations, we need to recall some basic definitions and results in this subsection. Consider ${\mathcal {F}}$ a foliation in M, B a ${\mathcal {F}}$ -foliated box and m the Lebesgue measure in M. Denote by $\mathrm {Vol}_{L_x}$ the Lebesgue measure of leaf $L_x$ and $m_{L_x}$ the disintegration of the Lebesgue measure m along the leaf $L_x$ . The disintegrated measures are in fact a projective class of measures. However, whenever we fix a compact foliated box B, then we can use the Rohlin disintegration theorem for the normalized restriction of m on B to get probability conditional measures. So in what follows, after fixing a foliated box B by $m_{L_x}$ and $\mathrm {Vol}_{L_x}$ , we understand probability measures whose support is inside the plaque of $L_x$ which contains $x \in B$ and is inside B.

We also mention the result of Ya. Pesin for non-uniformly hyperbolic systems, see [Reference Barreira and Pesin2, Theorem 4.3.1], which shows absolute continuity of local Pesin laminations.

3.1. Proof of Theorem B

We use Theorem 3.1 and the following proposition to conclude the proof of Theorem B.

3.2. Proof of Theorem A

From [Reference Roldan19, Proposição 3.1.2], we have the following proposition.

By the proposition above, f has a $d_u$ closed and non-degenerate form on $W^u$ , using the inverse $f^{-1}$ , we get a $d_s$ closed and non-degenerate form on $W^s$ . By Theorem 3.1, we conclude the first part of Theorem A. In the next subsection, we complete the proof.

3.2.1. Maximizing measures

In this section, we prove the last part of Theorem A. We write all proofs for the unstable bundle.

First, we recall the definition of topological and metric entropy along an expanding foliation. In [Reference Hu, Hua and Wu13], the authors define the notion of topological and metric entropy along unstable foliation and prove the variational principle.

Let $f\colon M\rightarrow M$ be a $C^1$ -partially hyperbolic diffeomorphism and $\mu $ is an f-invariant probability measure. For a partition $\alpha $ of M, denote $\alpha _0^{n-1}=\bigvee _{i=0}^{n-1}f^{-i}\alpha $ and by $\alpha (x)$ the element of $\alpha $ containing x. Given $\varepsilon>0$ , let $\mathcal {P}=\mathcal {P}_{\varepsilon }$ denote the set of finite measurable partitions of M whose elements have diameters smaller than or equal to $\varepsilon $ . For each $\alpha \in \mathcal {P}$ , we define a partition $\eta $ such that $\eta (x)=\alpha (x)\cap W^u_\mathrm{loc}(x)$ for each $x \in M$ , where $W^u_\mathrm{loc}(x)$ denotes the local unstable manifold at x whose size is greater than the diameter $\varepsilon $ of $\alpha $ . Let $\mathcal {P}^u=\mathcal {P}^u_{\varepsilon }$ denote the set of partitions $\eta $ obtained in this way. We define the conditional entropy of $\alpha $ given $\eta $ with respect to $\mu $ by

$$ \begin{align*} H_{\mu}(\alpha|\eta):=-\int_{M}\log\mu^{\eta}_{x}(\alpha(x))\,d\mu(x), \end{align*} $$

where $\mu ^{\eta }_{x}$ refer to the disintegration of $\mu $ along $\eta $ .

Definition 3.7. The conditional entropy of f with respect to a measurable partition $\alpha $ given $\eta \in \mathcal {P}^u$ is defined as

$$ \begin{align*} h_{\mu}(f,\alpha|\eta)=\limsup_{n\rightarrow \infty}\frac{1}{n}H_{\mu}(\alpha_{0}^{n-1}|\eta). \end{align*} $$

The conditional entropy of f given $\eta \in \mathcal {P}^u$ is defined as

$$ \begin{align*} h_{\mu}(f|\eta)=\sup_{\alpha\in\mathcal{P}}h_{\mu}(f,\alpha|\eta), \end{align*} $$

and the unstable metric entropy of f is defined as

$$ \begin{align*} h_{\mu}^u(f)=\sup_{\eta\in\mathcal{P}^u}h_{\mu}(f|\eta). \end{align*} $$

Now, we go to define the unstable topological entropy.

We denote by $\rho ^u$ the metric induced by the Riemannian structure on the unstable manifold and let $\rho ^u_n(x,y)=\max _{0\leq j\leq n-1}\rho ^u(f^j(x),f^j(y))$ . Let $W^u(x,\delta )$ be the open ball inside $W^u(x)$ centred at x of radius $\delta $ with respect to the metric $\rho ^u$ . Let $N^u(f,\epsilon ,n,x,\delta )$ be the maximal number of points in $\overline {W^u(x,\delta )}$ with pairwise $\rho ^u_ n$ -distances at least $\epsilon $ . We call such a set an $(n,\epsilon ) u$ -separated set of $\overline {W^u(x,\delta )}$ .

Definition 3.8. The unstable topological entropy of f on M is defined by

$$ \begin{align*} h^{u}_{\mathrm{top}}(f)=\lim_{\delta\to 0}\sup_{x\in M}h^u_{\mathrm{top}}(f,\overline{W^u(x,\delta)}), \end{align*} $$

where

$$ \begin{align*} h^u_{\mathrm{top}}(f,\overline{W^u(x,\delta)})=\lim_{\epsilon\to 0}\limsup_{n\to\infty}\frac{1}{n}\log N^u(f,\epsilon,n,x,\delta). \end{align*} $$

Let $\mathcal {M}_f(M)$ and $\mathcal {M}_f^e(M)$ denote the set of all f-invariant and ergodic probability measures on M, respectively.

Theorem 3.9. [Reference Hu, Hua and Wu13]

Let $f\colon M\rightarrow M$ be a $C^1$ -partially hyperbolic diffeomorphism. Then,

$$ \begin{align*} h^u_{\mathrm{top}}(f)=\sup\{h^u_{\mu}(f): \mu\in\mathcal{M}_f(M)\}. \end{align*} $$

Moreover,

$$ \begin{align*} h^u_{\mathrm{top}}(f)=\sup\{h^u_{\nu}(f): \nu\in\mathcal{M}_f^e(M)\}. \end{align*} $$

Furthermore, the authors proved that unstable topological entropy coincides with unstable volume growth.

Theorem 3.10. [Reference Hu, Hua and Wu13]

Unstable topological entropy coincides with unstable volume growth, i.e. $h^u_{\mathrm {top}}(f)=\chi _u(f)$ .

As in the hypotheses of Theorem A, the diffeomorphisms f and A are homotopic, so using Theorem 3.2, we have

$$ \begin{align*} h^u_{\mathrm{top}}(f)=\chi_u(f)=\log sp(f_{\ast},u)=\log sp(A_{\ast},u)=\chi_u(A)=h^u_{\mathrm{top}}(A). \end{align*} $$

Thus,

$$ \begin{align*} \sum_{i = 1}^{d_u} \unicode{x3bb}^u_i(f, x)=h^u_m(f)\leq h^u_{\mathrm{top}}(f)=h^u_{\mathrm{top}}(A)= \sum_{i = 1}^{d_u} \unicode{x3bb}^u_i(A). \end{align*} $$

Then we conclude that $h^u_m(f)= h^u_{\mathrm {top}}(f)$ .

3.3. Proof of Theorem C

We remember that $A\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ is a linear Anosov diffeomorphism and $0<\unicode{x3bb} <1<\gamma $ are rates for hyperbolicity, $d_s =\mathrm {dim} (E^s_A)$ and ${d_u =\mathrm {dim} (E^u_A).}$ Let $W^s_A, W^u_A$ be the stable and unstable foliation of A and by $\widetilde {W}^{s}_A$ , $\widetilde {W}^{u}_A$ , we denote their lifts to the universal cover $\mathbb {R}^d$ . These foliations are stable and unstable foliations of $\widetilde {A}$ which is a lift of A. We use ‘ $\sim $ ’ for objects in the universal cover. The norm and distance on $\mathbb {R}^d$ are the lift of adapted norm and corresponding distance where the hyperbolicity conditions of A are satisfied, see §2.2 before the announcement of Theorem C.

Similar to [Reference Hammerlindl12, Proposition 2.5 and Corollary 2.6], we show the following proposition.

Corollary 3.12. Fixing $x \in \mathbb {R}^n$ , if $ \|x-y\|\rightarrow \infty $ and $y\in \mathcal {E}(x)$ , then $({x-y})/{\|x-y\|}\rightarrow \widetilde {E}^{s}_A(x)$ uniformly. More precisely, for $\varepsilon>0$ , there is $M>0$ such that if $x\in {\mathbb {R}}^d$ , $y\in \mathcal {E}(x)$ and $\|x-y\|>M$ , then

$$ \begin{align*} \|\pi_A^{u}(x-y)\|<\varepsilon\|\pi^{s}_A(x-y)\|, \end{align*} $$

where $\pi _A^{s}$ is the orthogonal projection on the subspace $E^s_A$ along $E^u_A$ , and $\pi _A^{u}$ is the projection on the subspace $E^{u}_A$ along $E^s_A.$

Analogous statements hold for $\mathcal {F}$ . The following proposition is a topological remark (see [Reference Hammerlindl12]) and comes from the fact that f and A are homotopic and A is not singular.

Proposition 3.13. Let $f\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ be a homeomorphism with linearization A, then for each $k \in {\mathbb {Z}}$ and $C>1$ , there is an $M>0$ such that for all $x,y\in {\mathbb {R}}^d$ ,

$$ \begin{align*} \|x-y\|>M \Rightarrow \frac{1}{C}<\frac{\|\tilde{f}^k(x)-\tilde{f}^k(y)\|}{\|\tilde{A}^k(x)-\tilde{A}^k(y)\|}<C. \end{align*} $$

More generally, for each $k\in {\mathbb {Z}}$ , $C>1$ and any linear projection $\pi \colon {\mathbb {R}}^d\rightarrow {\mathbb {R}}^d$ , with A-invariant image, there is $M> 0$ such that for $x,y\in {\mathbb {R}}^d$ , with $\|x-y\|>M$ ,

$$ \begin{align*} \frac{1}{C}<\frac{\|\pi(\tilde{f}^k(x)-\tilde{f}^k(y))\|}{\|\pi(\tilde{A}^k(x)-\tilde{A}^k(y))\|}<C. \end{align*} $$

Proof of Proposition 3.11

To prove the first claim (the second one is similar), it is enough to show that $\|\pi ^u_A(x-y)\|$ is uniformly bounded for all $y\in \mathcal {E}(x)$ . Let $C>1$ close enough to $1$ such that $C \hat {\gamma } < \gamma $ and $1 < {\gamma }/{C}.$ Put $k=1$ and C chosen as above in Proposition 3.13, and get appropriate M. By contradiction, suppose that $\|\pi ^u_A(x-y)\|$ is not bounded. So, there is $y\in \mathcal {E}(x)$ with $\|\pi ^u_A(x-y)\|>M$ and consequently,

$$ \begin{align*} \|\pi^u_A(\tilde{f}(x)-\tilde{f}(y))\|> \frac{1}{C}\|\pi^u_A(\tilde{A}(x-y))\| = \frac{1}{C} \|(\tilde{A}( \pi^u_A(x-y))\| \geq \frac{\gamma}{C} \|\pi^u_A(x-y) \| > M, \end{align*} $$

which implies that we can use induction and for any $n \geq 1$ , obtain

$$ \begin{align*} \|\pi^u_A(\tilde{f}^{n}(x)-\tilde{f}^{n}(y))\|>\frac{\gamma^{n}}{C^n} M. \end{align*} $$

Finally, there is a constant $\eta> 0$ such that

(3.3) $$ \begin{align} \|\tilde{f}^{n}(x)-\tilde{f}^{n}(y)\|> \eta \frac{\gamma^{n}}{C^n} M. \end{align} $$

Now consider a smooth curve $\alpha \colon [a,b]\rightarrow \mathcal {E}(x)$ with $\alpha (a)=x$ and $\alpha (b)=y$ whose length is $d_{\mathcal {E}}(x,y)$ . Then,

$$ \begin{align*} \|\tilde{f}^{n}(x)-\tilde{f}^{n}(y)\|&\leq d_{\mathcal{E}}(\tilde{f}^{n}(x),\tilde{f}^{n}(y))\leq l(\tilde{f}^{n}(\alpha(t))) =\int_a^b\bigg\|\frac{d}{dt}(\tilde{f}^{n}(\alpha(t))\bigg\|\,dt\\ &\leq \int_a^b\|D\tilde{f}^{n}|_{W^{cs}}\|\cdot\|(\alpha'(t))\|\,dt< \int_a^b\hat{\gamma}^{n} \|\alpha'(t)\|\,dt, \end{align*} $$

implies that

(3.4) $$ \begin{align} \|\tilde{f}^{n}(x)-\tilde{f}^{n}(y)\|<\hat{\gamma}^{n}\,d_{\mathcal{E}}(x,y). \end{align} $$

Equations (3.3) and (3.4) and ${\gamma }/{C}> \hat {\gamma }$ give us a contradiction when n is large enough.

Let $\mathcal {R}$ be the regular set in Pesin sense such that $m(\mathcal {R}) = 1$ and $\mathcal {R} = \bigcup _{l=1}^{\infty } \mathcal {R}_l$ is the union of Pesin’s block where $m(\mathcal {R}_l) \rightarrow 1, \mathcal {R}_l \subset \mathcal {R}_{l+1}.$ Moreover, the size of Pesin stable manifolds of points in $\mathcal {R}_l$ is larger than some positive constant $r_l.$ In general, $r_l \rightarrow 0$ when $l \rightarrow \infty .$ We use an absolute continuity result due to Pesin [Reference Barreira and Pesin2, Theorem 4.3.1] which essentially shows the absolute continuity of Pesin’s stable laminations in each block.

Fix $x\in \mathbb {R}^n$ and denote by $U_r\subset \widetilde {W}^u_A (x)$ the ball of radius r with centre x.

Proposition 3.15. Let $f\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ as in Theorem C, then given $\varepsilon>0$ , there is $r_0>0$ and a constant $C_0>0$ such that

for all $n>0$ and $r \geq r_0$ , where $\pi _z$ is the orthogonal projection from $\mathcal {F}(z)$ to $\widetilde {W}_A^u(z)$ (along $\widetilde {E}_A^s$ ) for any $z \in \mathbb {R}^n$ .

Proof of Proposition 3.15

We prove the following claims.

Claim 1. For $z\in {\mathbb {R}}^d$ , the orthogonal projection $\pi _z\colon \mathcal {F}(z)\rightarrow \widetilde {W}^u_A(z)$ is a uniform bi-Lipschitz diffeomorphism.

Proof of Claim 1

By item (2) of Theorem C and continuity of E and F, there is $\alpha>0$ such that the angle $\angle (E,E^u_A)>\alpha $ and $\angle (F, E^s_A)>\alpha .$ So $\widetilde {W}_A^s$ is uniformly transversal to the foliation $\mathcal {F}$ and there is $\beta>0$ such that

(3.5) $$ \begin{align} \|d\pi_z(x)(v)\|\geq\beta\|v\| \end{align} $$

for any $x\in \mathcal {F}(z)$ and $v\in T_x\mathcal {F}(z)$ . This implies that $\mathrm {Jac} \pi _z(x)\neq 0$ . By the inverse function theorem, for each $x\in \mathcal {F}(z)$ , there is a ball $B(x, \delta )\subset \mathcal {F}(z)$ such that $\pi _z|_{B(x, \delta )}$ is a diffeomorphism. In fact, from the proof of the inverse function theorem and equation (3.5), $\delta $ can be taken independent of x. Again, by equation (3.5), there is $\varepsilon>0$ , independent of x such that $B(\pi _z(x), \varepsilon )\subset \pi _z(B(x, \delta ))$ . To prove that $\pi _z$ is surjective, we show that $\pi _z(\mathcal {F}(z))$ is an open and closed subset. As $\pi _z$ is a local homeomorphism, then $\pi _z(\mathcal {F}(z))$ is open. To verify that it is also closed, let $y_n\in \pi _z(\mathcal {F}(z))$ be a sequence converging to y, and hence there is $n_0$ large enough such that $y\in B(\varepsilon ,y_{n_0})$ and, therefore, $y\in {\pi _z}(\mathcal {F}(z))$ . So, $\pi _z$ is surjective. Moreover, $\pi _z$ is a covering map and injectivity follows from the fact that any covering map from a path-connected space to a simply connected space is a homeomorphism.

Let us prove that $\pi _z$ is bi-Lipschitz. In fact,

$$ \begin{align*}\|\pi_z(x)-\pi_z(y)\|\leq\|x-y\|\leq d_{\mathcal{F}}(x,y).\end{align*} $$

Let us show that $\pi _z^{-1}$ is also Lipschitz. This is immediate from equation (3.5). Indeed, let $[x,y]$ be the line segment in $\widetilde {W}^u_A (z)$ connecting x to $y,$ so the set $\pi _z^{-1}([x,y])=\Gamma $ is a smooth curve connecting the points $\pi _z^{-1}(x)$ and $\pi _z^{-1}(y)$ in $\mathcal {F}(z)$ . So,

$$ \begin{align*} d_{\mathcal{F}}(\pi_z^{-1}(x),\pi_z^{-1}(y))\leq \mathrm{length}\,(\Gamma)=\int_{[x,y]}|d\pi_z^{-1}(t)|\,dt\leq \frac{1}{\beta}\|x-y\|.\\[-42pt] \end{align*} $$

Claim 2. Given $\varepsilon>0$ , there is $r_0> 0$ such that $\tilde {f}^n(\pi _x^{-1}(U_r)) \subset \pi _{\tilde {f}^n(x)}^{-1}(U_{r, n})$ for every $r \geq r_0$ and $n \geq 1,$ where $U_{r, n} \subset \widetilde {W}_A^u (\tilde {f}^n(x))$ is a $d_u$ -dimensional ellipsoid with volume bounded above by $(1+\epsilon )^{nd_u} \mathrm {Vol}_{\tilde {W}_A^u}(A^n(U_r)).$

Proof. Let $ r \geq r_0 := ({2R+2K})/{\varepsilon m(A|_{E^u})}$ , where R comes from Proposition 3.11 and $\|\tilde {f}- \tilde {A}\|_{\infty } \leq K.$ We prove that

(3.6) $$ \begin{align} \tilde{f}(\pi_x^{-1}(U_r)) \subset \pi_{\tilde{f}(x)}^{-1}(U_{r, 1}), \end{align} $$

where $U_{r,1} \subset \widetilde {W}^u_A(\tilde {f}(x))$ is obtained from $\tilde {A}(U_r)$ by first applying a homothety of ratio $(1+\epsilon )$ centred at $\tilde {A}(x)$ and then translating by $\tilde {f}(x) - \tilde {A}(x)$ . Observe that $U_{r, 1}$ is an ellipsoid inside the affine $d_u$ dimensional subspace passing through $\tilde {f}(x).$ Take any y on the boundary of $U_r$ . Let $z:=\pi _{\tilde {f}(x)}(\tilde {f}(\pi _x^{-1}(y))) \in \widetilde {W}^u_A(\tilde {f}(x))$ . On the one hand, we have

(3.7) $$ \begin{align} \| z - (\tilde{A}(y) + \tilde{f}(x)- \tilde{A}(x)) \| \leq \| \tilde{f}(x)- \tilde{A}(x) \| + \| z - \tilde{A}(y) \| \leq K + \| z - \tilde{A}(y) \|. \end{align} $$

On the other hand,

(3.8) $$ \begin{align} \| z - \tilde{A}(y) \| &\leq \|z - \tilde{f}(\pi_x^{-1}(y))\| + \| \tilde{f}(\pi_x^{-1}(y))-\tilde{A}(\pi_x^{-1}(y)) \| \nonumber\\ &\quad+ \| \tilde{A}(\pi_x^{-1}(y)) - \tilde{A}(y) \| \nonumber\\ & \leq 2R+K. \end{align} $$

In the above inequalities, we have used Proposition 3.11 two times to get

$$ \begin{align*}\|z - \tilde{f}(\pi_x^{-1}(y)\| \leq R, \end{align*} $$

$$ \begin{align*}\| \tilde{A}(\pi_x^{-1}(y)) - \tilde{A}(y) \| \leq \|y - \pi_x^{-1}(y)\| \leq R, \end{align*} $$

(observe that $y - \pi _x^{-1}(y)$ belongs to the stable subspace of A) and finally

$$ \begin{align*} \| \tilde{f}(\pi_x^{-1}(y)))-\tilde{A}(\pi_x^{-1}(y)) \| \leq \|\tilde{f}-\tilde{A}\|_{\infty} \leq K.\end{align*} $$

Now, putting the inequalities in equations (3.7) and (3.8) together, we get

$$ \begin{align*} \|z - (\tilde{A}(y) + \tilde{f}(x)- \tilde{A}(x))\| \leq 2K + 2R.\end{align*} $$

Observe that $\tilde {A}(y) + \tilde {f}(x)- \tilde {A}(x)$ is the translation of the $\tilde {A}(y)$ and belongs to $\widetilde {W}_A^u(\tilde {f}(x)).$ Indeed, $\tilde {A}(y)- \tilde {A}(x)$ is a vector which belongs to the unstable bundle of $\tilde {A}$ and we identify the unstable bundle at $\tilde {f}(x)$ with the corresponding affine subspace of $\mathbb {R}^n$ passing through $\tilde {f}(x)$ . Now, as the distance between $(1+\epsilon ) \tilde {A}(U_r) $ and $ \tilde {A}(U_r)$ is $ \epsilon r m(\tilde {A}|_{E^u}) \geq 2R + 2K$ (by the choice of r), we conclude that z belongs to the ellipsoid $U_{r, 1} :=(\tilde {f}(x)- \tilde {A}(x)) + (1+\epsilon ) \tilde {A}(U_r)$ which proves equation (3.6). Moreover, observe that $\mathrm {Vol}_{\widetilde {W}^u_A} (U_{r, 1}) =(1+\epsilon )^{d_u} \mathrm {Vol}_{\widetilde {W}^u_A}(\tilde {A}(U_r)) = (1+\epsilon )^{d_u} e^{\sum _{i=1}^{d_u} \unicode{x3bb} ^u_i(A)} \mathrm {Vol}_{\widetilde {W}^u_A} (U_r).$

Figure 1

Volume comparison.

Now we apply again $\tilde {f}$ and obtain

$$ \begin{align*}\tilde{f}^2(\pi_x^{-1}(U_r)) \subset \tilde{f} (\pi_{\tilde{f}(x)}^{-1}(U_{r,1})). \end{align*} $$

As the distance between $\tilde {A}(U_{r, 1})$ and $(1+\epsilon )\tilde {A}(U_{r, 1})$ is larger than $\varepsilon r m(\tilde {A}|_{E^u})$ , similarly as above, we obtain

$$ \begin{align*} \tilde{f} (\pi_{\tilde{f}(x)}^{-1}(U_{r,1})) \subset \pi_{\tilde{f}^2(x)}^{-1}(U_{r, 2}),\end{align*} $$

where $U_{r, 2}$ is a translation of $(1+\varepsilon )^2 \tilde {A}^2(U_r).$ In fact, inductively, we obtain

(3.9) $$ \begin{align} \tilde{f}^{n+1}(\pi_x^{-1}(U_r)) \subset \tilde{f} (\tilde{f}^{n}(\pi_x^{-1}(U_r))) \subset \tilde{f} (\pi_{\tilde{f}^n(x)}^{-1}(U_{r, n})) \subset \pi_{\tilde{f}^{n+1}(x)}^{-1}(U_{r, n+1}), \end{align} $$

where $U_{r, n+1}$ is an ellipsoid with volume less than $(1+\varepsilon )^{(n+1)d_u} e^{(n+1) \sum _{i=1}^{d_u} \unicode{x3bb} ^u_i(A)} \mathrm {Vol}_{\tilde {W}_A^u} (U_r).$ The last inclusion in equation (3.9) follows using the same arguments as above to prove equation (3.6) substituting the ball $U_r$ by the ellipsoid $U_{r, n}.$

By Claim 1, for all $x\in M$ , we have $|\mathrm {Jac}\, \pi ^{-1}_x|$ is uniformly bounded and, consequently, there is a constant $C_0>0$ such that

$$ \begin{align*} \mathrm{Vol}_{\mathcal{F}}(\tilde{f}^n\pi_x^{-1}(U_r))&\leq\mathrm{Vol}_{\mathcal{F}}(\pi_{\tilde{f}^n(x)}^{-1}(U_{r, n}) \leq C_0(1+ \varepsilon)^{nd_u} e^{n \sum_{i = 1}^{d_u} \unicode{x3bb}^u_i(A)}\mathrm{Vol}_{\tilde{W}_A^u}(U_r). \end{align*} $$

It concludes the proof of the Proposition 3.15 (see Figure 1).

Proof of Theorem C

Suppose by contradiction that there is a positive volume set $Z\subset {\mathbb {T}}^d$ such that $\sum _{i=1}^{d_u}\unicode{x3bb} _i^{cu}(f,x)>\sum _{i=1}^{d_u}\unicode{x3bb} _i^u(A)$ for any $x\in Z$ .

Let $P\colon {\mathbb {R}}^d\rightarrow {\mathbb {T}}^d$ be the covering map, $D\subset {\mathbb {R}}^d$ a fundamental domain and ${\widetilde {Z}=P^{-1}(Z)\cap D}$ . We have $\mathrm {Vol}(\widetilde {Z})>0$ .

For each $q\in {\mathbb {N}}\setminus \{0\}$ , we define the set

We have $\bigcup _{q=1}^{\infty }Z_q=\widetilde {Z}$ , and thus there is q such that $m(Z_q)>0$ . For each $x\in Z_q$ , it follows that

So there is $n_0$ such that for $n\geq n_0$ , we have

This implies that

For every $n>0$ , we define

There is $N>0$ with $\mathrm {Vol}(Z_{q,N})>0$ .

For each $x\in D$ , consider $B_x\subset {\mathbb {R}}^d$ a foliated box of $\mathcal {F}.$ By compactness, there is finite cover $\{B_{x_i}\}_{i=1}^{j}$ covering $\overline {D}.$ Since $W^{cu}_f$ is absolutely continuous and the covering map is smooth, then $\mathcal {F}$ is absolutely continuous, and thus there is some i and $p\in B_{x_i}$ such that $\mathrm {Vol}_{\mathcal {F}}(B_{x_i}\cap \mathcal {F}(p)\cap Z_{q,N})>0$ .

There is a set $\pi _p^{-1}(U_r)\subset \mathcal {F}(p)$ , where $\pi _p^{-1}(U_r)$ is as in Proposition 3.15 containing p and r is large enough such that $\mathrm {Vol}_{\mathcal {F}}(\pi _p^{-1}(U_r)\cap Z_{q,N})>0.$ Let $\alpha>0$ be such that $\mathrm {Vol}_{\mathcal {F}}(\pi _p^{-1}(U_r)\cap Z_{q,N})=\alpha \mathrm {Vol}_{\mathcal {F}}(\pi _p^{-1}(U_r)),$ and by Proposition 3.15, we have

(3.10)

However,

(3.11)

Equations (3.10) and (3.11) give us a contradiction when n is large enough, and thus prove Theorem C.