Entropy along expanding foliations
Let \(f\) be a diffeomorphism on a manifold \(M\) and \(\mu\) be an invariant probability measure of \(f\). The continuous foliation \(\mathcal{F}\) with smooth leaves is expanding for \(f\) if it is invariant under the iteration of \(f\) and the derivative \(Df\) restricted to the tangent bundle of \(\mathcal{F}\) is uniformly expanding. Let \(\mathcal{F}\) be a foliation of \(M\) with dimension \(l\), that is, every leaf is a \(l\)-dimensional smooth immersed submanifold. An \(\mathcal{F}\)-foliation box is the image \(B\) of a topological embedding \(\Phi\;:\; \mathcal{D}^{d-l} \times \mathcal{D}^l \longrightarrow M\), where \(\mathcal{D}^l \) is the \(l\)-dimensional open unit disk of \(\mathbb{R}^l\), such that every plaque \(P_x = \Phi(\{x\} \times \mathcal{D}^l )\) is contained in a leaf of \(\mathcal{F}\), and every \(\Phi(x,.) \;:\; y \in\mathcal{D}^l \mapsto \Phi(x,y) \in M\) is a \(C^1\) embedding depending continuously on \(x\) in the \(C^1\) topology. We put \(D = \Phi( \mathcal{D}^{d-l} \times\{0\})\), and denote this foliation box by \((B, \Phi, D)\). We say that the sequence of \(l\)-dimensional foliations \(\mathcal{F}_n\) converge to \(\mathcal{F}\) if: (1) For each \(n\) there exists a finite cover of \(M\) by \(\mathcal{F}_n\)-foliation boxes \(\{(B_i^n , \Phi_i^n , D_i) \}_{i=1}^k\); (2) For each \(1 \leq i \leq k\) the topological embeddings \(\Phi_i^n \;:\; \mathcal{D}^{d-l} \times \mathcal{D}^l \longrightarrow M\) converge uniformly to \(\Phi_i\) in the \(C^0\) topology; (3) For every \(x \in D_i, 1 \leq i \leq k,\) \(\Phi^n(x,\cdot) \;:\; y \in\mathcal{D}^l \mapsto \Phi^n(x,y) \in M\) is a \(C^1\) embedding which converges to \(\Phi(x,.)\) in the \(C^1\) topology as \(n \longrightarrow +\infty\). A measurable partition \(\xi\) of \(M\) is \(\mu\)-subordinate to the \(\mathcal{F}\)-foliation if for \(\mu-a.e. x\), we have: (1) \(\xi(x) \subset \mathcal{F}(x)\) and \(\xi(x)\) has uniformly small diameter inside \(\mathcal{F}(x)\) (\(\xi\) the element of \(\xi\) which contains \(x\)); (2) \(\xi(x)\) contains an open neighborhood of \(x\) inside the leaf \(\mathcal{F}(x)\); (3) \(\xi\) is an increasing partition, that is, \(f\xi \prec \xi\). According to Ledrappier-Young's Lemma [\textit{F. Ledrappier} and \textit{L. S. Young}, Ann. Math. (2) 122, 509--539 (1985; Zbl 0605.58028); \textit{F. Ledrappier} and \textit{L. S. Young}, Ann. Math. (2) 122, 540--574 (1985; Zbl 1371.37012)], for any expanding foliation, its entropy \(h_\mu(f,\xi)\) with respect to any measurable partition \(\xi\) \(\mu\)-subordinate to \(\mathcal{F}(x)\) is independent from \(\xi\). We thus define the partial \(\mu\)-entropy \(h_\mu (f,\mathcal{F})\) of the expanding foliation \(\mathcal{F}\) by putting \(h_\mu (f,\mathcal{F})=h_\mu (f, \xi)\) for any \(\mu\)-subordinate partition. We recall that the diffeomorphism \(f\) is said to be partially hyperbolic if there exists a decomposition \(TM = E_s \oplus E_c \oplus E_u\) of the tangent bundle \(TM\) into three continuous invariant subbundles \(E_x^s\) and \(E_x^c\) and \(E_x^u\), such that \(Df|_{E s}\) is uniform contraction, \(Df|_{E_u}\) is uniform expansion and \(Df|_{E_c}\) lies in between: \[\frac{\big\|Df(x)v^s\big\|}{\big\|Df(x)v^c\big\|} \leq \frac12 ~~\textrm{and}~~ \frac{\big\|Df(x)v^c\big\|}{\big\|Df(x)v^u\big\|} \leq \frac12,\] for any unit vectors \(v^s \in E_x^s , v^c \in E_x^c , v^u \in E_x^u\) and any \(x \in M\). The stable subbundle \(E_s\) and the unstable subbundle \(E_u\) are uniquely integrable, that is, there are unique foliations: the stable foliation \(\mathcal{F}_s\) and the unstable foliation \(\mathcal{F}_s\), whose leaves are smooth immersed submanifolds of \(M\) tangent to \(E_s\) and \(E_u\), respectively, at every point. The unstable and stable foliations are expanding foliations for the partially hyperbolic diffeomorphism and its inverse respectively. We recall also that the Gibbs \(u\)-state is any invariant probability measure whose conditional probabilities along strong unstable leaves \(\mathcal{F}^u\) are absolutely continuous with respect to the Lebesgue measure on the leaves. In this setting, the author proves that the partial entropy is semi-continuous, that is, \[\limsup h_{\mu_n}(f_n,\mathcal{F}_n) \leq h_\mu(f,\mathcal{F}).\] (This is exactly Theorem A of the paper with a correction of a misprint). As a consequence, he establishes that the space of the Gibbs \(u\)-states varies upper semi-continuously among the \(C^{1+}\) partially hyperbolic diffeomorphisms in the \(C^1\) topology. This yields that the sets of partially hyperbolic diffeomorphisms with mostly contracting center or mostly expanding center are \(C^1\) open.
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