Entropy along expanding foliations
type: Item
instance of: scholarly article
title: Entropy along expanding foliations
zbMATH Open document ID: 1481.37031
author: Jiagang Yang
published in: Advances in Mathematics
publication date: 2021-08-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1601.05504
review text: 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 …
Mathematics Subject Classification ID: 37D30
Mathematics Subject Classification ID: 37D10
Mathematics Subject Classification ID: 37C05
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 28D20
zbMATH DE Number: 7384212
zbMATH Keywords: partially hyperbolic diffeomorphism
zbMATH Keywords: partial entropy
zbMATH Keywords: expanding foliation
zbMATH Keywords: Gibbs state
reviewed by: El Houcein El Abdalaoui
MaRDI profile type: Publication
OpenAlex ID: W3192983666
arXiv ID: 1601.05504
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DOI: 10.1016/J.AIM.2021.107893
description: scientific article