An analogue of the variational principle for group and pseudogroup actions (Q381139)

In this paper, the notion of a local measure entropy introduced by \textit{M. Brin} and \textit{A. Katok} [Lect. Notes Math. 1007, 30--38 (1983; Zbl 0533.58020)] for a single map is generalized to the case of finitely generated groups of homeomorphisms. The author applies the theory of dimensional type characteristics of a dynamical system elaborated by Pesin to obtain a relationship between the topological entropy of a pseudogroup and a group of homeomorphisms of a metric space, defined by \textit{E. Ghys} et al. [Acta Math. 160, No. 1--2, 105--142 (1988; Zbl 0666.57021)], and its local measure entropies. An analogue of the variational principle for group and pseudogroup actions, which allows us to study local dynamics of foliations, is obtained.NEWLINENEWLINELet \((X, d)\) be a compact metric space, and denote by \(\text{Homeo}(X)\) the family of all homeomorphisms of \(X\). For any finite set \(G_1 \subset \text{Homeo}(X)\), let \(G\) be the pseudogroup generated by \(G_1\). In such case, we say that \((G, G_1)\) is finitely generated. The main results of this paper are the following two theorems:NEWLINENEWLINETheorem 5.2. Let \((G, G_1)\) be a finitely generated group of homeomorphisms of a compact closed and oriented manifold \((M, d)\). Let \(E\) be a Borel subset of \(M\), \(s \in (0, \infty)\) and \(\mu_v\), the natural volume measure on \(M\). If \(h_{\mu_v}^G(x) \leq s\) for all \(x \in E\), then \(h_{\mathrm{top}}((G, G_1), E) \leq s\).NEWLINENEWLINETheorem 5.3. Let \((G, G_1)\) be a finitely generated group of homeomorphisms on a compact metric space \((X, d)\). Let \(E\) be a Borel subset of \(X\) and \(s \in (0, \infty)\). Denote by \(\mu\) a Borel probability measure on \(X\). If \(h_{\mu,G}(x) \geq s\) for all \(x \in E\) and \(\mu(E) > 0\), then \(h_{\mathrm{top}}((G, G_1), E) \geq s\).NEWLINENEWLINEThe results are a generalization of Theorem 1 of \textit{J.-H. Ma} and \textit{Z.-Y. Wen} [C. R., Math., Acad. Sci. Paris 346, No. 9--10, 503--507 (2008; Zbl 1138.37007)].