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Stability of the maximal measure for piecewise monotonic interval maps. (English) Zbl 0898.58015
The paper is devoted to the study of stability of the maximal measure for piecewise monotonic interval maps. Let \(T: X\to\mathbb{R}\) be a piecewise monotonic interval map, where \(X\) is a finite union of closed intervals. This means that there exists a finite partition \(\mathcal Z\) of \(X\) into pairwise disjoint open intervals \(Z\) with \(\cup _{Z\in {\mathcal Z}}{\overline Z}=X\) such that \(T| _Z\) is bounded, strictly monotone and continuous for all \( Z\in {\mathcal Z}\). The map \(T\) generates a dynamical system \((R(T),T)\), where \(R(T)=\cap _{n=0}^{\infty}\overline{T^{-n}X}\) is the set for which \(T^{n}\) is defined for all \( n\in\mathbb{N}\). A \(T\)-invariant Borel probability measure \(\mu\) concentrated on \(R(T)\) is called a maximal measure of \(T\) if \(h_{\mu}(R(T),T)=h_{\text{top}}(R(T),T)\), where \(h_{\mu}\) and \(h_{\text{top}}\) are measure-theoretic and topological entropies of \(T\), respectively. Assuming that \(T\) has a unique maximal measure, the influence of small perturbations of \(T\) on the maximal measure is investigated. It is shown that if \((R(T),T)\) has positive topological entropy, and if a certain stability condition is satisfied, then every piecewise monotonic map \(\widetilde T\) sufficiently close to \(T\) has a unique maximal measure \({\widetilde \mu}\) and the map \(\widetilde T \mapsto {\widetilde \mu}\) is continuous at \(T\). The main tool in studying the problem is the Markov diagram introduced by F. Hofbauer [Probab. Theory Relat. Fields 72, 359-386 (1986; Zbl 0591.60064)] that allows to calculate maximal measures.

MSC:
37E99 Low-dimensional dynamical systems
37A99 Ergodic theory
54C70 Entropy in general topology
28D20 Entropy and other invariants
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