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On the continuity of the pressure for monotonic mod one transformations. (English) Zbl 1034.37021
Consider an increasing continuous map \(f:[0,1]\rightarrow \mathbb{R}\), and define \(T_fx=f(x)\pmod 1\) so that \(T_f\) is a piecewise monotone piecewise continuous map of the interval. Another such map \(\widetilde T\) is \(\varepsilon \)-close to \(T_f\) if there is an increasing continuous map \(\widetilde f:[0,1]\rightarrow \mathbb{R}\) such that \(\widetilde Tx=\widetilde f(x)\pmod 1\) and \(\| \widetilde f - f\| _\infty <\varepsilon \). The author investigates the influence of small perturbations of \(T_f\) on the topological pressure \(p(T_f,g)\) and the topological entropy \(h_{\text{top}}(T_f)\). He shows that the topological pressure is lower semi-continuous, gives upper bounds for the jump up, and provides conditions sufficient for the continuity of the maximal measure. Next he proves that the topological pressure is upper semi-continuous for any continuous \(g:[0,1]\rightarrow \mathbb{R}\) if and only if either 0 or 1 is not periodic. Finally, he shows that \(h_{\text{top}}\) is continuous at \(T_f\) if \(h_{\text{top}}(T_f)>0\).

37E05 Dynamical systems involving maps of the interval
37B40 Topological entropy
54H20 Topological dynamics (MSC2010)
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