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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 $$h_{\text{top}}(T_f)>0$$.

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