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Continuity of the entropy for monotonic mod one transformations. (English) Zbl 0906.54016
The author considers the problem of continuity of the topological entropy $$h_{\text{top}}$$ on the space of piecewise monotone transformations of $$I=[0,1]$$. Two piecewise monotone transformations of $$I$$ are near if the closures of their graphs in $$I^2$$ are near with respect to the Hausdorff distance. It is proven that for a continuous strictly increasing function $$f:I\to\mathbb R$$ the topological entropy $$h_{\text{top}}$$ is continuous at the piecewise monotone transformation $$T:=f\;(\text{mod} 1)$$ (such transformations $$T$$ are called monotone mod one transformations), provided $$h_{\text{top}}(T)>0$$. The author presents two examples of monotone mod one transformations of $$I$$ at which the entropy is discontinuous. Also, he gives an example of a monotonic mod one transformation $$T$$ with positive (and thus continuous at $$T$$) entropy such that the maximal measure and the pressure $$p(T,g)$$ are discontinuous at $$T$$ for some continuous function $$g:I\to\mathbb R$$.

##### MSC:
 54C70 Entropy in general topology 54H20 Topological dynamics (MSC2010)
##### Keywords:
topological pressure
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