If $f:[0,1]\to{\Bbb R}$ is strictly increasing and continuous define $T_fx=f(x)\, (\operatorname{mod} 1)$. A transformation $\tilde{T}:[0,1]\to [0,1]$ is called $\varepsilon$-close to $T_f$, if $\tilde{T}x=\tilde{f}(x)\, (\operatorname{mod} 1)$ for a strictly increasing and continuous function $\tilde{f}:[0,1]\to{\Bbb R}$ with $\|\tilde{f}-f\|_{\infty}<\varepsilon$. It is proved that the topological pressure $p(T_f,g)$ is lower semi-continuous, and an upper bound for the jumps up is given. Furthermore the continuity of the maximal measure is shown, if a certain condition is satisfied. Then it is proved that the topological pressure is upper semi-continuous for every continuous function $g:[0,1]\to{\Bbb R}$, if and only if $0$ is not periodic or $1$ is not periodic. Finally it is shown that the topological entropy is continuous, if $h_{\text{\rm top}}(T_f)>0$.
@article{119141,
author = {Peter Raith},
title = {On the continuity of the pressure for monotonic mod one transformations},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {41},
year = {2000},
pages = {61-78},
zbl = {1034.37021},
mrnumber = {1756927},
language = {en},
url = {http://dml.mathdoc.fr/item/119141}
}
Raith, Peter. On the continuity of the pressure for monotonic mod one transformations. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) pp. 61-78. http://gdmltest.u-ga.fr/item/119141/
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