On the continuity of the pressure for monotonic mod one transformations

Raith, Peter

Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 1, pp. 61-78.

Résumé

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

Détail

Classification : 37B40, 37D35, 37E05, 37E99, 54H20
Keywords: mod one transformation; topological pressure; topological entropy; maximal measure; perturbation