Peter Raith
On the continuity of the pressure for monotonic mod one transformations

Comment.Math.Univ.Carolinae 41,1 (2000) 61-78.

Abstract:If $f:[0,1]\to {\Bbb R}$ is strictly increasing and continuous define $T_fx=f(x) (mod 1)$. A transformation $\tilde {T}:[0,1]\to [0,1]$ is called $\varepsilon $-close to $T_f$, if $\tilde {T}x=\tilde {f}(x) (mod 1)$ 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 $h_{\text {top}}(T_f)>0$.

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