On the continuity of the pressure for monotonic mod one transformations

@article{Raith2000,
abstract = {If $f:[0,1]\rightarrow \{\mathbb \{R\}\}$ is strictly increasing and continuous define $T_fx=f(x)\, (\operatorname\{mod\} 1)$. A transformation $\tilde\{T\}:[0,1]\rightarrow [0,1]$ is called $\varepsilon $-close to $T_f$, if $\tilde\{T\}x=\tilde\{f\}(x)\, (\operatorname\{mod\} 1)$ for a strictly increasing and continuous function $\tilde\{f\}:[0,1]\rightarrow \{\mathbb \{R\}\}$ with $\Vert \tilde\{f\}-f\Vert _\{\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 $g:[0,1]\rightarrow \{\mathbb \{R\}\}$, if and only if $0$ is not periodic or $1$ is not periodic. Finally it is shown that the topological entropy is continuous, if $h_\{\text\{\rm top\}\}(T_f)>0$.},
author = {Raith, Peter},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {mod one transformation; topological pressure; topological entropy; maximal measure; perturbation; mod one transformation; topological pressure; topological entropy; maximal measure; perturbation},
language = {eng},
number = {1},
pages = {61-78},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the continuity of the pressure for monotonic mod one transformations},
url = {http://eudml.org/doc/248628},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Raith, Peter
TI - On the continuity of the pressure for monotonic mod one transformations
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 1
SP - 61
EP - 78
AB - If $f:[0,1]\rightarrow {\mathbb {R}}$ is strictly increasing and continuous define $T_fx=f(x)\, (\operatorname{mod} 1)$. A transformation $\tilde{T}:[0,1]\rightarrow [0,1]$ is called $\varepsilon $-close to $T_f$, if $\tilde{T}x=\tilde{f}(x)\, (\operatorname{mod} 1)$ for a strictly increasing and continuous function $\tilde{f}:[0,1]\rightarrow {\mathbb {R}}$ with $\Vert \tilde{f}-f\Vert _{\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 $g:[0,1]\rightarrow {\mathbb {R}}$, if and only if $0$ is not periodic or $1$ is not periodic. Finally it is shown that the topological entropy is continuous, if $h_{\text{\rm top}}(T_f)>0$.
LA - eng
KW - mod one transformation; topological pressure; topological entropy; maximal measure; perturbation; mod one transformation; topological pressure; topological entropy; maximal measure; perturbation
UR - http://eudml.org/doc/248628
ER -