# On the continuity of the pressure for monotonic mod one transformations

Commentationes Mathematicae Universitatis Carolinae (2000)

- Volume: 41, Issue: 1, page 61-78
- ISSN: 0010-2628

## Access Full Article

top## Abstract

top## How to cite

topRaith, Peter. "On the continuity of the pressure for monotonic mod one transformations." Commentationes Mathematicae Universitatis Carolinae 41.1 (2000): 61-78. <http://eudml.org/doc/248628>.

@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 -

## References

top- Alsedà Ll., Ma nosas F., Mumbrú P., Continuity of entropy for bimodal maps, J. London Math. Soc. 52 (1995), 547-567. (1995) MR1363820
- Hofbauer F., On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math. 34 (1979), 213-237 Part 2 Israel J. Math. 38 (1981), 107-115. (1981) Zbl0456.28006MR0570882
- Hofbauer F., Monotonic mod one transformations, Studia Math. 80 (1984), 17-40. (1984) Zbl0506.54034MR0781724
- Misiurewicz M., Jumps of entropy in one dimension, Fund. Math. 132 (1989), 215-226. (1989) Zbl0694.54019MR1002409
- Misiurewicz M., Shlyachkov S.V., Entropy of piecewise continuous interval maps, European Conference on Iteration Theory (ECIT 89), Batschuns, 1989 Ch. Mira, N. Netzer, C. Simó, Gy. Targoński 239-245 World Scientific Singapore (1991). (1991) Zbl1026.37504MR1184170
- Raith P., Hausdorff dimension for piecewise monotonic maps, Studia Math. 94 (1989), 17-33. (1989) Zbl0687.58013MR1008236
- Raith P., Continuity of the Hausdorff dimension for piecewise monotonic maps, Israel J. Math. 80 (1992), 97-133. (1992) Zbl0768.28010MR1248929
- Raith P., Continuity of the Hausdorff dimension for invariant subsets of interval maps, Acta Math. Univ. Comenian. 63 (1994), 39-53. (1994) Zbl0828.58014MR1342594
- Raith P., Continuity of the entropy for monotonic mod one transformations, Acta Math. Hungar. 77 (1997), 247-262. (1997) Zbl0906.54016MR1485848
- Raith P., Stability of the maximal measure for piecewise monotonic interval maps, Ergodic Theory Dynam. Systems 17 (1997), 1419-1436. (1997) Zbl0898.58015MR1488327
- Raith P., The dynamics of piecewise monotonic maps under small perturbations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24 (1997), 783-811. (1997) MR1627314
- Raith P., Perturbations of a topologically transitive piecewise monotonic map on the interval, Proceedings of the European Conference on Iteration Theory (ECIT 96), Urbino, 1996 (L. Gardini et al., eds.), Grazer Math. Ber. 339 (1999), 301-312. Zbl0948.37026MR1748832
- Raith P., Discontinuities of the pressure for piecewise monotonic interval maps, Ergodic Theory Dynam. Systems, to appear preprint, Wien, 1997. Zbl0972.37024MR1826666
- Walters P., An introduction to ergodic theory, Graduate Texts in Mathematics 79 Springer New York (1982). (1982) Zbl0475.28009MR0648108

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.