On the continuity of the pressure for monotonic mod one transformations

Peter Raith

Commentationes Mathematicae Universitatis Carolinae (2000)

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

Abstract

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If f : [ 0 , 1 ] → ℝ is strictly increasing and continuous define T f x = f ( x ) ( mod 1 ) . A transformation T ˜ : [ 0 , 1 ] → [ 0 , 1 ] is called ε -close to T f , if T ˜ x = f ˜ ( x ) ( mod 1 ) for a strictly increasing and continuous function f ˜ : [ 0 , 1 ] → ℝ with ∥ f ˜ - f ∥ ∞ < ε . 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 ] → 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 top ( T f ) > 0 .

How to cite

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Raith, 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

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  1. Alsedà Ll., Ma nosas F., Mumbrú P., Continuity of entropy for bimodal maps, J. London Math. Soc. 52 (1995), 547-567. (1995) MR1363820
  2. 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
  3. Hofbauer F., Monotonic mod one transformations, Studia Math. 80 (1984), 17-40. (1984) Zbl0506.54034MR0781724
  4. Misiurewicz M., Jumps of entropy in one dimension, Fund. Math. 132 (1989), 215-226. (1989) Zbl0694.54019MR1002409
  5. 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
  6. Raith P., Hausdorff dimension for piecewise monotonic maps, Studia Math. 94 (1989), 17-33. (1989) Zbl0687.58013MR1008236
  7. Raith P., Continuity of the Hausdorff dimension for piecewise monotonic maps, Israel J. Math. 80 (1992), 97-133. (1992) Zbl0768.28010MR1248929
  8. Raith P., Continuity of the Hausdorff dimension for invariant subsets of interval maps, Acta Math. Univ. Comenian. 63 (1994), 39-53. (1994) Zbl0828.58014MR1342594
  9. Raith P., Continuity of the entropy for monotonic mod one transformations, Acta Math. Hungar. 77 (1997), 247-262. (1997) Zbl0906.54016MR1485848
  10. Raith P., Stability of the maximal measure for piecewise monotonic interval maps, Ergodic Theory Dynam. Systems 17 (1997), 1419-1436. (1997) Zbl0898.58015MR1488327
  11. Raith P., The dynamics of piecewise monotonic maps under small perturbations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24 (1997), 783-811. (1997) MR1627314
  12. 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
  13. Raith P., Discontinuities of the pressure for piecewise monotonic interval maps, Ergodic Theory Dynam. Systems, to appear preprint, Wien, 1997. Zbl0972.37024MR1826666
  14. Walters P., An introduction to ergodic theory, Graduate Texts in Mathematics 79 Springer New York (1982). (1982) Zbl0475.28009MR0648108

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