# Article

Full entry | PDF   (0.3 MB)
Keywords:
mod one transformation; topological pressure; topological entropy; maximal measure; perturbation
Summary:
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 $h_{\text{\rm top}}(T_f)>0$.
References:
[1] Alsedà Ll., Ma nosas F., Mumbrú P.: Continuity of entropy for bimodal maps. J. London Math. Soc. 52 (1995), 547-567. MR 1363820
[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. MR 0570882 | Zbl 0456.28006
[3] Hofbauer F.: Monotonic mod one transformations. Studia Math. 80 (1984), 17-40. MR 0781724 | Zbl 0506.54034
[4] Misiurewicz M.: Jumps of entropy in one dimension. Fund. Math. 132 (1989), 215-226. MR 1002409 | Zbl 0694.54019
[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). MR 1184170 | Zbl 1026.37504
[6] Raith P.: Hausdorff dimension for piecewise monotonic maps. Studia Math. 94 (1989), 17-33. MR 1008236 | Zbl 0687.58013
[7] Raith P.: Continuity of the Hausdorff dimension for piecewise monotonic maps. Israel J. Math. 80 (1992), 97-133. MR 1248929 | Zbl 0768.28010
[8] Raith P.: Continuity of the Hausdorff dimension for invariant subsets of interval maps. Acta Math. Univ. Comenian. 63 (1994), 39-53. MR 1342594 | Zbl 0828.58014
[9] Raith P.: Continuity of the entropy for monotonic mod one transformations. Acta Math. Hungar. 77 (1997), 247-262. MR 1485848 | Zbl 0906.54016
[10] Raith P.: Stability of the maximal measure for piecewise monotonic interval maps. Ergodic Theory Dynam. Systems 17 (1997), 1419-1436. MR 1488327 | Zbl 0898.58015
[11] Raith P.: The dynamics of piecewise monotonic maps under small perturbations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24 (1997), 783-811. MR 1627314
[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. MR 1748832 | Zbl 0948.37026
[13] Raith P.: Discontinuities of the pressure for piecewise monotonic interval maps. Ergodic Theory Dynam. Systems, to appear preprint, Wien, 1997. MR 1826666 | Zbl 0972.37024
[14] Walters P.: An introduction to ergodic theory. Graduate Texts in Mathematics 79 Springer New York (1982). MR 0648108 | Zbl 0475.28009

Partner of