# Article

Full entry | PDF   (0.9 MB)
Keywords:
piecewise monotonic map; nonwandering set; topologically transitive subset
Summary:
In this paper piecewise monotonic maps $T [0,1]\to[0,1]$ are considered. Let $Q$ be a finite union of open intervals, and consider the set $R(Q)$ of all points whose orbits omit $Q$. The influence of small perturbations of the endpoints of the intervals in $Q$ on the dynamical system $(R(Q),T)$ is investigated. The decomposition of the nonwandering set into maximal topologically transitive subsets behaves very unstably. Nonetheless, it is shown that a maximal topologically transitive subset cannot be completely destroyed by arbitrary small perturbations of $Q$. Furthermore it is shown that every sufficiently "big" maximal topologically transitive subset of a sufficiently small perturbation of $(R(Q),T)$ is "dominated" by a topologically transitive subset of $(R(Q),T)$.
References:
[1] F. Hofbauer: The structure of piecewise monotonic transformations. Ergodic Theory Dynam. Systems 1 (1981), 159-178. MR 0661817 | Zbl 0474.28007
[2] F. Hofbauer: Piecewise invertible dynamical systems. Probab. Theory Related Fields 72 (1986), 359-386. DOI 10.1007/BF00334191 | MR 0843500 | Zbl 0578.60069
[3] F. Hofbauer P. Raith: Topologically transitive subsets of piecewise monotonic maps, which contain no periodic points. Monatsh. Math. 107 (1989), 217-239. DOI 10.1007/BF01300345 | MR 1008681
[4] M. Misiurewicz S. V. Shlyachkov: Entropy of piecewise continuous interval maps. European conference on iteration theory (ECIT 89), Batschuns, 1989 (Ch. Mira, N. Netzer, C. Simó, Gy. Targoński, eds.). World Scientific, Singapore, 1991, pp. 239-245. MR 1184170
[5] Z. Nitecki: Topological dynamics on the interval. Ergodic theory and dynamical systems, Vol. 2, Proceedings of the Special Year at the University of Maryland, 1979/1980 (A. Katok, ed.). Progress in Mathematics 21, Birkhauser, Boston, 1982, pp. 1-73. MR 0670074
[6] P. Raith: Hausdorff dimension for piecewise monotonic maps. Studia Math. 94 (1989), 17-33. MR 1008236 | Zbl 0687.58013
[7] P. Raith: Continuity of the Hausdorff dimension for piecewise monotonic maps. Israel J. Math. 80 (1992), 97-133. DOI 10.1007/BF02808156 | MR 1248929 | Zbl 0768.28010
[8] P. Raith: 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] P. Raith: The dynamics of piecewise monotonic maps under small perturbations. Preprint, Warwick, 1994. MR 1627314
[10] M. Urbański: Hausdorff dimension of invariant sets for expanding maps of a circle. Ergodic Theory Dynam. Systems 6 (1986), 295-309. MR 0857203
[11] M. Urbański: Invariant subsets of expanding mappings of the circle. Ergodic Theory Dynam. Systems 7 (1987), 627-645. MR 0922369
[12] P. Walters: An introduction to ergodic theory. Graduate Texts in Mathematics 79, Springer, New York, 1982. DOI 10.1007/978-1-4612-5775-2 | MR 0648108 | Zbl 0475.28009

Partner of