Topological transitivity for expanding monotonic mod one transformations with two monotonic pieces
Abstract
Consider a continuous and strictly increasing function f: [0,1]→[0,2], and define Tfx = f(x)(mod 1). Then Tf is a monotonie mod one transformation with two monotonic pieces, if and only if f(0) < 1 < f(1). It is proved that Tf is topologically transitive, if f is piecewise differentiable and infx∈[0,1]f'(x) ≥ √2.
References
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2. F. Hofbauer, Monotonie mod one transformations, Studia Math. 80 (1984), 17-40.
3. F. Hofbauer, Piecewise invertible dynamical systems, Probab. Theory Related Fields 72 (1986), 359-386.
4. F. Hofbauer, P. Raith, Density of periodic orbit measures for transformations on the interval with two monotonic pieces, Fund. Math. 157 (1998), 221-234.
5. S. Kolyada, L'. Snoha, Some aspects of topological transitivity - a survey, in Proceedings of the European conference on iteration theory (ECIT 94), Opava, 1994, (L. Reich, J. Smítal, Gy. Targoński, eds.), Grazer Math. Ber. 334 (1997), 3-35.
6. P. Raith, Continuity of the Hausdorff dimension for piecewise monotonic maps, Israel J. Math. 80 (1992), 97-133.
7. P. Raith, Continuity of the entropy for monotonic mod one transformations, Acta Math. Hungar. 77 (1997), 247-262.
8. P. Raith, Dynamical properties of a piecewise monotonie interval map and its small perturbations, in Proceedings of the European conference on iteration theory (ECIT 94), Opava, 1994, (L. Reich, J. Smítal, Gy. Targoński, eds.), Grazer Math. Ber. 334 (1997), 37-65.
9. P. Raith, Perturbations of a topologically transitive piecewise monotonic map on the interval, in Proceedings of the Europan conference on iteration theory (ECIT 96), Urbino, 1996 (to appear).
10. P. Raith, Discontinuities of the pressure for piecewise monotonic interval maps, Ergodic Theory Dynam. Systems (to appear).
11. P. Raith, On the continuity of the pressure for monotonic mod one transformations, Preprint, Wien 1997.
12. P. Raith, Topological transitivity for expanding piecewise monotonic maps on the interval, Aequationes Math. 57 (1999), 303-311.
13. P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics 79, Springer, New York 1982.
RaithP. (1999). Topological transitivity for expanding monotonic mod one transformations with two monotonic pieces. Annales Mathematicae Silesianae, 13, 233-241. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14152
Peter Raith
Peter.Raith@univie.ac.at
Institut für Mathematik, Universität Wien, Austria Austria
Institut für Mathematik, Universität Wien, Austria Austria
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