On entropy of dynamical systems with almost specification
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We construct a family of shift spaces with almost specification and multiple measures of maximal entropy. This answers a question from Climenhaga and Thompson [Israel J. Math. 192 (2012), 785–817]. Elaborating on our examples we prove that sufficient conditions for every shift factor of a shift space to be intrinsically ergodic given by Climenhaga and Thompson are in some sense best possible; moreover, the weak specification property neither implies intrinsic ergodicity, nor follows from almost specification. We also construct a dynamical system with the weak specification property, which does not have the almost specification property. We prove that the minimal points are dense in the support of any invariant measure of a system with the almost specification property. Furthermore, if a system with almost specification has an invariant measure with non-trivial support, then it also has uniform positive entropy over the support of any invariant measure and cannot be minimal.
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- L. S. Block and W. A. Coppel, One-Dimensional Dynamics, Lecture Notes in Mathematics, Vol. 1513, Springer-Verlag, Berlin, 1992.Google Scholar
- E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, Providence, RI, 2003.Google Scholar
- P. Kurka, Topological and Symbolic Dynamics, Cours Spécialisés, Vol. 11, Société Mathématique de France, Paris, 2003.Google Scholar
- D. Kwietniak, M. Łącka and P. Oprocha, A panorama of specificationlike properties and their consequences, Contemporary Mathematics, to appear, arXiv:1503.07355[math.DS], http://dx.doi.org/10.1090/conm/669.Google Scholar
- R. Pavlov, On weakenings of the specification property and intrinsic ergodicity, Advances in Mathematics, to appear.Google Scholar
- P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, Springer-Verlag, New York–Berlin, 1982.Google Scholar
- X. Wu, P. Oprocha and G. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity, to appear.Google Scholar