Israel Journal of Mathematics

, Volume 213, Issue 1, pp 475–503 | Cite as

On entropy of dynamical systems with almost specification

Article

Abstract

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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Copyright information

© Hebrew University of Jerusalem 2016

Authors and Affiliations

  • Dominik Kwietniak
    • 1
  • Piotr Oprocha
    • 2
    • 3
  • Michał Rams
    • 4
  1. 1.Faculty of Mathematics and Computer ScienceJagiellonian University in KrakówKrakówPoland
  2. 2.AGH University of Science and TechnologyFaculty of Applied MathematicsKrakówPoland
  3. 3.National Supercomputing Centre IT4InnovationsDivision of the University of Ostrava Institute for Research and Applications of Fuzzy Modeling 30. dubna 22OstravaCzech Republic
  4. 4.Institute of MathematicsPolish Academy of SciencesWarszawaPoland

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