Specification and thermodynamical properties of semigroup actions

2015/02/28 by Fagner B. Rodrigues, Paulo Varandas · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Bicyclic semigroup #Cellular Automata and Applications #Chaos control and synchronization #Entropy (arrow of time) #Exponential function #Exponential growth #Extension (predicate logic) #Lipschitz continuity #Mathematical Dynamics and Fractals #Semigroup #Topological entropy #math-ph #math.DS #math.GR #math.MP #msc:22F05 #msc:37B05 #msc:37B40 #msc:37C50 #msc:37C85 #msc:37D20 #msc:37D35

paper · pdf · doi:10.1063/1.4950928

26 pages; Final version to appear in Journal of Mathematical Physics

openalex publication_date 2016/05/01 · arxiv created 2016/05/08 · arxiv updated 2016/06/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

In the present paper, we study the thermodynamical properties of finitely generated continuous subgroup actions. We propose a notion of topological entropy and pressure functions that do not depend on the growth rate of the semigroup and introduce strong and orbital specification properties, under which the semigroup actions have positive topological entropy and all points are entropy points. Moreover, we study the convergence and Lipschitz regularity of the pressure function and obtain relations between topological entropy and exponential growth rate of periodic points in the context of semigroups of expanding maps, obtaining a partial extension of the results obtained by Ruelle for ℤd-actions [D. Ruelle, Trans. Am. Math. Soc., 187, 237–251 (1973)]. The specification properties for semigroup actions and the corresponding one for its generators and the action of push-forward maps are also discussed.

Citations

• Topological entropy of continuous actions of compactly generated groups
• On the weaker forms of the specification property and their applications
• Irregular sets, the 𝛽-transformation and the almost specification property
• Non-uniform hyperbolicity and non-uniform specification
• Directional uniformities, periodic points, and entropy

Cited by

• Specification and Thermodynamic Properties of Topological Time-Dependent Dynamical Systems
• On topological entropy and topological pressure of non-autonomous iterated function systems

Related

• On the measure-theoretic entropy and topological pressure of free semigroup actions(to appear in ETDS)
• Pressures for multi-potentials in semigroup dynamics
• Discrete dynamical systems in group theory
• Entropy of Isometries Semi-groups of Hyperbolic space
• Amenability and measure of maximal entropy for semigroups of rational maps