Equilibrium states of intermediate entropies. Issue 1 (2nd January 2021)

• Record Type: Journal Article
• Authors: Sun, Peng
• Is Part Of: Dynamical systems. Volume 36:Issue 1(2021)
• Page Start: 69
• Page End: 78
• Publication Date: 2021-01-02
• Subjects: Equilibrium state -- intermediate entropy -- Mañé diffeomorphism -- ergodic optimization -- thermodynamic formalism; 37A35 -- 37D35; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2020.1823320
• Languages: English
• ISSNs: 1468-9367

Abstract

Abstract : We explore a new approach to the conjecture of Katok on intermediate entropies, which shows that the conjecture can be a consequence of certain results on equilibrium states. Suppose that the system ( X, f ) has finite topological entropy and the entropy function is upper semi-continuous. Let ϕ be a continuous potential such that t ϕ has a unique equilibrium state μ t for every t ≥ 0 and the maximizing measures of ϕ have zero entropy. Then the entropies of these equilibrium states can assume any intermediate value between zero and the topological entropy of the system. As an application, we prove Katok's conjecture for Mañé diffeomorphisms.