Equilibrium states of intermediate entropies

TL;DR

This paper investigates the conjecture of Katok regarding intermediate entropies by leveraging the uniqueness of equilibrium states and applies this framework to prove the conjecture for Mañé diffeomorphisms.

Contribution

It introduces a novel approach connecting equilibrium state uniqueness with entropy conjectures and confirms Katok's conjecture for a class of dynamical systems.

Findings

Proves Katok's conjecture for Mañé diffeomorphisms.

Establishes a link between upper semi-continuity of entropy and equilibrium states.

Provides a new method for studying intermediate entropies in dynamical systems.

Abstract

We explore an approach to the conjecture of Katok on intermediate entropies that based on uniqueness of equilibrium states, provided the entropy function is upper semi-continuous. As an application, we prove Katok's conjecture for Ma\~n\'e diffeomorphisms.