Entropy Flexibility of Dynamical Systems

TL;DR

This paper introduces the concept of entropy flexibility in dynamical systems, demonstrating its prevalence in various classes and establishing it as a typical property in certain vector fields and surface diffeomorphisms.

Contribution

It defines entropy flexibility for dynamical systems and proves its generic presence in vector fields on 3-manifolds and surface diffeomorphisms.

Findings

Entropy flexibility exists in several classes of systems.

Any intermediate entropy value can be realized on a strictly ergodic subsystem.

Entropy flexibility is a typical property in certain dynamical contexts.

Abstract

Inspired by Katok's intermediate entropy property [Inst. Hautes \'Etudes Sci. Publ. Math. 51 (1980), 137-173], we introduce and study the notion of entropy flexibility for discrete-time and continuous-time dynamical systems. By using renewal systems techniques, we show that this property is present in several classes of systems where any intermediate value of entropy can be attained on a strictly ergodic sub-system. In addition, we prove an entropy flexibility analogue of Katok's conjecture: Entropy flexibility is a typical property for vector fields on 3-manifolds and surface diffeomorphisms.