Entropy and periodic orbits for equivalent smooth flows

TL;DR

The paper constructs equivalent smooth flows with contrasting entropy and periodic orbit growth rates, demonstrating complex dynamical behaviors and growth phenomena in low-dimensional systems.

Contribution

It introduces new examples of equivalent flows with divergent entropy and periodic orbit growth, including a smooth flow on the sphere with super-exponential orbit growth.

Findings

Existence of equivalent flows with positive entropy and zero exponential orbit growth

Construction of a smooth flow on the sphere with super-exponential orbit growth

Any two-dimensional flow has zero topological entropy

Abstract

Given any $K>0$, we construct two equivalent $C^2$ flows, one of which has positive topological entropy larger than $K$ and admits zero as the exponential growth of periodic orbits, in contrast, the other has zero topological entropy and super-exponential growth of periodic orbits. Moreover we establish a $C^{\infty}$ flow on $\mathbb{S}^2$ with super-exponential growth of periodic orbits, which is also equivalent to another flow with zero exponential growth of periodic orbits. On the other hand, any two dimensional flow has only zero topological entropy.