Slow entropy for some smooth flows on surfaces

TL;DR

This paper investigates slow entropy in smooth mixing flows on surfaces, revealing how different singularities influence orbit growth rates and distinguishing these flows from local rank one systems.

Contribution

It provides new results on slow entropy for Arnol'd and Kochergin flows with specific singularities, showing their distinct orbit growth behaviors.

Findings

Arnol'd flows have slow entropy 1 in the scale n(log n)^t

Kochergin flows have slow entropy 1+γ in the scale n^t

Local rank one flows have zero slow entropy in the scale n(log n)^t

Abstract

We study slow entropy in some classes of smooth mixing flows on surfaces. The flows we study can be represented as special flows over irrational rotations and under roof functions which are $C^2$ everywhere except one point (singularity). If the singularity is logarithmic asymmetric (Arnol'd flows) we show that in the scale $a_n(t)=n(logn)^t$ slow entropy equals 1 (the speed of orbit growth is nlogn) for a.e. irrational $\alpha$. If the singularity is of power type ($x^{-\gamma}$, $\gamma\in (0,1)$) (Kochergin flows) we show that in the scale $a_n(t)=n^t$ slow entropy equals $1+\gamma$ for a.e. $\alpha$. We show moreover that for local rank one flows slow entropy equals $0$ in the scale $n(logn)^t$. As a consequence we get that a.e. Arnol'd and a.e. Kochergin flow is never of local rank one.