Slow entropy for some smooth flows on surfaces

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 C2 everywhere except one point (singularity). If the singularity is logarithmic asymmetric (Arnol’d flows), we show that in the scale an(t) = n(log n)t slow entropy equals 1 (the speed of orbit growth is n log n) for a.e. irrational α. If the singularity is of power type (x−γ, γ ∈ (0, 1)) (Kochergin flows), we show that in the scale an(t) = nt slow entropy equals 1 + γ for a.e. α. We show moreover that for local rank one flows, slow entropy equals 0 in the n(log n)t scale and is at most 1 for scale nt. As a consequence we get that a.e. Arnol’d and a.e Kochergin flow is never of local rank one.