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 $C^2$ 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.