Cherry flow: physical measures and perturbation theory

TL;DR

This paper proves that Cherry flows on a torus with specific singularities have a unique physical measure with full basin volume, and analyzes how perturbations depend on the divergence at the saddle.

Contribution

It establishes the existence and uniqueness of physical measures for Cherry flows and characterizes their perturbations based on divergence at the saddle.

Findings

Unique physical measure with full basin volume for Cherry flows.

Perturbation behavior depends on the divergence at the saddle.

Flow can be approximated by flows with multiple sinks when divergence is non-negative.

Abstract

In this article we consider Cherry flows on torus which have two singularities: a source and a saddle, and no periodic orbits. We show that every Cherry flow admits a unique physical measure, whose basin has full volume. This proves a conjecture given by R. Saghin and E. Vargas in~\cite{SV}. We also show that the perturbation of Cherry flow depends on the divergence at the saddle: when the divergence is negative, this flow admits a neighborhood, such that any flow in this neighborhood belongs to the following three cases: (a) has a saddle connection; (b) a Cherry flow; (c) a Morse-Smale flow whose non-wandering set consists two singularities and one periodic sink. In contrary, when the divergence is non-negative, this flow can be approximated by non-hyperbolic flow with arbitrarily larger number of periodic sinks.