Invariant measures for Cherry flows

TL;DR

This paper studies invariant measures for Cherry flows on the two-torus, identifying conditions under which measures are supported at fixed points or on the quasi-minimal set, and exploring their physical relevance.

Contribution

It characterizes invariant measures for Cherry flows, especially distinguishing cases with dissipative, conservative, or other saddle behaviors, and introduces techniques for analyzing return times and near-saddle dynamics.

Findings

Only fixed point measures in dissipative or conservative cases.

Existence of an invariant measure on the quasi-minimal set in other cases.

The saddle's Dirac measure is the physical measure in certain cases.

Abstract

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.