Cherry flows with non trivial attractors

TL;DR

This paper constructs a specific Cherry flow on a 2D torus featuring a non-trivial attractor, with a detailed analysis of its first return map showing complex dynamics including wandering intervals.

Contribution

It provides the first example of a Cherry flow with a quasi-minimal set that acts as an attractor, expanding understanding of flow dynamics on the torus.

Findings

Existence of Cherry flow with a quasi-minimal attractor

First return map is a smooth circle map with a flat interval

Flow exhibits complex dynamics including wandering intervals

Abstract

We provide an example of Cherry flow (i.e. smooth flow on the $2$-dimensional torus with a sink and a saddle) having quasi-minimal set which is an attractor. The first return map for such a flow, constructed also in the paper, is a smooth circle map having a flat interval and a non-trivial wandering interval.