A Phase Transition for Circle Maps and Cherry Flows

TL;DR

This paper investigates phase transitions in circle maps with flat intervals, revealing a sharp change from degenerate to bounded geometry that affects the Hausdorff dimension of the non-wandering set and impacts Cherry flow dynamics.

Contribution

It establishes a precise phase transition in circle maps with flat intervals, linking geometric properties to the Hausdorff dimension and Cherry flow behavior.

Findings

Zero Hausdorff dimension in degenerate geometry

Positive Hausdorff dimension in bounded geometry

Sharp transition in Cherry flow dynamics

Abstract

We study $C^{2}$ weakly order preserving circle maps with a flat interval. The main result of the paper is about a sharp transition from degenerate geometry to bounded geometry depending on the degree of the singularities at the boundary of the flat interval. We prove that the non-wandering set has zero Hausdorff dimension in the case of degenerate geometry and it has Hausdorff dimension strictly greater than zero in the case of bounded geometry. Our results about circle maps allow to establish a sharp phase transition in the dynamics of Cherry flows.