A Phase Transition for Circle Maps and Cherry Flows

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.

Cited by in corpus (5)

• Cherry flow: physical measures and perturbation theory
• Asymmetric unimodal maps with non-universal period-doubling scaling laws
• Invariant Manifolds for Non-differentiable Operators
• Quasi-Symmetric Conjugacy for Circle Maps with a Flat Interval
• Cherry Maps with Different Critical Exponents: Bifurcation of Geometry