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.

Unbounded regime for circle maps with a flat interval

“…Using Theorem 1.2 and following the strategies in [10], we are now able to generalize Theorem 1.6 in [10] and give an example of Cherry flow with a metrically non-trivial quasi-minimal set in the general case of unbounded regime 3 . More precisely: Theorem 1.3.…”

“…assumption was then removed in [10]. In these papers, it is proved that the geometry is degenerate when the critical exponent is less than or equal to 2 and becomes bounded when the critical exponent passes 2.…”

“…This paper is a part of the recent reawakening of interest in this class of mappings, principally motivated by a deeper understanding of their connections with Cherry flows. This line of research has also been pursued recently for example in [10] and [13].…”

Quasi-symmetric conjugacy for circle maps with a flat interval

“…We contribute to the area of the understanding the dynamics of a class of continuous degree one circle maps with a flat interval, see [21], [19], [13], [14], [5], [4], [15], [16]. In this paper we address the problem of quasi-symmetric conjugation.…”

“…If f −j and F j are contained in the same gap of the partition F n−1 , then f −j and F j are comparable.Proof. See Lemma 5.1 in[15].…”