Beau bounds for multicritical circle maps

TL;DR

This paper establishes universal geometric bounds for multicritical circle maps, extending previous results from single to multiple critical points, which are fundamental for understanding their dynamical behavior.

Contribution

It introduces real a-priori bounds for multicritical circle maps, generalizing previous single-critical point results to multiple critical points.

Findings

Proves asymptotically universal bounds on orbit geometry

Extends techniques from single to multiple critical points

Provides foundational results for multicritical circle dynamics

Abstract

Let $f: S^1\to S^1$ be a $C^3$ homeomorphism without periodic points having a finite number of critical points of power-law type. In this paper we establish real a-priori bounds, on the geometry of orbits of $f$, which are beau in the sense of Sullivan, i.e. bounds that are asymptotically universal at small scales. The proof of the beau bounds presented here is an adaptation, to the multicritical setting, of the one given by the second author and de Melo, for the case of a single critical point.