Complex a priori bounds for multicritical circle maps with bounded type rotation number

TL;DR

This paper establishes complex a priori bounds for multicritical circle maps with bounded type rotation numbers, leading to $C^{1+eta}$ rigidity results for bi-cubic maps.

Contribution

It introduces complex a priori bounds for multicritical circle maps with bounded type rotation numbers, advancing understanding of their rigidity properties.

Findings

Proved complex a priori bounds for multicritical circle maps.

Demonstrated $C^{1+eta}$ rigidity for bi-cubic maps with the same rotation number.

Extended rigidity results to maps with multiple critical points.

Abstract

In this paper we study homeomorphisms of the circle with several critical points and bounded type rotation number. We prove complex a priori bounds for these maps. As an application, we get that bi-cubic circle maps with same bounded type rotation number are $C^{1+\alpha}$ rigid.