Dynamics of multicritical circle maps

TL;DR

This survey reviews recent advances in understanding the geometric, ergodic, and complex-analytic properties of smooth circle homeomorphisms with multiple critical points, highlighting key results and open problems.

Contribution

It compiles and discusses recent and classical results on the structure and dynamics of multicritical circle maps, emphasizing their geometric and ergodic properties.

Findings

Detailed analysis of orbit structures in multicritical circle maps

Connections between geometric and ergodic properties of these systems

Identification of open conjectures and questions in the field

Abstract

This paper presents a survey of recent and not so recent results concerning the study of smooth homeomorphisms of the circle with a finite number of non-flat critical points, an important topic in the area of One-dimensional Dynamics. We focus on the analysis of the fine geometric structure of orbits of such dynamical systems, as well as on certain ergodic-theoretic and complex-analytic aspects of the subject. Finally, we review some conjectures and open questions in this field.