Dynamics of multicritical circle maps

2021/03/01 by Edson de Faria, Pablo Guarino · 3 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Computer science #Dynamical systems theory #Dynamics (music) #Ergodic theory #Field (mathematics) #Focus (optics) #Mathematical Dynamics and Fractals #Mathematics #Multicritical point #Phase diagram #Physics #Pure mathematics #Quantum chaos and dynamical systems #Statistical physics #Subject (documents) #math.DS

paper · pdf · doi:10.1007/s40863-021-00236-1

published in São Paulo Journal of Mathematical Sciences 16(1), 340-395 (Springer Nature) · 50 pages, 6 figures

arxiv created 2021/03/01 · openalex publication_date 2021/05/20 · arxiv updated 2021/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals
• The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets
• Herman’s theory revisited
• Renormalization of unicritical analytic circle maps
• The Attractor of Renormalization and Rigidity of Towers of Critical Circle Maps
• Renormalization of bi-cubic circle maps
• Complex a priori bounds for multicritical circle maps with bounded type rotation number