Dynamics of Circle Mappings

This book explores recent developments in the dynamics of invertible circle maps, a rich and captivating topic in one-dimensional dynamics. It focuses on two main classes of invertible dynamical systems on the circle: global diffeomorphisms and smooth homeomorphisms with critical points. The latter is the book's core, reflecting the authors' recent research interests.

Organized into four parts and 14 chapters, the content covers rigid rotations, circle homeomorphisms, and the concept of rotation number in the first part. The second part delves into circle diffeomorphisms, presenting classical results. The third part introduces multicritical circle maps—smooth homeomorphisms of the circle with a finite number of critical points. The fourth and final part centers on renormalization theory, analyzing the fine geometric structure of orbits of multicritical circle maps. Complete proofs for several fundamental results in circle dynamics are provided throughout. The book concludes with a list of open questions.

Primarily intended for graduate students and young researchers in dynamical systems, this book is also suitable for mathematicians from other fields with an interest in the subject. Prerequisites include familiarity with the content of a standard graduate course in real analysis, along with some understanding of ergodic theory and dynamical systems. Basic knowledge of complex analysis is needed for specific chapters.

Preface.- Part I - Basic Theory: 1 Rotations.- 2 Homeomorphisms of the Circle.- Part II - Diffeomorphisms: 3 Diffeomorphisms: Denjoy Theory.- 4 Smooth Conjugacies to Rotations.- Part III - Multicritical Circle Maps.- 5 Cross-ratios and Distortion Tools.- 6 Topological Classification and the Real Bounds.- 7 Quasisymmetric Rigidity.- 8 Ergodic Aspects.- 9 Orbit Flexibility.- Part IV - Renormalization Theory: 10 Smooth Rigidity and Renormalization..- 11 Quasiconformal Deformations.- 12 Lipschitz Estimates for Renormalization.- 13 Exponential Convergence: the Smooth Case.- 14 Renormalization: Holomorphic Methods.- Epilogue.- Appendices.- Bibliography.

Edson de Faria graduated with a B.Sc. in Physics at the University of São Paulo (USP) and received his Ph.D. in Mathematics from the City University of New York (CUNY), studying with Dennis Sullivan. He held postdoctoral positions at the Institute for Advanced Study in Princeton, ETH-Zürich, IHES and SUNY at Stony-Brook, and later visiting professorships at CUNY and Imperial College London. He is currently a Full Professor at USP.  His research deals primarily with low-dimensional dynamical systems.

Pablo Guarino graduated in Mathematics in Uruguay, where he began to study Dynamical Systems. He then moved to Rio de Janeiro, to work on his Ph.D. at the Institute of Pure and Applied Mathematics (IMPA). He learned one-dimensional dynamics from his advisor, Welington de Melo, and his interest in the area persists to this day. He was a postdoctoral fellow at the Institute of Mathematics and Statistics of the University of São Paulo (IME-USP). His Ph.D. thesis was awarded in 2013 with the CAPES Best Thesis Prize and the Carlos Gutierrez Prize, given by ICMC-USP, Brazil. He is currently an Assistant Professor at Fluminense Federal University (UFF), in the state of Rio de Janeiro.