This book provides an overview of recent advances in the theory of dynamical systems, with a particular emphasis on their connections to other areas of mathematical research, including number theory, geometry, mathematical physics, complex analysis, and celestial mechanics. Compiling the lecture notes from some of the contributions presented at the C.I.M.E. school "Modern Aspects of Dynamical Systems" held in Cetraro in August 2021, the contributions are the following: Homogeneous dynamics and Diophantine problems by Manfred Einsiedler, Effective ergodic theory for translation flow by Giovanni Forni, Integrability and rigidity for convex billiards by Vadim Kaloshin, Holomorphic dynamics by Jasmin Raissy and Exponentially small phenomena and its role in the dynamics by Tere Martinez-Seara.

These notes are suitable for graduate students and young researchers interested in an introduction to some of the modern research areas within the field of dynamical systems.

Provides an overview of some modern aspects of the theory of dynamical systems Written by world's leading experts in the field Gives special emphasis to the connections with other areas of mathematical research

Autorentext

Manfred Einsiedler studied mathematics at the University of Vienna and has been a Professor at the ETH Zürich since 2009. He was an invited speaker at the 2008 European Mathematical Congress in Amsterdam and the 2010 International Congress of Mathematicians in Hyderabad. His primary research area is ergodic theory with connections to number theory. In cooperation with Lindenstrauss and Katok, Einsiedler made significant progress towards the Littlewood conjecture. Thomas Ward studied mathematics at the University of Warwick and is Deputy Vice-Chancellor for student education at the University of Leeds. He works in ergodic theory and number theory, and has written several monographs, including Heights of Polynomials and Entropy in Algebraic Dynamics with Graham Everest and Ergodic Theory: with a view towards Number Theory with Manfred Einsiedler.

Inhalt

• Homogeneous Dynamics and its connection to Diophantine Approximation.- Effective unique ergodicity and weak mixing of translation flows.- Lecture notes on Birkhoff billiards: dynamics, integrability and spectral rigidity.- Introduction to Fatou components in holomorphic dynamics.