- 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.

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.