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.

Autorentext

Claudio Bonanno is an associate professor of mathematical physics at the Department of Mathematics of the University of Pisa. He has obtained the PhD in Mathematics at the University of Pisa in 2003, and held positions at the École Poytechnique and the INdAM. His research interests focus on infinite ergodic theory and the applications to continued fraction algorithms. Since November 2020, he has been in charge of the group "DinAmicI" inside the Italian Mathematical Union.

Alfonso Sorrentino is a full professor of mathematical analysis at the Department of Mathematics of the University of Rome Tor Vergata. His research interests focus on the study of Hamiltonian dynamical systems, with expertise spanning nonlinear analysis, differential geometry, and mathematical physics. After obtaining his Ph.D. in mathematics from Princeton University in 2008, he held academic positions at the Fondation des Sciences Mathématiques in Paris (2008-2009) and the University of Cambridge (2009-2012). He has received several awards, including the "Fubini Prize" in 2018, the "Barcelona Dynamical Systems Prize" in 2019, the "Best paper award (Gold medal)" from the International Consortium of Chinese Mathematicians in 2020, and the "Frontiers of Science Award" in 2023.

Corinna Ulcigrai is a professor at Institute for Mathematics of the Universität Zürich, Switzerland. She obtained her PhD in Mathematics at Princeton University (USA) and held positions at the School of Mathematics at the University of Bristol (UK). Ulcigrai's research focusses on the mathematical investigation of chaos, in particular systems which are slowly chaotic or parabolic. For her achievements, Corinna Ulcigrai was awarded several prizes, including the European Mathematical Society prize, the Whitehead Prize, the Leverhulme Prize and the Brin Prize for Dynamical Systems. She was the recipient of an ERC Starting Grant and currently holds a SNSF Consolidator grant. Corinna Ulcigrai was also an invited speaker at the International Congress of Mathematicians in 2022, in the Dynamical Systems session.