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Parabolic dynamics and renormalization: an introduction 4/6

By Corinna Ulcigrai

Appears in collection : Hadamard Lectures 2022 - Corinna Ulcigrai - Parabolic dynamics and renormalization: an introduction

Parabolic dynamical systems are mathematical models of the many phenomena which display a "slow" form of chaotic evolution, in the sense that nearby trajectories diverge polynomially in time. In contrast with hyperbolic and elliptic dynamical systems, there is no general theory which describes parabolic dynamics. In recent years, a lot of progress has been done in understanding the chaotic features of several classes of such systems, as well as in identifying key mechanisms and techniques which play a central role in their study. In these lectures we will give a self-contained introduction to some results on chaotic features of parabolic flows, some classical as well as many very recent. We will in particular discuss: -renormalizable linear flows; -horocycle flows on compact hyperbolic surfaces and their time-changes; -the Heisenberg nilflow, nilflows on nilmanifolds and their time-changes; -smooth area preserving (also known as 'locally Hamiltonian') flows on surfaces. We will define the mathematical objects as well as the dynamical properties we will discuss to keep the lectures accessible to a wide audience and try to highlight throughout the importance of phenomena such as shearing and techniques based on renormalization. Connections between parabolic flows and mathematical physics, spectral theory and Teichmueller dynamics will also be mentioned.

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