Slow chaos in surface flows
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Slow chaos in surface flows Corinna Ulcigrai1 Received: 22 June 2020 / Accepted: 10 October 2020 © The Author(s) 2020
Abstract Flows on surfaces describe many systems of physical origin and are one of the most fundamental examples of dynamical systems, studied since Poincará. In the last decade, there have been a lot of advances in our understanding of the chaotic properties of smooth areapreserving flows (a class which include locally Hamiltonian flows), thanks to the connection to Teichmueller dynamics and, very recently, to the influence of the work of Marina Ratner in homogeneous dynamics. We motivate and survey some of the recent breakthroughs on their mixing and spectral properties and the mechanisms, such as shearing, on which they are based, which exploit analytic, arithmetic and geometric techniques.
1 Slowly chaotic dynamical systems Deterministic chaos and the butterfly effect Dynamical systems provide mathematical models of systems which evolve in time. Many systems phenomena in our world, from the evolution of the weather to the motion of an electron in a metal, can be described by a dynamical system. While in a model one can include a random component, or external noise, we will restrict ourselves to fully deterministic systems, whose evolution is completely described by a system of pre-determined rules or equations. We will furthermore consider continuous time dynamical systems, namely systems for which the time variable is a real parameter t ∈ R, described by a flow on a space X , namely a 1-parameter group1 ϕR = (ϕt )t∈R of maps ϕR : X → X (diffeomorphisms if X is a smooth manifold). Deterministic dynamical systems often display chaotic features (see Sect. 2 for examples), which make their behaviour as time grows hard to predict. This is a phenomenon known as deterministic chaos. One of the best known features of chaotic behaviour is sensitive
1 Assuming that ϕ = (ϕ ) t t∈R is a group of diffeomorphisms under composition is equivalent to requiring R
that ϕt+s (x) = ϕt (ϕs (x)) for every x ∈ X (or almost every in the measure-preserving set up introduced below) and every t, s ∈ R. The typical example of a flow is given by solutions to differential equations. The precise definition of the type of flows on which we will focus here, namely area-preserving flows on a surface, will be given below.
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Corinna Ulcigrai [email protected] XXI Congresso dell’Unione Matematica Italiana, Pavia, Italy
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C. Ulcigrai
dependence on initial conditions (SDIC for short), a property which was popularized as the butterfly effect. In a system which displays SDIC, a small variation of the initial condition can lead to a macroscopically very different evolution after a long time. In particular, given a flow ϕR : X → X on a metric space (X , d) with SFIC and a point x ∈ X (the initial condition) one can find arbitrarily close initial conditions y ∈ X such that the (forward) trajectories of x and y, namely the orbits (ϕt (x))t≥0 and (ϕt (y))t≥0 drift apart.2
Fast chaos versus slow chaos Dynamical system
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