Horocycles, shrinking targets and Extremal theory

There is a rich theory in dynamical systems involving the study of "shrinking targets".  Given an ergodic probability measure the famous Birkhoff ergodic theorem shows that typically for every point x the orbit (T^nx) enters a ball of fixed radius  (a "target") infinitely often. However, if  we allow the radius of the ball to shrink as n increases (a "shrinking target") then we can ask about the size of the set of x whose orbits still enter these sets infinitely often.    Extremal theory helps to quantify this behaviour.  Many of the classical examples involve hyperbolic maps T.  However, horocycle flows are a very simple class of (non-hyperbolic) dynamical systems that also give a gentle insight into the world of homogeneous dynamics. Somewhat  surprisingly  analogous extremal theory results can be shown in this context, via a remarkably simple argument.    No prior knowledge of these topics will be assumed and this will be a blackboard talk.   This is joint work with J. Marklof (Bristol)

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There will be a coffee break after the seminar.