Abstract: We justify the averaging principle for slow fast systems in the case where the frozen motion is ergodic on almost every level set. We then discuss the rate of convergence in the important case where the fast motion is quasiperiodic
Abstract: We consider orbits in the planar circular restricted 3-body problem containing repeated near-collisions with the smaller primary. A theorem of Font, Nunes, and Simó states that there exists such an orbit shadowing any sequence of collision orbits satisfying certain conditions. This is proven by estimating the Poincaré map from one near-collision to the next and showing that it is a horseshoe map. It follows that there is an invariant hyperbolic set on which this map is conjugated to a subshift on an alphabet of collision orbits.
Abstract: Motivated by the question of M. Kac: Can you hear the shape of a drum? We introduce the Laplace spectrum and the length spectrum for convex planar domains. After showing a connection between the two spectra we shall discuss problems of spectral rigidity, namely, if it is possible to smoothly deform a domain without changing its spectra.
Abstract: In a work in progress with Artur Avila and Mikhail Lyubich, we show that there are maps in the complex Hénon family with a stable homoclinic tangency. We will first review Newhouse’s phenomenon for real surface dynamics and related results. Then we explain how this phenomenon is related to stable intersections between Cantor sets. Then we will explain a new mechanism on the stable intersections between two dynamical Cantor sets generated by two classes of conformal IFSs on the complex plane.
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