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Tuesday, August 27, 2019

**Abstract:** I will give an overview of work carried out together with T. Bedford, M. Urbanski, P. Arnoux, and M. Talet. We consider the result of zooming toward a point of a geometric object embedded in an ambient Euclidean space. For a smooth submanifold this just converges to the tangent space at the point, a fixed point for the flow. But for irregular objects like fractal sets the scenery at small scales keeps changing. The collection of the asymptotic limiting objects can be thought of as the tangent bundle of the fractal, these ``scenes'' are related to Furstenberg's ``microsets''. This collection of sets is acted upon by the flow of zooming exponentially fast toward small scales. A mathematical challenge, then, is to precisely define this ``scenery flow'' and to study its dynamical properties. As we sketch, for the limit set of a Kleinian group, the scenery flow is isomorphic to the geodesic frame flow of the associated hyperbolic three-manifold. Now as Sullivan proved, this flow entropy equals the Hausdorff dimension of the limit set. On the other hand, Bowen showed the Hausdorff dimension of the limit set is given in terms of a zero of the pressure function for log of the derivative. This formula for dimension was extended by Ruelle to hyperbolic Julia sets. With Bedford and Urbanski, we constructed the scenery flow for hyperbolic Julia sets, leading to a unification of Bowen's formula with Sullivan's: in both cases, "Hausdorff dimension equals scenery flow entropy". In this sense the scenery flow thus provides an analogue for Julia sets of the geodesic flow. These ideas extend to other situations. In work with M. Talet and P. Arnoux we have studied the scenery flow for Brownian motion paths, and for the nested tilings associated with circle rotations and interval exchange transformations and defined by renormalization. We sketch these ideas, indicating the methods of proof.