Journal of Modern Dynamics (JMD)

Forty years of unimodal dynamics: On the occasion of Artur Avila winning the Brin Prize

Pages: 183 - 203, Issue 2, April 2012      doi:10.3934/jmd.2012.6.183

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Mikhail Lyubich - Mathematics Department, Stony Brook University, Stony Brook, NY, 11794-3651, USA Government (email)

Abstract: The field of one-dimensional dynamics, real and complex, emerged from obscurity in the 1970s and has been intensely explored ever since. It combines the depth and complexity of chaotic phenomena with a chance to fully understand it in probabilistic terms: to describe the dynamics of typical orbits for typical maps. It also revealed fascinating universality features that had never been noticed before. The interplay between real and complex worlds illuminated by beautiful pictures of fractal structures adds special charm to the field. By now, we have reached a full probabilistic understanding of real analytic unimodal dynamics, and Artur Avila has been the key player in the final stage of the story (which roughly started with the new century). To put his work into perspective, we will begin with an overview of the main events in the field from the 1970s up to the end of the last century. Then we will describe Avila's work on unimodal dynamics that effectively closed up the field. We will finish by describing his results in the closely related direction, the geometry of Feigenbaum Julia sets, including a recent construction of a new class of Julia sets of positive area.

Keywords:  Unimodal maps, hyperbolicity, regular and stochastic dynamics, renormalization, quadratic-like maps, hybrid lamination, Julia sets, Mandelbrot set.
Mathematics Subject Classification:  Primary: 37E05 37E20 37F10 37F35 37F45 37F50; Secondary: 37D20 37D25 37D45 37C40 37C45 37C70 37C75 30C62.

Available Online: August 2012.