15 octobre 2021
Entropy along the Mandelbrot set
15 octobre 2021 de 15 h 30 à 16 h 30 (heure de Montréal/HNE)
Colloque par Giulio Tiozzo (University of Toronto)
Lauréat 2021 du prix de mathématiques André-Aisenstadt
The notion of topological entropy, arising from information theory, is a fundamental tool to understand the complexity of a dynamical system. When the dynamical system varies in a family, the natural question arises of how the entropy changes with the parameter.
In the last decade, W. Thurston introduced these ideas in the context of complex dynamics by defining the "core entropy" of a quadratic polynomials as the entropy of a certain forward-invariant set of the Julia set (the Hubbard tree).
As we shall see, the core entropy is a purely topological/combinatorial quantity which nonetheless captures the richness of the fractal structure of the Mandelbrot set. In particular, we will relate the variation of such a function to the geometry of the Mandelbrot set. We will also prove that the core entropy on the space of polynomials of a given degree varies continuously, answering a question of Thurston.
Finally, we will provide a new interpretation of core entropy in terms of measured laminations, and discuss its finer regularity properties such as its Holder exponent.