# Giulio Tiozzo

University of Toronto

October 15, 2021 from 15:30 to 16:30 (Montreal/EST time)

Colloquium presented by **Giulio Tiozzo (University of Toronto)**

2021 André Aisenstadt Prize in Mathematics Recipient

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.