**Title:** The core entropy of quadratic polynomials

**Speaker:** Giulio Tiozzo

**Speaker Info:** Yale University

**Brief Description:**

**Special Note**:

**Abstract:**

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.Recently, W. Thurston has 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 shall see how to relate the variation of such a function to the geometry of the Mandelbrot set. We will also prove that the core entropy of quadratic polynomials varies continuously as a function of the external angle, answering a question of Thurston.

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