The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree, was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.

Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations defining the Julia set.

On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the "other side" of Sullivan's dictionary. Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space. Speaker(s): Giulio Tiozzo (Toronto)