**Title:** Energies of graph maps and a positive characterization of expanding rational maps

**Speaker:** Kevin Pilgrim

**Speaker Info:** Indiana University

**Brief Description:**

**Special Note**:

**Abstract:**

A rational map $f$ defines a dynamical system on the Riemann sphere; its chaotic locus is called the Julia set, $J(f)$. If $f$ is expanding on $J(f)$, then for all sufficiently small $\epsilon$, the restriction of $f$ to an $\epsilon$-neighborhood of $J(f)$ looks like a map between planar graphs. This leads to a new, positive characterization of expanding rational maps among topological self-branched covers: a topological map is equivalent to a rational map if and only if there exists an associated graph map for which a certain energy is less than one. Ingredients in the proof include novel graph map energies, relating extremal length on surfaces to similar quantities on graphs, standard quasiconformal surgery, and a characterization of when one compact Riemann surface with boundary conformally embeds inside another. The fact that this is a positive criterion suggests a new algorithm for the decidability of rationality among such topological maps.The general program is due to Dylan Thurston, with portions joint with Jeremy Kahn, Dylan Thurston, and myself.

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