Speaker: DAVID SIMMONS (UNT)

Title: Theromodynamic Formalism for Meromorphic Maps on Complex Manifolds: Existence and Uniqueness of Equilibria for Finite-to-One Meromorphic Endomorphisms of Compact Kähler Manifolds with Large Topological Degree, Part 1

Abstract: Let X be a compact Kähler manifold of dimension k, and let T : X - - → X be a dominant meromorphic map whose topological degree d_k is strictly greater than its (k - 1)th dynamical degree d_k-1 (we say that T has large topological degree). It is known that if X is projective and if either
k ≤ 3 or X admits a transitive automorphism group, then T possesses a unique measure of maximal entropy. We extend this result to all compact Kähler manifolds, and eliminate the other assumption on X. Next we study existence and uniqueness of equilibria for a Hölder continuous potential function ᶲ : X → R. To do this we introduce the assumptions of flatness and regularity of T. We prove that if T is finite-to-one and regular, then there exists K > 0 so that if ᶲ : X → R is Holder continuous and satisfies max(ᶲ) - min(ᶲ) < K, then ᶲ and T have a unique equilibrium state.