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Title: The specification approach to equilibrium states for parabolic rational maps

Abstract: We develop the specification and orbit-decomposition approach to equilibrium states in the setting of parabolic rational maps; that is, for analytic endomorphisms of the Riemann sphere with rationally indifferent periodic points and no critical points in their Julia set.  Our result extends the well-known results on uniqueness of equilibrium states in this setting, notably the results of Denker, Przytycki and Urbanski. We give a pressure gap condition which is sharp in the class of potentials we consider.  In the family of geometric potentials, our approach gives a simple new proof of uniqueness of equilibrium states up to the phase transition that occurs at the Hausdorff dimension of the Julia set. To the best of our knowledge, this is the first time that the specification approach to equilibrium states has been developed in a complex dynamics setting. We anticipate that this approach will be suitable for application to other classes of maps in complex dynamics, and our analysis is intended to lay the groundwork for further progress. The talk will serve as an accessible introduction to the specification/orbit-decomposition approach to thermodynamic formalism and will include pictures of Julia sets. This is joint work with Katelynn Huneycutt.