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Title:
Factors of Gibbs measures on subshifts (1 of 2)

Speaker:
Sophie Macdonald

Abstract:
Classical results of Dobrushin and Lanford-Ruelle establish, in rough terms, that for a local energy function on a subshift without too much long-range order, the translation-invariant Gibbs measures are precisely the equilibrium measures. There are multiple definitions of a Gibbs measure in the literature, which do not always coincide. We will discuss two of these definitions, one introduced by Capocaccia and the other used by Dobrushin-Lanford-Ruelle, and outline a proof (available at [arxiv.org/abs/2003.05532]) that they are equivalent. We will also discuss forthcoming work, in which we show that Gibbsianness is preserved by pushforward through a certain kind of almost invertible factor map. As an application in one dimension, we show that for a sufficiently regular potential, any equilibrium measure on an irreducible sofic shift is Gibbs. As far as we know, this is the first reasonably general result of the Lanford-Ruelle type for a class of subshifts without the topological Markov property. Joint work with LuĂ­sa Borsato, with extensive advice from Brian Marcus and Tom Meyerovitch. This lecture was given in two parts. The video on this page was distributed as a pre-recorded session ahead of a second live lecture.

Link:
https://mathtube.org/lecture/video/factors-gibbs-measures-subshifts-1-2

Workshop:
Mathtube- Pacific Dynamics Seminar