Factors of Gibbs measures on subshifts (2 of 2)

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 Luísa Borsato, with extensive advice from Brian Marcus and Tom Meyerovitch.

This lecture was given in two parts. The video on this page was given as a follow up to a pre-recorded video.