On factors of Gibbs measures for almost additive potentials (Q2793109)

A factor map is a continuous, surjective semi-conjugacy between dynamical systems. In particular, the author considers one-sided subshifts \((X, \sigma_X), (Y, \sigma_Y)\) and a factor map \(\pi : X \to Y\). As a factor map, \(\pi \circ \sigma_X = \sigma_Y \circ \pi\). One may consider what relations there are between thermodynamical properties of \((X,\sigma_X)\) and those of \((Y, \sigma_Y)\). The author investigates this question thoroughly for almost-additive potentials (potentials which vary with ``time''). There are related results in [\textit{J. Barral} and \textit{D.-J. Feng}, Asian J. Math. 16, No. 2, 319--352 (2012; Zbl 1261.37016)]. For a fixed potential, see [\textit{M. Pollicott} and \textit{T. Kempton}, Lond. Math. Soc. Lect. Note Ser. 385, 246--257 (2011; Zbl 1342.37011)].NEWLINENEWLINEFrom the abstract: ``Let \((X, \sigma_X), (Y, \sigma_Y)\) be one-sided subshifts and \(\pi :X \to Y\) a factor map. Suppose that \(X\) has the specification property. Let \(\mu\) be a unique invariant Gibbs measure for a sequence of continuous functions \(\mathcal{F} = \{f_n\}_{n=1}^\infty\) on \(X\), which is an almost additive potential with bounded variation. We show that \(\pi \mu\) is a unique invariant Gibbs measure for a sequence of continuous functions \(\mathcal{G} = \{g_n\}_{n=1}^\infty\) on \(Y\). When \((X, \sigma_X)\) is a full shift, we characterize \(\mathcal{G}\) and \(\pi \mu\) by using relative pressure. This \(\mathcal{G}\) is a generalization of a continuous function found by Pollicott and Kempton in their work on factors of Gibbs measures for continuous functions. We also consider the following question: given a unique invariant Gibbs measure \(\nu\) for a sequence of continuous functions \(\mathcal{F}_2\) on \(Y\), can we find an invariant Gibbs measure \(\mu\) for a sequence of continuous functions \(\mathcal{F}_1\) on \(X\) such that \(\pi \mu = \nu\)? We show that such a measure exists under a certain condition. In particular, if \((X, \sigma_X)\) is a full shift and \(\nu\) is a unique invariant Gibbs measure for a function in the Bowen class, then there exists a preimage \(\mu\) of \(\nu\) which is a unique invariant Gibbs measure for a function in the Bowen class.''