Weak Gibbs measures for the natural extension of $(1/β, β)$-shifts

TL;DR

This work extends the Pfister–Sullivan framework for weak Gibbs measures from $\alpha=0$ to the positive-$\alpha$ regime by analyzing the natural extension of the $(\tfrac{1}{\beta},\beta)$-shift with $\beta>3$ and $\alpha=1/\beta$. It introduces a combinatorial toolkit based on a labeled graph for $X^{\alpha,\beta}$ and a sequence of word-transformations that modify admissible words while controlling follower structures, enabling precise upper and lower bounds for cylinder measures. The main result gives a necessary-and-sufficient criterion on the growth of $\bar z^{\alpha,\beta}(n)$ (and thus on $\beta$) that determines when any equilibrium measure for a potential with bounded total oscillations is a weak Gibbs measure for $\psi=\varphi-p(\varphi)$, and when it is not. The proof leverages decoupling/specification arguments and distortion control via the graph description to connect cylinder probabilities to the thermodynamic pressure, shedding light on the statistical properties of non-Bernoulli $\beta$-shifts and their natural extensions in the thermodynamic formalism context.

Abstract

In this paper we consider the weak Gibbs measures for $(α, β)$-shifts. In the case of $α=0$, Pfister and Sullivan have given a necessary and sufficient condition on $β$ such that any equilibrium measure for a function of bounded total oscillations is a weak Gibbs measure in the natural extension of a $β$-shift. So it is natural to ask what happens when $α>0$. However, their proof cannot be applied to general $(α, β)$-shifts in a similar way. In this paper we consider the case of $α=1/β$ and give a criterion for the weak Gibbs property of equilibrium measures for $(1/β, β)$-shifts.