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arXiv · 2509.25621

Weak Gibbs measures for the natural extension of $(\kappa/\beta, \beta)$-shifts

Miki Yamashita
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Abstract

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

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Miki Yamashita. 2025-09-30. Weak Gibbs measures for the natural extension of $(\kappa/\beta, \beta)$-shifts. https://arxiv.org/abs/2509.25621

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