Bowen Lemma in the Countable Symbolic Space

Abstract

We consider the sets of quasi-regular points in the countable symbolic space. We measure the sizes of the sets by Billingsley-Hausdorff dimension defined by Gibbs measures. It is shown that the dimensions of those sets, always bounded from below by the convergence exponent of the Gibbs measure, are given by a variational principle, which generalizes Li and Ma’s result and Bowen’s result.

Keywords

• measure theoretic entropy
• Gibbs measure
• Hausdorff dimension
• symbolic dynamical system
• Bowen Lemma
• quasi-regular point