Generic Points of shift-Invariant Measures in the Countable Symbolic Space

TL;DR

This paper investigates the size of sets of generic points for shift-invariant measures in countable symbolic spaces, revealing a new variational principle involving convergence exponents and entropy dimensions.

Contribution

It introduces a novel variational principle for the dimension of generic point sets in countable symbolic spaces, extending previous finite-symbol results.

Findings

Dimension characterized by a variational principle involving convergence exponent and entropy dimension

Different from finite-symbol case where convergence exponent is zero

Application to expanding interval dynamical systems

Abstract

We are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving the convergence exponent of the Gibbs measure and the relative entropy dimension of the Gibbs measure with respect to the invariant measure. This variational principle is different from that of the case of finite symbols, where the convergent exponent is zero and is not involved. An application is given to a class of expanding interval dynamical systems.