A multifractal analysis for uniform level sets with constraints on word appearance

Abstract

Given a topologically transitive subshift of finite type, the $u$-dimension of the $\alpha$-level set of the quotient of Birkhoff sums is a well-studied topic. In this paper, we consider the uniform $\alpha$-level set, which is a subset of the $\alpha$-level set. We shall show that the $\alpha$-level set and the uniform $\alpha$-level set have the same $u$-dimension. Moreover, we also consider two types of subsets of the $\alpha$-level set, whose elements are sequences satisfying some constraints on their subwords. We shall show that for $\alpha$ not on the boundary of the dimension spectrum, these subsets also have the same $u$-dimension as the $\alpha$-level set. As an application, we shall perform a multifractal analysis of the Hoelder regularity of a Gibbs measure on a one-dimensional manifold.