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