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 level sets of the quotient of Birkhoff sums is a well-studied topic. In this paper, we consider what we shall call the uniform level sets, which are subsets of the level sets. 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 uniform level sets, whose elements satisfy certain constraints on their subwords in addition. We shall show that for α not on the boundary of the dimension spectrum, these subsets of the uniform α-level set also have the same u-dimension as the α-level set. As an application, we shall perform a multifractal analysis of the Hölder regularity of a Gibbs measure on a one-dimensional manifold.