Hausdorff dimension of frequency sets in beta-expansions

Abstract

We obtain an exact formula of the Hausdorff dimension of the level sets in beta-expansions for pseudo-golden ratios by using a variation formula. Before this, we prove that the Hausdorff dimension of an arbitrary set in the shift space is equal to its projection in [0, 1], and we clarify that for calculating the Hausdorff dimension of the level sets, one only needs to focus on the Markov measures when β ∈ (1, 2) and the beta-expansion of 1 is finite.

Random walks associated to beta-shifts

“…by definition we know that µ is a k-step Markov measure. , it suffices to prove that λ is a unique (m + 1)-step Markov measure (see [10,11] for definition) in M σ (Σ β ) taking value a on [0].…”

“…We employ the random walks on Σ * β above to extend the work [9] to more general numbers and obtain the following extension: Theorem 1.2. For 1 < β < 2 such that ε(1, β) = 10 m 10 ∞ with some m ∈ {0, 1, 2, 3, • • • }, the following exact formulas of the Hausdorff dimension of F p , F p and F p hold: For calculating the Hausdorff dimension of the level set F p , there is a variation formula in [15] says that we only need to calculate the measure-theoretic entropy of T β with respect to the invariant probability Borel measure with maximal entropy taking value p on [0, 1 β ) (see also [11,Proposition 4.2]). The following two examples show that if we assume that β has the form assumed in Theorem 1.2, then m p , the T β -ergodic invariant probability Borel measure we study in Section 4, is a measure with maximal entropy: Example 1.3.…”