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Dimension spectrum of digit frequency sets for beta-expansions
For any beta-shift $(X_β,σ)$ on two symbols, i.e., the symbolic coding of the beta-map for $1<β\leq2$, we give an exact formula for the Hausdorff dimension $\dim_{H} Λ_{α(t)}$ as a function of $t\in\mathbb{R}$, where $Λ_α$ denotes the frequency set of the digit $1$ defined by \[Λ_α=\Biggl\{(x_i)_{i=
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