Subshifts of finite type and matching for intermediate $\beta$-transformations

Abstract

We focus on the relationships between matching and subshift of finite type for intermediate $\beta$-transformations $T_{\beta ,\alpha }(x)=\beta x+\alpha$ ($\mathrm{mod}$ 1), where $x\in [0,1]$ and $(\beta ,\alpha )\in \Delta :=\{(\beta ,\alpha )\in {\mathbb{R}}^2:\beta \in (1,2)\mathrm{and}0<\alpha <2-\beta \}$. We prove that if the kneading space ${\Omega }_{\beta ,\alpha }$ is a subshift of finite type, then $T_{\beta ,\alpha }$ has matching. Moreover, each $(\beta ,\alpha )\in \Delta$ with $T_{\beta ,\alpha }$ has matching corresponds to a matching interval, and there are at most countable different matching intervals on the fiber. Using combinatorial approach, we construct a pair of linearizable periodic kneading invariants and show that, for any $\epsilon >0$ and $(\beta ,\alpha )\in \Delta$ with $T_{\beta ,\alpha }$ has matching, there exists $(\beta ,{\alpha }^{\prime })$ on the fiber with $|\alpha -{\alpha }^{\prime }|<\epsilon$, such that ${\Omega }_{\beta ,{\alpha }^{\prime }}$ is a subshift of finite type. As a result, the set of $(\beta ,\alpha )$ for which ${\Omega }_{\beta ,\alpha }$ is a subshift of finite type is dense on the fiber if and only if the set of $(\beta ,\alpha )$ for which $T_{\beta ,\alpha }$ has matching is dense on the fiber.

arXiv comment19 pp

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Cited by in corpus (2)

• Topological expansive Lorenz maps with a hole at critical point
• Two bifurcation sets of expansive Lorenz maps with a hole at the critical point