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- W2937431637 abstract "Given $betain(1,2],$ the $beta$-transformation $T_beta: xmapsto beta xpmod 1$ on the circle $[0, 1)$ with a hole $[0, t)$ was investigated by Kalle et al.~(2019). They described the set-valued bifurcation set [ mathcal E_beta:={tin[0, 1): K_beta(t')ne K_beta(t)~forall t'>t}, ] where $K_beta(t):={xin[0, 1): T_beta^n(x)ge t~forall nge 0}$ is the survivor set. In this paper we investigate the dimension bifurcation set [ mathcal B_beta:={tin[0, 1): dim_H K_beta(t')ne dim_H K_beta(t)~forall t'>t}, ] where $dim_H$ denotes the Hausdorff dimension. We show that if $betain(1,2]$ is a multinacci number then the two bifurcation sets $mathcal B_beta$ and $mathcal E_beta$ coincide. Moreover we give a complete characterization of these two sets. As a corollary of our main result we prove that for $beta$ a multinacci number we have $dim_H(mathcal E_betacap[t, 1])=dim_H K_beta(t)$ for any $tin[0, 1)$. This confirms a conjecture of Kalle et al.~for $beta$ a multinacci number." @default.
- W2937431637 created "2019-04-25" @default.
- W2937431637 creator A5056602178 @default.
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- W2937431637 date "2019-04-15" @default.
- W2937431637 modified "2023-09-27" @default.
- W2937431637 title "Two bifurcation sets arising from the beta transformation with a hole at $0$" @default.
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- W2937431637 doi "https://doi.org/10.48550/arxiv.1904.07007" @default.
- W2937431637 hasPublicationYear "2019" @default.
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