The [STIX]x1D6FD-transformation with a hole at 0. (September 2020) - KALLE, CHARLENE; KONG, DERONG; LANGEVELD, NIELS; LI, WENXIA
Abstract : For [STIX]x1D6FD(1, 2] the [STIX]x1D6FD -transformation T_[STIX]x1D6FD:[0, 1)→ [0, 1) is defined by T_[STIX]x1D6FD(x)=[STIX]x1D6FDx0.6em(mod0.2em1) . For t[0, 1) let K_[STIX]x1D6FD(t) be the survivor set of T_[STIX]x1D6FD with hole (0, t) given by eqnarrayK_[STIX]x1D6FD(t):={x[0, 1):T_[STIX]x1D6FD^n(x)(0, t) for all n≥ 0}.eqnarray In this paper we characterize the bifurcation set E_[STIX]x1D6FD of all parameters t[0, 1) for which the set-valued function tK_[STIX]x1D6FD(t) is not locally constant. We show that E_[STIX]x1D6FD is a Lebesgue null set of full Hausdorff dimension for all [STIX]x1D6FD(1, 2) . We prove that for Lebesgue almost every [STIX]x1D6FD(1, 2) the bifurcation set E_[STIX]x1D6FD contains infinitely many isolated points and infinitely many accumulation points arbitrarily close to zero. On the other hand, we show that the set of [STIX]x1D6FD(1, 2) for which E_[STIX]x1D6FD contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for E_2, the bifurcation set of the doubling map. Finally, we give for each [STIX]x1D6FD(1, 2) a lower and an upper bound for the value [STIX]x1D70F_[STIX]x1D6FD such that the Hausdorff dimension of K_[STIX]x1D6FD(t) is positive if and only if t<[STIX]x1D70F_[STIX]x1D6FD . We show that [STIX]x1D70F_[STIX]x1D6FD≤ 1-(1/[STIX]x1D6FD) for all [STIX]x1D6FD(1, 2) .