The β-transformation with a hole at 0

For the-transformation is defined by. For let be the survivor set of with hole given by In this paper we characterize the bifurcation set of all parameters for which the set-valued function is not locally constant. We show that is a Lebesgue null set of full Hausdorff dimension for all. We prove that for Lebesgue almost every the bifurcation set contains infinitely many isolated points and infinitely many accumulation points arbitrarily close to zero. On the other hand, we show that the set of for which contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for, the bifurcation set of the doubling map. Finally, we give for each a lower and an upper bound for the value such that the Hausdorff dimension of is positive if and only if <![CDATA[t. We show that for all.