Let $0<a<b<1$ and let $T$ be the doubling map. Set $\mathcal J(a,b):=\{ x\in[0,1] : T^nx\notin (a,b), n\ge0\}$. In this paper we completely characterize the holes $(a,b)$ for which any of the following scenarios holds: { enumerate} $\mathcal J(a,b)$ contains a point $x\in(0,1)$; $\mathcal J(a,b)\cap [\de,1-\de]$ is infinite for any fixed $\de>0$; $\mathcal J(a,b)$ is uncountable of zero Hausdorff dimension; $\mathcal J(a,b)$ is of positive Hausdorff dimension. { enumerate} In particular, we show that (iv) is always the case if \[ b-a<\frac14\prod_{ n=1}^\infty \bigl(1-2^{ -2^n}\bigr)\approx 0.175092 \] and that this bound is sharp. As a corollary, we give a full description of first and second order critical holes introduced in \cite{ SSC} for the doubling map. Furthermore, we show that our model yields a continuum of "routes to chaos" via arbitrary sequences of products of natural numbers, thus generalizing the standard route to chaos via period doubling.
Categories: math.DS