AbstractLet$0\lt a\lt b\lt 1$and let$T$be the doubling map. Set$ \mathcal{J} (a, b): = \{ x\in [0, 1] : {T}^{n} x\not\in (a, b), n\geq 0\} $. In this paper we completely characterize the holes$(a, b)$for which any of the following scenarios hold: (i)$ \mathcal{J} (a, b)$contains a point$x\in (0, 1)$; (ii)$ \mathcal{J} (a, b)\cap [\delta , 1- \delta ] $is infinite for any fixed$\delta \gt 0$; (iii)$ \mathcal{J} (a, b)$is uncountable of zero Hausdorff dimension; (iv)$ \mathcal{J} (a, b)$is of

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... tive Hausdorff dimension. In particular, we show that (iv) is always the case if$$\begin{eqnarray*}b- a\lt \frac{1}{4} { \mathop{\prod }\nolimits}_{n= 1}^{\infty } (1- {2}^{- {2}^{n} } )\approx 0. 175\hspace{0.167em} 092\end{eqnarray*}$$and that this bound is sharp. As a corollary, we give a full description of first- and second-order critical holes introduced by N. Sidorov [Supercritical holes for the doubling map.Preprint, seehttp://arxiv.org/abs/1204.1920] 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.