The author studies the case \(X=[0,1]\), and \(Tx=2x\pmod1\). Let \(H\subset X\) be a hole and denote \[ \mathcal J_H(T)=X\backslash \bigcup_{n=0}^\infty T^{-n}H. \]
Definition 1.4. We say that a hole \(H_0\) is supercritical for \(T\) if

(1)

for any hole \(H\) such that \(\overline H_0\subset H\) we have \(\mathcal J_H(T)\subset\text{Fix}(T)\); for any hole \(H\) such that \(\overline H\subset H_0\) we have \(\dim_H(\mathcal J_H(T))>0\).

(2)

The purpose of this article is to completely characterize all supercritical holes.

Theorem 3.13. Each supercritical hole for \(T\) is one of the following:

(1)

\((\alpha,1/2)\) or \((1/2,1-\alpha)\) for any \(\alpha\in[0,1/4]\);

(2)

\((\alpha,\alpha+1/4)\) or \((3/4-\alpha,1-\alpha)\), where \(\alpha\) is one of the following:
\(\bullet\) \(1/3\);
\(\bullet\) \(\pi(01s_\infty(\gamma))\) and \(\gamma<1/2\) irrational;
\(\bullet\) \(\pi(01(s_n)^\infty)\) or \(\pi(01(s_n^{a_n-1}s_{n-2}s_{n-1})^\infty)\), where \((s_k)_{k=-1}^n\) is the sequence of standard words parametrized by an arbitrary \(n\)-tuple \((a_1,\dots,a_n)\in\mathbb N^n\) with \(a_n\geq2\). The binary expansion of \(\alpha\) is an eventually periodic sequence of period \(q\), where \(p/q=[a_1+1,\dots,a_n]\).

Let us explain the notations. Let \(\Sigma=\{0,1\}^{\mathbb N}\) and \(\pi:\Sigma\to[0,1]\) corresponds to a binary expansion, and \(\sigma\) is the shift on \(\Sigma\). For an irrational number \(0<\gamma<1/2\), denote its continued expansion by \([d_1+1,d_2,d_3,\ldots]\). Then define a sequence \(s_{-1}=1\), \(s_0=0\), \(s_{n+1}=s_n^{d_n+1}s_{n-1}\), a word \(s_n\) is called the \(n\)th standard word given by \(\gamma\). Define \(s_\infty(\gamma)=\lim_{n\to\infty}s_n\).