Supercritical holes for the doubling map. (English) Zbl 1340.37007
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
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.
- (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 \(p/q=[a_1+1,\dots,a_n]\).
Reviewer: Makoto Mori (Tokyo)