Gaps in N-expansions

TL;DR

This paper studies the structure of gaps in N-expansions for certain continued fraction maps, identifying conditions under which the interval is gapless or contains gaps, with implications for the orbit structure.

Contribution

It provides clear criteria for when the interval $I_eta$ is gapless, especially when it contains at least five cylinder sets, and analyzes the rare cases with gaps for fewer cylinder sets.

Findings

Intervals with at least five cylinder sets are always gapless.

Gaps are rare and occur only in specific configurations with four cylinder sets.

Gaplessness depends on the position of endpoints relative to fixed points.

Abstract

For a natural number $N\geq 2$ and a real $\alpha$ such that $0 < \alpha \leq \sqrt{N}-1$, we define $I_\alpha:=[\alpha,\alpha+1]$ and $I_\alpha^-:=[\alpha,\alpha+1)$ and investigate the continued fraction map $T_\alpha:I_\alpha \to I_\alpha^-$, which is defined as $T_\alpha(x):= N/x-d(x),$ where $d(x):=\left \lfloor N/x -\alpha\right \rfloor$. For all natural $N \geq 7$, for certain values of $\alpha$, open intervals $(a,b) \subset I_\alpha$ exist such that for almost every $x \in I_{\alpha}$ there is an natural number $n_0$ for which $T_\alpha^n(x) \notin (a,b)$ for all $n\geq n_0$. These \emph{gaps} $(a,b)$ are investigated in the square $\Upsilon_\alpha:=I_\alpha \times I_\alpha^-$, where the \emph{orbits} $T_\alpha^k(x), k=0,1,2,\ldots$ of numbers $x \in I_\alpha$ are represented as cobwebs. The squares $\Upsilon_\alpha$ are the union of \emph{fundamental regions}, which are related to the cylinder sets of the map $T_\alpha$, according to the finitely many values of $d$ in $T_\alpha$. In this paper some clear conditions are found under which $I_\alpha$ is gapless. When $I_\alpha$ consists of at least five cylinder sets, it is always gapless. In the case of four cylinder sets there are usually no gaps, except for the rare cases that there is one, very wide gap. Gaplessness in the case of two or three cylinder sets depends on the position of the endpoints of $I_\alpha$ with regard to the fixed points of $I_\alpha$ under $T_\alpha$.