论文标题
与某些$ n $ expansions的轨道匹配与有限的数字
Matching of orbits of certain $N$-expansions with a finite set of digits
论文作者
论文摘要
在本文中，我们考虑了一类持续的分数扩展：所谓的$ n $扩展，带有有限的数字集，其中$ n \ geq 2 $是整数。这些\ emph {$ n $ - 具有有限数字集}的}在[kl，l]中引入，并在[djkn，s]中进一步研究。对于$ n $固定的，它们是由参数$α\在（0，\ sqrt {n} -1] $中的参数。 $ n \ in \ mathbb n $，$ n \ geq 2 $，为了表明熵在这种高原上是不变的，我们获得了地图$t_α$的基本平面自然扩展同构，同构明确给出了一个名为匹配的属性。
In this paper we consider a class of continued fraction expansions: the so-called $N$-expansions with a finite digit set, where $N\geq 2$ is an integer. These \emph{$N$-expansions with a finite digit set} were introduced in [KL,L], and further studied in [dJKN,S]. For $N$ fixed they are steered by a parameter $α\in (0,\sqrt{N}-1]$. In [KL], for $N=2$ an explicit interval $[A,B]$ was determined, such that for all $α\in [A,B]$ the entropy $h(T_α)$ of the underlying Gauss-map $T_α$ is equal. In this paper we show that for all $N\in \mathbb N$, $N\geq 2$, such plateaux exist. In order to show that the entropy is constant on such plateaux, we obtain the underlying planar natural extension of the maps $T_α$, the $T_α$-invariant measure, ergodicity, and we show that for any two $α,α'$ from the same plateau, the natural extensions are metrically isomorphic, and the isomorphism is given explicitly. The plateaux are found by a property called matching.
