Ultrafilter-limit points in metric dynamical systems

García-Ferreira, S. ; Sanchis, M.

Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 3, pp. 465-485

Résumé

Given a free ultrafilter $p$ on $\Bbb N$ and a space $X$, we say that $x\in X$ is the $p$-limit point of a sequence $(x_n)_{n\in \Bbb N}$ in $X$ (in symbols, $x = p$-$\lim_{n\to \infty}x_n$) if for every neighborhood $V$ of $x$, $\{n\in \Bbb N : x_n\in V\}\in p$. By using $p$-limit points from a suitable metric space, we characterize the selective ultrafilters on $\Bbb N$ and the $P$-points of $\Bbb N^* = \beta (\Bbb N)\setminus \Bbb N$. In this paper, we only consider dynamical systems $(X,f)$, where $X$ is a compact metric space. For a free ultrafilter $p$ on $\Bbb N^*$, the function $f^p: X\to X$ is defined by $f^p(x) = p$-$\lim_{n\to \infty}f^n(x)$ for each $x\in X$. These functions are not continuous in general. For a dynamical system $(X,f)$, where $X$ is a compact metric space, the following statements are shown: 1. If $X$ is countable, $p\in \Bbb N^*$ is a $P$-point and $f^p$ is continuous at $x\in X$, then there is $A\in p$ such that $f^q$ is continuous at $x$, for every $q\in A^*$. 2. Let $p\in \Bbb N^*$. If the family $\{f^{p+n} : n\in \Bbb N\}$ is uniformly equicontinuous at $x\in X$, then $f^{p+q}$ is continuous at $x$, for all $q\in \beta (\Bbb N)$. 3. Let us consider the function $F: \Bbb N^* \times X\to X$ given by $F(p,x) = f^p(x)$, for every $(p,x)\in \Bbb N^* \times X$. Then, the following conditions are equivalent. • $f^p$ is continuous on $X$, for every $p\in \Bbb N^*$. • There is a dense $G_\delta$-subset $D$ of $\Bbb N^*$ such that $F|_{D \times X}$ is continuous. • There is a dense subset $D$ of $\Bbb N^*$ such that $F|_{D \times X}$ is continuous.

Détail

Classification : 22A99, 54C20, 54D80, 54G20, 54H11, 54H20
Keywords: ultrafilter; $P$-limit point; dynamical system; selective ultrafilter; $P$-point; compact metric