Ultrafilter-limit points in metric dynamical systems

García-Ferreira, Salvador, and Sanchis, Manuel. "Ultrafilter-limit points in metric dynamical systems." Commentationes Mathematicae Universitatis Carolinae 48.3 (2007): 465-485. <http://eudml.org/doc/250204>.

@article{García2007,
abstract = {Given a free ultrafilter $p$ on $\mathbb \{N\}$ and a space $X$, we say that $x\in X$ is the $p$-limit point of a sequence $(x_n)_\{n\in \mathbb \{N\}\}$ in $X$ (in symbols, $x = p$-$\lim _\{n\rightarrow \infty \}x_n$) if for every neighborhood $V$ of $x$, $\lbrace n\in \mathbb \{N\} : x_n\in V\rbrace \in p$. By using $p$-limit points from a suitable metric space, we characterize the selective ultrafilters on $\mathbb \{N\}$ and the $P$-points of $\mathbb \{N\}^* = \beta (\mathbb \{N\})\setminus \mathbb \{N\}$. In this paper, we only consider dynamical systems $(X,f)$, where $X$ is a compact metric space. For a free ultrafilter $p$ on $\mathbb \{N\}^*$, the function $f^p: X\rightarrow X$ is defined by $f^p(x) = p$-$\lim _\{n\rightarrow \infty \}f^n(x)$ 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, $p\in \mathbb \{N\}^*$ 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 $p\in \mathbb \{N\}^*$. If the family $\lbrace f^\{p+n\} : n\in \mathbb \{N\}\rbrace $ is uniformly equicontinuous at $x\in X$, then $f^\{p+q\}$ is continuous at $x$, for all $q\in \beta (\mathbb \{N\})$. 3. Let us consider the function $F: \mathbb \{N\}^* \times X\rightarrow X$ given by $F(p,x) = f^p(x)$, for every $(p,x)\in \mathbb \{N\}^* \times X$. Then, the following conditions are equivalent. • $f^p$ is continuous on $X$, for every $p\in \mathbb \{N\}^*$. • There is a dense $G_\delta $-subset $D$ of $\mathbb \{N\}^*$ such that $F|_\{D \times X\}$ is continuous. • There is a dense subset $D$ of $\mathbb \{N\}^*$ such that $F|_\{D \times X\}$ is continuous.},
author = {García-Ferreira, Salvador, Sanchis, Manuel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {ultrafilter; $P$-limit point; dynamical system; selective ultrafilter; $P$-point; compact metric; ultrafilter; -point; dynamical system; limit along an ultrafilter},
language = {eng},
number = {3},
pages = {465-485},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Ultrafilter-limit points in metric dynamical systems},
url = {http://eudml.org/doc/250204},
volume = {48},
year = {2007},
}

TY - JOUR
AU - García-Ferreira, Salvador
AU - Sanchis, Manuel
TI - Ultrafilter-limit points in metric dynamical systems
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 3
SP - 465
EP - 485
AB - Given a free ultrafilter $p$ on $\mathbb {N}$ and a space $X$, we say that $x\in X$ is the $p$-limit point of a sequence $(x_n)_{n\in \mathbb {N}}$ in $X$ (in symbols, $x = p$-$\lim _{n\rightarrow \infty }x_n$) if for every neighborhood $V$ of $x$, $\lbrace n\in \mathbb {N} : x_n\in V\rbrace \in p$. By using $p$-limit points from a suitable metric space, we characterize the selective ultrafilters on $\mathbb {N}$ and the $P$-points of $\mathbb {N}^* = \beta (\mathbb {N})\setminus \mathbb {N}$. In this paper, we only consider dynamical systems $(X,f)$, where $X$ is a compact metric space. For a free ultrafilter $p$ on $\mathbb {N}^*$, the function $f^p: X\rightarrow X$ is defined by $f^p(x) = p$-$\lim _{n\rightarrow \infty }f^n(x)$ 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, $p\in \mathbb {N}^*$ 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 $p\in \mathbb {N}^*$. If the family $\lbrace f^{p+n} : n\in \mathbb {N}\rbrace $ is uniformly equicontinuous at $x\in X$, then $f^{p+q}$ is continuous at $x$, for all $q\in \beta (\mathbb {N})$. 3. Let us consider the function $F: \mathbb {N}^* \times X\rightarrow X$ given by $F(p,x) = f^p(x)$, for every $(p,x)\in \mathbb {N}^* \times X$. Then, the following conditions are equivalent. • $f^p$ is continuous on $X$, for every $p\in \mathbb {N}^*$. • There is a dense $G_\delta $-subset $D$ of $\mathbb {N}^*$ such that $F|_{D \times X}$ is continuous. • There is a dense subset $D$ of $\mathbb {N}^*$ such that $F|_{D \times X}$ is continuous.
LA - eng
KW - ultrafilter; $P$-limit point; dynamical system; selective ultrafilter; $P$-point; compact metric; ultrafilter; -point; dynamical system; limit along an ultrafilter
UR - http://eudml.org/doc/250204
ER -