# Ultrafilter-limit points in metric dynamical systems

• Volume: 48, Issue: 3, page 465-485
• ISSN: 0010-2628

top Access to full text Full (PDF) Access to full text

## Abstract

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

## Citations in EuDML Documents

top

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.