| Title:
|
Ultrafilter-limit points in metric dynamical systems (English) |
| Author:
|
García-Ferreira, S. |
| Author:
|
Sanchis, M. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
48 |
| Issue:
|
3 |
| Year:
|
2007 |
| Pages:
|
465-485 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
ultrafilter |
| Keyword:
|
$P$-limit point |
| Keyword:
|
dynamical system |
| Keyword:
|
selective ultrafilter |
| Keyword:
|
$P$-point |
| Keyword:
|
compact metric |
| MSC:
|
22A99 |
| MSC:
|
54C20 |
| MSC:
|
54D80 |
| MSC:
|
54G20 |
| MSC:
|
54H11 |
| MSC:
|
54H20 |
| idZBL:
|
Zbl 1199.54194 |
| idMR:
|
MR2374128 |
| . |
| Date available:
|
2009-05-05T17:04:11Z |
| Last updated:
|
2012-05-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119673 |
| . |
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| . |
