# Article

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Keywords:
ultrafilter; $P$-limit point; dynamical system; selective ultrafilter; $P$-point; compact metric
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.
References:
[1] Akin E.: Recurrence in Topological Dynamics. Furstenberg Families and Ellis Actions. The University Series in Mathematics, Plenum Press, New York, 1997. MR 1467479 | Zbl 0919.54033
[2] Arkhangel'skii A.V.: Topological Function Spaces. Mathematics and its Applications (Soviet Series), vol. 78, Kluwer Academic Publishers, Dordrecht, 1992. MR 1144519
[3] Auslander J., Furstenberg H.: Product recurrence and distal points. Trans. Amer. Math. Soc. 343 (1994), 221-232. MR 1170562 | Zbl 0801.54031
[4] Bernstein A.R.: A new kind of compactness for topological spaces. Fund. Math. 66 (1970), 185-193. MR 0251697 | Zbl 0198.55401
[5] Blass A.: Ultrafilters: where topological dynamics = algebra = combinatorics. Topology Proc. 18 (1993), 33-56. MR 1305122 | Zbl 0856.54042
[6] Blass A., Shelah S.: There may be simple $P_{\aleph_1}$- and $P_{\aleph_2}$-points and the Rudin-Keisler ordering may be downward directed. Ann. Pure Appl. Logic 33 (1987), 3 213-243. MR 0879489
[7] Comfort W., Negrepontis S.: The Theory of Ultrafilters. Springer, Berlin, 1974. MR 0396267 | Zbl 0298.02004
[8] Engelking R.: General Topology. Sigma Series in Pure Mathematics, Vol. 6, Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[9] Furstenberg H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, 1981. MR 0603625 | Zbl 0459.28023
[10] Gillman L., Jerison M.: Rings of Continuous Functions. Graduate Texts in Mathematics, No. 43, Springer, New York-Heidelberg, 1976. MR 0407579 | Zbl 0327.46040
[11] Hindman N., Strauss D.: Algebra in the Stone-Čech Compactification. Walter de Gruyter, Berlin, 1998. MR 1642231 | Zbl 0918.22001
[12] Namioka I.: Separate continuity and joint continuity. Pacific J. Math. 51 (1974), 515-531. MR 0370466 | Zbl 0294.54010
[13] Rudin W.: Homogeneity problems in the theory of Čech compactifications. Duke Math. J. 23 (1956), 409-419. MR 0080902 | Zbl 0073.39602
[14] Sanchis M.: Continuous functions on locally pseudocompact groups. Topology Appl. 86 (1998), 5-23. MR 1619340 | Zbl 0922.22001
[15] Whitehead J.H.C.: A note on a theorem due to Borsuk. Bull. Amer. Math. Soc. 54 (1948), 1125-1132. MR 0029503

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