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abstract dynamical system; pointwise periodic system; symbolic dynamics; $\bold Z^2$-action
Let $T$ be a permutation of an abstract set $X$. In ZFC, we find a necessary and sufficient condition it terms of cardinalities of the $T$-orbits that allows us to topologize $(X,T)$ as a topological dynamical system on a compact Hausdorff space. This extends an early result of H. de Vries concerning compact metric dynamical systems. An analogous result is obtained for ${\bold Z}^2$-actions without periodic points.
Edelstein M.: On the representation of mappings of compact metrizable spaces as restrictions of linear transformations. Canad. J. Math 22 (1970), 372-375. MR 0263040 | Zbl 0195.24604
Iwanik A.: Period structure for pointwise periodic isometries of continua. Acta Univ. Carolin. - Math. Phys. 29 2 (1988), 19-21. MR 0983446 | Zbl 0674.54020
Iwanik A., Janos L., Kowalski Z.: Periods in equicontinuous topological dynamical systems. in: Nonlinear Analysis, Th. M. Rassias Ed., World Scientific Publ. Co. Singapore, 1987, pp.355-365. MR 0934109 | Zbl 0696.54028
Janos L.: Compactification and linearization of abstract dynamical systems. preprint. MR 1489853 | Zbl 0928.54036
Kowalski Z.S.: A characterization of periods in equicontinuous topological dynamical systems. Bull. Polish Ac. Sc. 38 (1990), 121-124. MR 1194254 | Zbl 0769.54044
de Vries H.: Compactification of a set which is mapped onto itself. Bull. Acad. Polon. Sci. 5 (1957), 943-945. MR 0092144 | Zbl 0078.04202
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