Enveloping semigroups and mappings onto the two-shift

Berg, Kenneth; Gove, David; Haddad, Kamel

doi:10.1090/S0002-9939-98-04185-9

Enveloping semigroups and mappings onto the two-shift
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by Kenneth Berg, David Gove and Kamel Haddad

Proc. Amer. Math. Soc. 126 (1998), 899-905

DOI: https://doi.org/10.1090/S0002-9939-98-04185-9

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Abstract:

Enveloping semigroups of topological actions of semigroups $G$ on compact spaces are studied. For zero dimensional spaces, and under modest conditions on $G$, the enveloping semigroup is shown to be the Stone-Čech compactification if and only if some Cartesian product has the two-shift as a factor. Examples are discussed showing that, unlike in the measure theory case, positive entropy does not imply the existence of such a factor even if the Cartesian product has large entropy.

References

Joseph Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. MR 956049
Mike Boyle, Lower entropy factors of sofic systems, Ergodic Theory Dynam. Systems 3 (1983), no. 4, 541–557. MR 753922, DOI 10.1017/S0143385700002133
Robert Ellis, A semigroup associated with a transformation group, Trans. Amer. Math. Soc. 94 (1960), 272–281. MR 123636, DOI 10.1090/S0002-9947-1960-0123636-3
Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625, DOI 10.1515/9781400855162
S. Glasner and B. Weiss, Interpolation sets for subalgebras of $l^{\infty }(\textbf {Z})$, Israel J. Math. 44 (1983), no. 4, 345–360. MR 710239, DOI 10.1007/BF02761993
Frank Hahn and Yitzhak Katznelson, On the entropy of uniquely ergodic transformations, Trans. Amer. Math. Soc. 126 (1967), 335–360. MR 207959, DOI 10.1090/S0002-9947-1967-0207959-1