Journal of Symbolic Logic

Topological dynamics and definable groups

Anand Pillay

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We give a commentary on Newelski's suggestion or conjecture [8] that topological dynamics, in the sense of Ellis [3], applied to the action of a definable group $G(M)$ on its “external type space” $S_{G,\textit{ext}}(M)$, can explain, account for, or give rise to, the quotient $G/G^{00}$, at least for suitable groups in NIP theories. We give a positive answer for measure-stable (or $fsg$) groups in NIP theories. As part of our analysis we show the existence of “externally definable” generics of $G(M)$ for measure-stable groups. We also point out that for $G$ definably amenable (in a NIP theory) $G/G^{00}$ can be recovered, via the Ellis theory, from a natural Ellis semigroup structure on the space of global $f$-generic types.

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J. Symbolic Logic, Volume 78, Issue 2 (2013), 657-666.

First available in Project Euclid: 15 May 2013

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Pillay, Anand. Topological dynamics and definable groups. J. Symbolic Logic 78 (2013), no. 2, 657--666. doi:10.2178/jsl.7802170.

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