Topological dynamics of definable group actions

Abstract

We interpret the basic notions of topological dynamics in the model-theoretic setting, relating them to generic types of definable group actions and their generalizations.

Topological dynamics and definable groups

“…We will be repeating some observations from [8], [9], but hopefully this will help to popularize the nice ideas.…”

“…This relationship has been explored in a series of papers by Newelski, including [8] and [9] which are most relevant to the considerations of this paper. For stable groups, namely groups definable in stable theories, there is a good match, which we will briefly recall below, and the issue is whether this extends to more general contexts.…”

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AbstractWe 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 type space S G (M ), can explain, account for, or give rise to, the quotient G/G 00 , at least for suitable groups in N IP theories. We give a positive answer for measure-stable (or f sg) groups in N IP theories.

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On minimal flows, definably amenable groups, and o-minimality

“…The notion "generic" is well-known in topological dynamics where it goes under the name syndetic, whereas the notion weak generic was introduced in [10] Definition 1.6. (i) A subset Y of X is generic if finitely many G-translates cover X.…”

“…An example was given where the two classes differ and the problem was explicitly raised (Problem 5.4 of [10]) of finding an o-minimal or even just NIP example. This is what we address in the current paper.…”