Generically stable and smooth measures in NIP theories

Hrushovski, Ehud; Pillay, Anand; Simon, Pierre

doi:10.1090/s0002-9947-2012-05626-1

Cited by 51 publications

(130 citation statements)

References 19 publications

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“…The following summarizes the situation, where the reader is referred to Proposition 4.2 of [2] for (i) and Theorem 7.7 of [3] and Theorem 4.3 of [4] for (ii), (iii), and (iv).…”

Section: Resultsmentioning

confidence: 99%

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A Note on Generically Stable Measures and fsg Groups

Hrushovski¹,

Pillay²,

Simon³

2012

Notre Dame J. Formal Logic

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We prove (Proposition 2.1) that if µ is a generically stable measure in an N IP theory, and µ(φ(x, b)) = 0 for all b then for some n, µ (n) (∃y(φ(x 1 , y) ∧ .. ∧ φ(x n , y))) = 0. As a consequence we show (Proposition 3.2) that if G is a definable group with f sg in an N IP theory, and X is a definable subset of G then X is generic if and only if every translate of X does not fork over ∅, precisely as in stable groups, answering positively Problem 5.5 from [3].

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“…The following summarizes the situation, where the reader is referred to Proposition 4.2 of [2] for (i) and Theorem 7.7 of [3] and Theorem 4.3 of [4] for (ii), (iii), and (iv).…”

Section: Resultsmentioning

confidence: 99%

“…We briefly recall some definitions and properties of Keisler measures, refer-ring the reader to [4] for more details.…”

Section: Introductionmentioning

confidence: 99%

A Note on Generically Stable Measures and fsg Groups

Hrushovski¹,

Pillay²,

Simon³

2012

Notre Dame J. Formal Logic

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“…For the other direction we will make use of "generic compact domination" from [6], as well as the following result proved in [7] …”

Section: Measure-stable Groupsmentioning

confidence: 99%

Topological dynamics and definable groups

Pillay

2013

J. symb. log.

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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 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. As part of our analysis we show the existence of "externally definable" generics of G(M ) for measurestable groups. We also point out that for G definably amenable (in a N IP 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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“…If µ(x) and λ(y) are both global Aut(M /M 0 )-invariant measures, then so are µ(x) ⊗ λ(y) and λ(y) ⊗ µ(x). Moreover from [5], if at least one of µ(x), λ(y) is generically stable then…”

Section: Preliminariesmentioning

confidence: 99%

“…One direction of the proof of Theorem 1.1 will make heavy use of a special class of generically stable measures, which we call average measures and were introduced in [5]. So we will give the definition again here and record a few facts concerning nonforking products (or amalgams) which will be needed later.…”

Section: Average Measuresmentioning

confidence: 99%