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arXiv · 2511.22945

On idempotent measure conjecture and decomposition of invariant measures

Daniel Max Hoffmann·Tomasz Rzepecki
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Abstract

We continue the study of the semigroup of global invariant types introduced by Gannon, Hoffmann, and Krupi\'nski and the associated convolution semigroup of invariant Keisler measures. The first part of the paper concerns the Idempotent Measure Conjecture, studied in [CGK24] and [GHK25], which predicts that idempotent fim Keisler measures should be precisely the invariant Haar measures on relatively type-definable subgroups. We identify structural properties underlying all previously known positive instances of the conjecture and prove a conditional version under natural semigroup-theoretic hypotheses. In particular, we show that left simplicity of the support and a local invariance condition are sufficient for the conjectural characterization. The second part studies invariant Keisler measures in amenable NIP theories. We prove that the semigroup of global invariant types has a unique minimal left ideal consisting of f-generic types and identify its Ellis groups. Using normalized Haar measure on the Kim-Pillay group of T, we associate canonically an invariant Keisler measure to every Ellis group. We show that the resulting measures are supported on minimal subflows, prove that the flow (S_m(C), Aut(C)) is hereditarily amenable, establish unique ergodicity of every minimal subflow in the countable case, and characterize all ergodic Aut(C)-invariant Keisler measures as the measures arising from this construction.

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BibTeXRIS

Daniel Max Hoffmann, Tomasz Rzepecki. 2025-11-28. On idempotent measure conjecture and decomposition of invariant measures. https://arxiv.org/abs/2511.22945

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