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On idempotent measure conjecture and decomposition of invariant measures

Daniel Max Hoffmann, Tomasz Rzepecki

TL;DR

The paper develops a model-theoretic dynamical framework for invariant types and Keisler measures using a definable convolution *-product, focusing on idempotent measures and their decompositions in amenable NIP theories. It proves a weak variant of the Idempotent Measure Conjecture under additional hypotheses and leverages f-generics to obtain a canonical ergodic description in countable amenable NIP theories, showing that invariant measures decompose into components supported on Ellis groups of the minimal left ideal. A key technical engine is the almost pullback map to GalKP(T) via a Borel isomorphism, which allows transfer of Haar measures and a precise description of KP-invariant components. The results extend the definable-group programs of CGK, GHK, and ArtemPierre to automorphism-group actions, providing a unified path to understanding ergodic and idempotent phenomena in the broader automorphism-dynamics setting.

Abstract

We work with the *-product introduced in [GHK25] and f-generic types to describe the minimal ideals of invariant types and to classify ergodic Keisler measures in amenable NIP theories. Moreover, we analyze the situation around the so-called Idempotent Measure Conjecture studied in [CGK24] and [GHK25].

On idempotent measure conjecture and decomposition of invariant measures

TL;DR

The paper develops a model-theoretic dynamical framework for invariant types and Keisler measures using a definable convolution *-product, focusing on idempotent measures and their decompositions in amenable NIP theories. It proves a weak variant of the Idempotent Measure Conjecture under additional hypotheses and leverages f-generics to obtain a canonical ergodic description in countable amenable NIP theories, showing that invariant measures decompose into components supported on Ellis groups of the minimal left ideal. A key technical engine is the almost pullback map to GalKP(T) via a Borel isomorphism, which allows transfer of Haar measures and a precise description of KP-invariant components. The results extend the definable-group programs of CGK, GHK, and ArtemPierre to automorphism-group actions, providing a unified path to understanding ergodic and idempotent phenomena in the broader automorphism-dynamics setting.

Abstract

We work with the *-product introduced in [GHK25] and f-generic types to describe the minimal ideals of invariant types and to classify ergodic Keisler measures in amenable NIP theories. Moreover, we analyze the situation around the so-called Idempotent Measure Conjecture studied in [CGK24] and [GHK25].

Paper Structure

This paper contains 19 sections, 38 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Definable groups versus automorphism groups
  4. Results of this paper
  5. Structure
  6. Acknowledgements
  7. Preliminaries
  8. Idempotent measure conjecture
  9. Supports and stabilizers
  10. Restrictions and summary
  11. F-generics
  12. f-generics and minimal ideals
  13. Idempotent measures with f-generics
  14. Decomposing invariant measures
  15. Almost pullback
  16. ...and 4 more sections

Key Result

Lemma 2.4

If $\mu,\nu\in\mathfrak{M}^{\mathrm{inv}}_{\bar{n}}(\mathfrak{C},M)$ are Borel $M$-definable, then $\mu\ast\nu\in\mathfrak{M}^{\mathrm{inv}}_{\bar{n}}(\mathfrak{C},M)$.

Theorems & Definitions (101)

  • Conjecture 1.1
  • Conjecture 1.2: Idempotent Measure Conjecture
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6
  • proof
  • ...and 91 more