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Time change rigidity for unipotent flows

Elon Lindenstrauss, Daren Wei

TL;DR

This work studies monotone Kakutani equivalence for unipotent flows on quotients $\mathbf{G}/\Gamma$ of semisimple groups and establishes a dichotomy: either the flow is loosely Kronecker (Ratner invariant zero) or monotone equivalence rigidity holds, forcing an algebraic isomorphism between the ambient groups and a time-change that is cohomologous to a trivial one up to renormalization. The authors develop a comprehensive framework based on two-sided $(\delta,\epsilon,R)$-matching and Kakutani-Bowen balls, derive a main lemma that propagates local matchings to global rigidity, and deploy a detailed renormalization argument together with an SL$_2(\mathbb{R})$ ergodic theorem to obtain convergence to a genuine isomorphism. Key technical innovations include a refined analysis of polynomial divergence for unipotent subgroups, Brudnyi-type measure estimates for exceptional returns, and a careful study of the normal core to localize obstructions. The results generalize and sharpen prior rigidity phenomena for unipotent flows, showing that, beyond the loosely Kronecker case, monotone equivalence implies algebraic isomorphism and a rigid time-change, with implications for the classification of homogeneous-parabolic systems in ergodic theory.

Abstract

We prove a dichotomy regarding the behavior of one-parameter unipotent flows on quotients of semisimple lie groups under time change. We show that if $u^{(1)}_t$ acting on $\mathbf{G}_{1}/Γ_1$ is such a flow it satisfies exactly one of the following: (1) The flow is loosely Kronecker, and hence measurably isomorphic after an appropriate time change to any other loosely Kronecker system. (2) The flow exhibits the following rigid behavior: if the one-parameter unipotent flow $u^{(1)} _ t$ on $\mathbf{G}_1/Γ_1$ is measurably isomorphic after time change to another such flow $u^{(2)} _ t$ on $\mathbf{G}_2/Γ_ 2$, then $\mathbf{G}_1/Γ_1 $ is isomorphic to $\mathbf{G}_2/ Γ_2$ with the isomorphism taking $u^{(1)}_t$ to $u^{(2)}_t$ and moreover the time change is cohomologous to a trivial one up to a renormalization.

Time change rigidity for unipotent flows

TL;DR

This work studies monotone Kakutani equivalence for unipotent flows on quotients of semisimple groups and establishes a dichotomy: either the flow is loosely Kronecker (Ratner invariant zero) or monotone equivalence rigidity holds, forcing an algebraic isomorphism between the ambient groups and a time-change that is cohomologous to a trivial one up to renormalization. The authors develop a comprehensive framework based on two-sided -matching and Kakutani-Bowen balls, derive a main lemma that propagates local matchings to global rigidity, and deploy a detailed renormalization argument together with an SL ergodic theorem to obtain convergence to a genuine isomorphism. Key technical innovations include a refined analysis of polynomial divergence for unipotent subgroups, Brudnyi-type measure estimates for exceptional returns, and a careful study of the normal core to localize obstructions. The results generalize and sharpen prior rigidity phenomena for unipotent flows, showing that, beyond the loosely Kronecker case, monotone equivalence implies algebraic isomorphism and a rigid time-change, with implications for the classification of homogeneous-parabolic systems in ergodic theory.

Abstract

We prove a dichotomy regarding the behavior of one-parameter unipotent flows on quotients of semisimple lie groups under time change. We show that if acting on is such a flow it satisfies exactly one of the following: (1) The flow is loosely Kronecker, and hence measurably isomorphic after an appropriate time change to any other loosely Kronecker system. (2) The flow exhibits the following rigid behavior: if the one-parameter unipotent flow on is measurably isomorphic after time change to another such flow on , then is isomorphic to with the isomorphism taking to and moreover the time change is cohomologous to a trivial one up to a renormalization.

Paper Structure

This paper contains 76 sections, 59 theorems, 559 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview of the argument
  3. Reader's guide
  4. Global and local notations
  5. Notations and preliminaries
  6. --two sides matching
  7. Preliminaries of homogeneous space
  8. Algebraic groups
  9. Metric, measure and adjoint action
  10. $\mathfrak{sl}_2$-triplet and chain basis
  11. Injective radius and matching function
  12. Kakutani-Bowen balls
  13. Auxiliary lemmas for Kakutani-Bowen balls
  14. An SL2(R)-ergodic theorem
  15. Kakutani equivalence in homogeneous spaces
  16. ...and 61 more sections

Key Result

Theorem 1.1

For $i=1,2$, let $\mathbf{G}_i$ be the identity component of a real semisimple linear algebraic group without compact factors, $\Gamma_i<\mathbf{G}_i$ a lattice, $m_i$ the probability measure on $\mathbf{G}_i/\Gamma_i$ induced by Haar measure on $\mathbf{G}_i$, $u_t^{(i)}$ a one-parameter unipoten

Figures (3)

  • Figure 1: Structure of paper
  • Figure 2: Sketch of proof of Proposition \ref{['prop:GeoTri']}
  • Figure :

Theorems & Definitions (118)

  • Theorem 1.1
  • Definition 3.1: cf. ratner1981some*Def. 1
  • Lemma 3.2: Smooth matching function lemma
  • Lemma 3.3
  • Theorem 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • Definition 3.9: Kakutani-Bowen balls
  • ...and 108 more