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Vector-valued horofunction boundaries and Patterson--Sullivan measures

Dongryul M. Kim, Andrew Zimmer

Abstract

In higher rank, there is a well-studied theory of Patterson--Sullivan measures supported on partial flag manifolds. However, establishing the existence and uniqueness of such measures is a difficult question. In this paper, we develop a theory for Patterson--Sullivan measures supported on certain vector-valued horofunction boundaries of the associated symmetric space, where existence is straightforward. We also introduce a notion of shadows for this compactification and establish a shadow lemma. For transverse groups, we prove uniqueness and ergodicity results.

Vector-valued horofunction boundaries and Patterson--Sullivan measures

Abstract

In higher rank, there is a well-studied theory of Patterson--Sullivan measures supported on partial flag manifolds. However, establishing the existence and uniqueness of such measures is a difficult question. In this paper, we develop a theory for Patterson--Sullivan measures supported on certain vector-valued horofunction boundaries of the associated symmetric space, where existence is straightforward. We also introduce a notion of shadows for this compactification and establish a shadow lemma. For transverse groups, we prove uniqueness and ergodicity results.

Paper Structure

This paper contains 22 sections, 28 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Compactifications
  3. Patterson--Sullivan measures
  4. Transverse groups
  5. Outline of Paper
  6. Patterson--Sullivan systems
  7. Notations for semisimple Lie groups
  8. Parabolic subgroups and flag manifolds
  9. Projection to the flag manifold
  10. The partial Iwasawa cocycle
  11. Linear Representations
  12. Irreducible actions
  13. Limit sets and transverse groups
  14. Compactifications and Patterson--Sullivan measures
  15. Shadows and contracting conical limit sets
  16. ...and 7 more sections

Key Result

Proposition 1.2

The space $\overline{X}^{\theta}: = X \sqcup \partial_{\theta} X$ has a topology which makes it a compactification of $X$, that is $\overline{X}^{\theta}$ is a compact metrizable space and the inclusion $X \hookrightarrow \overline{X}^{\theta}$ is a topological embedding with open dense image.

Theorems & Definitions (61)

  • Definition 1.1
  • Proposition 1.2: see Proposition \ref{['prop:compactification']} below
  • Definition 1.3
  • Proposition 1.4: see Proposition \ref{['prop:PS meas exists']} below
  • Proposition 1.5: see Proposition \ref{['prop:embedding of partial flags']} below
  • Theorem 1.6: see Theorem \ref{['thm:uniqueness for transverse']} below
  • Definition 2.1
  • Proposition 2.2: Shadow Lemma, KimZimmer1
  • Theorem 2.3: KimZimmer1
  • Remark 2.4
  • ...and 51 more