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On the geodesic flow on CAT(0) spaces

Charalampos Charitos, Ioannis Papadoperakis, Georgios Tsapogas

TL;DR

The paper proves that the geodesic flow on certain CAT(0) quotient spaces $X=\widetilde{X}/\Gamma$ is topologically mixing under Standing Assumptions, without requiring finite Bowen–Margulis measure. It extends classical approaches by employing the generalized Busemann function and stable/strong-stable manifolds, and leverages duality and boundary-density results to obtain transitivity and mixing, even when $\Lambda(\Gamma)=\partial\widetilde{X}$ and the Bowen–Margulis measure may be infinite. Concrete CAT(0) examples are constructed, including Euclidean surfaces with conical singularities and glued CAT(0) surfaces, to illustrate the applicability of the results and to demonstrate cases where the Bowen–Margulis measure is infinite. These findings broaden the scope of dynamical conclusions about geodesic flows beyond finite-measure settings and deepen the understanding of mixing phenomena in nonpositively curved geometry.

Abstract

Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen-Margulis' measure finiteness assumption used in recent work of Ricks is removed. We also construct examples of CAT(0) spaces which do not admit finite Bowen-Margulis measure.

On the geodesic flow on CAT(0) spaces

TL;DR

The paper proves that the geodesic flow on certain CAT(0) quotient spaces is topologically mixing under Standing Assumptions, without requiring finite Bowen–Margulis measure. It extends classical approaches by employing the generalized Busemann function and stable/strong-stable manifolds, and leverages duality and boundary-density results to obtain transitivity and mixing, even when and the Bowen–Margulis measure may be infinite. Concrete CAT(0) examples are constructed, including Euclidean surfaces with conical singularities and glued CAT(0) surfaces, to illustrate the applicability of the results and to demonstrate cases where the Bowen–Margulis measure is infinite. These findings broaden the scope of dynamical conclusions about geodesic flows beyond finite-measure settings and deepen the understanding of mixing phenomena in nonpositively curved geometry.

Abstract

Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen-Margulis' measure finiteness assumption used in recent work of Ricks is removed. We also construct examples of CAT(0) spaces which do not admit finite Bowen-Margulis measure.

Paper Structure

This paper contains 9 sections, 31 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Properties of geodesics and geodesic rays
  4. Mixing of the geodesic flow
  5. Topological transitivity
  6. Proof of topological mixing
  7. Examples and Applications
  8. Euclidean Surfaces and their properties
  9. Examples of CAT(0) surfaces

Key Result

Theorem 3

If $f,g:\mathbb{R}\rightarrow \widetilde{X}$ are two geodesics with $f\left( \infty \right)$$=g\left( \infty \right)$ and $f\left( - \infty \right) =g\left( - \infty \right)$ then $f$ and $g$ bound a flat strip, that is, a convex region isometric to the convex hull of two parallel lines in the

Figures (1)

  • Figure 1: The triangle $(x,y,x)$ in $\widetilde{X}$ and the thin triangle $( x_{z} , z_{y} , y_{x} )$ in $\widetilde{X}_{0}$

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Theorem 3: Flat Strip Theorem
  • Definition 4
  • Proposition 5: Prop. 1.7 Chapter II in [Coo]
  • Definition 6
  • Definition 7
  • Theorem 8
  • Proposition 9: Cor. 4.2 and Cor. 6.3 Chapter II in [Coo]
  • Proposition 10: Cor. 5.1, Chapter II in [Coo]
  • ...and 31 more