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Patterson-Sullivan distributions of finite regular graphs

Christian Arends, Guendalina Palmirotta

Abstract

On finite regular graphs, we construct Patterson-Sullivan distributions associated with eigenfunctions of the discrete Laplace operator via their boundary values on the phase space. These distributions are closely related to Wigner distributions defined via a pseudo-differential calculus on graphs, which appear naturally in the study of quantum chaos. Using a pairing formula, we prove that Patterson-Sullivan distributions are also related to invariant Ruelle distributions arising from the transfer operator of the geodesic flow on the shift space. Both relationships provide discrete analogues of results for compact hyperbolic surfaces obtained by Anantharaman-Zelditch and by Guillarmou-Hilgert-Weich.

Patterson-Sullivan distributions of finite regular graphs

Abstract

On finite regular graphs, we construct Patterson-Sullivan distributions associated with eigenfunctions of the discrete Laplace operator via their boundary values on the phase space. These distributions are closely related to Wigner distributions defined via a pseudo-differential calculus on graphs, which appear naturally in the study of quantum chaos. Using a pairing formula, we prove that Patterson-Sullivan distributions are also related to invariant Ruelle distributions arising from the transfer operator of the geodesic flow on the shift space. Both relationships provide discrete analogues of results for compact hyperbolic surfaces obtained by Anantharaman-Zelditch and by Guillarmou-Hilgert-Weich.
Paper Structure (29 sections, 24 theorems, 169 equations, 3 figures)

This paper contains 29 sections, 24 theorems, 169 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Harmonic analysis on finite regular graphs
  3. Boundary at infinity of a graph and its topology
  4. Horocycles and regular graphs as homogeneous spaces
  5. The horocycle bracket
  6. Homogeneous spaces
  7. Open cells and measures
  8. Normalization of measures
  9. Boundary values and Poisson transforms
  10. Poisson transforms for trees
  11. Regularity of boundary values
  12. From trees to finite regular graphs
  13. Phase spaces
  14. Fundamental domains and cutoff functions
  15. $\Gamma$-invariant boundary values
  16. ...and 14 more sections

Key Result

Lemma 2.1

Let $g, g' \in G,\ x \in \mathfrak{X},\ k \in K$ and $\omega \in \Omega$. Then

Figures (3)

  • Figure 1: Existence of a lcfd-function $\Xi$ by means of an example.
  • Figure 2: Illustration of the set $S_1$: In this case, $x$ together with the orange geodesic is not part of $S_1$, since the distance $d(x, ]\omega , \omega '[)$ is $2$. On the other hand, $x$ together with the blue geodesic is part of $S_1$, as here $d(x, ]\omega , \omega '[)$ is 1.
  • Figure 3: The gray line represents the horocycle $H_{\omega_+}(o)$. All the green vertices $y$ have $d(o,y)=6$, the orange vertices have $d(o,y)<6$.

Theorems & Definitions (70)

  • Remark 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.2
  • ...and 60 more