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Transition probability of discrete geodesic flow on the standard non-uniform quotient of $PGL_3$

Sanghoon Kwon

Abstract

We describe the local transition probability of a singular diagonal action on the standard non-uniform quotient of $PGL_3$ associated to the type 1 geodesic flow. As a consequence, we deduce the strongly positive recurrence property of the geodesic flow.

Transition probability of discrete geodesic flow on the standard non-uniform quotient of $PGL_3$

Abstract

We describe the local transition probability of a singular diagonal action on the standard non-uniform quotient of associated to the type 1 geodesic flow. As a consequence, we deduce the strongly positive recurrence property of the geodesic flow.
Paper Structure (3 sections, 5 theorems, 39 equations, 5 figures)

This paper contains 3 sections, 5 theorems, 39 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Reduction to countable Markov Shift
  3. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

The type 1 discrete geodesic flow system $(\Gamma\backslash G/M,T_a)$ is strongly positive recurrent in the sense that

Figures (5)

  • Figure 1: admissible geodesic paths
  • Figure 2: $\Gamma\backslash G/K$
  • Figure 3: Star of a vertex in $\mathcal{B}$
  • Figure 4: Local transition probability
  • Figure 5: Description of $\mathcal{D}$ by directed graph with weights

Theorems & Definitions (10)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • proof
  • Remark 3.4