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North-South dynamics of hyperbolic free group automorphisms on the space of currents

Martin Lustig, Caglar Uyanik

Abstract

Let $\varphi$ be a hyperbolic outer automorphism of a non-abelian free group $F_N$ such that $\varphi$ and $\varphi^{-1}$ admit absolute train track representatives. We prove that $\varphi$ acts on the space of projectivized geodesic currents on $F_N$ with generalized uniform North-South dynamics.

North-South dynamics of hyperbolic free group automorphisms on the space of currents

Abstract

Let be a hyperbolic outer automorphism of a non-abelian free group such that and admit absolute train track representatives. We prove that acts on the space of projectivized geodesic currents on with generalized uniform North-South dynamics.

Paper Structure

This paper contains 17 sections, 37 theorems, 122 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. Paths
  5. Graph maps
  6. Markings and topological representatives
  7. Geodesic Currents on Free Groups
  8. North-South dynamics
  9. Train-Tracks
  10. Goodness
  11. Illegal turns and iteration of the train track map
  12. Goodness versus illegal turns
  13. Uniform goodness in the future or the past
  14. Perron Frobenius Theorem for reducible substitutions and automorphisms
  15. Symbolic dynamics, substitutions and quotes from LU1
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\varphi\in\hbox{Out}(F_N)$ be a hyperbolic outer automorphism with the property that both $\varphi$ and $\varphi^{-1}$ admit (absolute) train track representatives. Then $\varphi$ acts on $\mathbb{P}\hbox{Curr}(F_N)$ with "generalized uniform North-South dynamics from $\Delta_-(\varphi)$ to $\D

Figures (2)

  • Figure 1:
  • Figure 2: $a_i$'s are legal ends of maximal bad segments, $\gamma$ is a good (legal) segment

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5: Bounded Cancellation Lemma
  • Remark 2.6
  • Definition 2.7
  • Proposition 2.8
  • Lemma 2.9
  • ...and 73 more