Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Path-connectivity of Thick Laminations, and Markov Processes with Thick Limit Sets

Jon Chaika, Sebastian Hensel

Abstract

A lamination $λ$ is $ε$-thick (with respect to a basepoint $X$), if the Teichmüller ray from $X$ in the direction of $λ$ stays in the $ε$-thick part. We show that, for surfaces of high enough genus, any two $ε$-thick laminations can be joined by a path of $δ$-thick laminations. As a consequence, we show that the Morse boundary of the mapping class group is path-connected. Furthermore, we construct a subshift of finite type on the mapping class group, whose limit set consists only of thick laminations and is path-connected.

Path-connectivity of Thick Laminations, and Markov Processes with Thick Limit Sets

Abstract

A lamination is -thick (with respect to a basepoint ), if the Teichmüller ray from in the direction of stays in the -thick part. We show that, for surfaces of high enough genus, any two -thick laminations can be joined by a path of -thick laminations. As a consequence, we show that the Morse boundary of the mapping class group is path-connected. Furthermore, we construct a subshift of finite type on the mapping class group, whose limit set consists only of thick laminations and is path-connected.

Paper Structure

This paper contains 17 sections, 36 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Outline of proof
  3. Outline of paper
  4. Acknowledgements:
  5. Tools
  6. Train Tracks
  7. Curve Graphs
  8. Teichmüller Spaces
  9. Splitting Packages
  10. Building Thick Paths
  11. Morse Boundaries
  12. A Markov Version
  13. Appendix: A type of connected subset that can't be thick
  14. Proof of Theorem \ref{['thm:main']}
  15. Proof of Proposition \ref{['prop:unif sc bound']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Suppose $X_0\in\mathcal{T}(S)$ is basepoint. For every $\epsilon>0$ there is a $\delta > 0$ with the following property. If $\lambda, \eta$ are two laminations which are $\epsilon$--thick relative to $X_0$, then there is a path in $\mathcal{PML}$ connecting $\lambda$ to $\eta$, so that any laminatio

Theorems & Definitions (59)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • ...and 49 more