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On the distribution of the components of multicurves of given type

Viveka Erlandsson, Juan Souto

Abstract

We study the distribution of the individual components of a random multicurve under the action of the mapping class group.

On the distribution of the components of multicurves of given type

Abstract

We study the distribution of the individual components of a random multicurve under the action of the mapping class group.
Paper Structure (11 sections, 19 theorems, 140 equations)

This paper contains 11 sections, 19 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Some background
  3. Currents
  4. Curve counting
  5. $k$-multicurrents
  6. A first convergence result
  7. Subconvergence and disintegration of sublimits
  8. Proof of Theorem \ref{['main']}
  9. Assymptotic distributions of length vectors
  10. Asymptotic distribution of vectors of ratios
  11. Examples

Key Result

Theorem 1.1

For every $k$-multicurve $\vec{\gamma}^o=(\gamma_1,\dots,\gamma_k)$ in $\Sigma$ there is a probability measure $\mathfrak p_{\vec{\gamma}^o}$ on $\Delta_k$ such that for any continuous, positive, homogenous function $\phi:\mathcal{C}(\Sigma)\to\mathbb R_{\geqslant 0}$ on the space of currents on $\S where $d{\bf t}$ stands for the standard Lebesgue measure on $\mathbb R_{\geqslant 0}$. In particul

Theorems & Definitions (45)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark
  • Theorem 1.4
  • Theorem : Erlandsson-Mondello
  • Proposition 2.1
  • Remark
  • Theorem
  • Remark
  • ...and 35 more