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Simple Braids Tend toward Positive Entropy

Luke Robitaille, Minh-Tâm Quang Trinh

Abstract

A simple braid is a positive braid that can be drawn so that any two strands cross at most once. We prove that as $n \to \infty$, the proportion of simple braids on $n$ strands that have positive topological entropy tends toward $100\%$. Notably, such braids are either pseudo-Anosov or reducible with a pseudo-Anosov component. Our proof involves a method of reduction from simple braids to non-simple $3$-strand braids that may be of independent interest.

Simple Braids Tend toward Positive Entropy

Abstract

A simple braid is a positive braid that can be drawn so that any two strands cross at most once. We prove that as , the proportion of simple braids on strands that have positive topological entropy tends toward . Notably, such braids are either pseudo-Anosov or reducible with a pseudo-Anosov component. Our proof involves a method of reduction from simple braids to non-simple -strand braids that may be of independent interest.
Paper Structure (5 sections, 16 theorems, 39 equations)

This paper contains 5 sections, 16 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Topological Entropy
  3. Simple Braids
  4. Permutations with Many Long Cycles
  5. Conclusion

Key Result

Theorem 1

The proportion of simple braids on $n$ strands that have positive topological entropy tends to 100% as $n$ tends to infinity.

Theorems & Definitions (29)

  • Theorem 1
  • Conjecture 2
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 5: Fried, Kolev
  • Corollary 6
  • proof
  • Corollary 7
  • proof
  • ...and 19 more