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Minimum dilatations of pseudo-Anosov braids

Chi Cheuk Tsang, Xiangzhuo Zeng

TL;DR

This work determines the asymptotic behavior of the minimal dilatation δ_n for pseudo-Anosov braids with n strands, proving δ_n^n converges to (2+√3)^2 and identifying n for which HK06 and Ven08 constructions realize the minimum. The authors introduce floral train tracks to tightly control the associated curve complex growth rate, connecting it to clique polynomials and allowing explicit lower bounds on dilatations via short periodic data. They prove a general lower bound λ(f) ≥ min{14.5^{1/n}, δ̄_n^n} for pseudo-Anosov braids and show, for large n, the minimum is attained by known examples, thereby solving the minimum dilatation problem on the n-punctured sphere except for a finite set of n. The framework also yields corollaries about hyperelliptic maps and sphere punctures, and provides a computational approach (via a notebook) to handle remaining cases. The results advance our understanding of extremal pseudo-Anosov dynamics and offer a concrete path toward full classification of least-dilatation braids and related mapping class problems.

Abstract

We determine the minimum dilatation $δ_n$ among pseudo-Anosov braids with $n$ strands, for large enough values of $n$. These are the dilatations attained by the examples of Hironaka-Kin and Venzke, and they satisfy $\lim_{n \to \infty} δ_n^n = (2+\sqrt{3})^2 \approx 13.928$. Together with previous work, this result confirms conjectures by Kin-Takasawa and Venzke, and solves the minimum dilatation problem on the $n$-punctured sphere, for all but $6$ values of $n$.

Minimum dilatations of pseudo-Anosov braids

TL;DR

This work determines the asymptotic behavior of the minimal dilatation δ_n for pseudo-Anosov braids with n strands, proving δ_n^n converges to (2+√3)^2 and identifying n for which HK06 and Ven08 constructions realize the minimum. The authors introduce floral train tracks to tightly control the associated curve complex growth rate, connecting it to clique polynomials and allowing explicit lower bounds on dilatations via short periodic data. They prove a general lower bound λ(f) ≥ min{14.5^{1/n}, δ̄_n^n} for pseudo-Anosov braids and show, for large n, the minimum is attained by known examples, thereby solving the minimum dilatation problem on the n-punctured sphere except for a finite set of n. The framework also yields corollaries about hyperelliptic maps and sphere punctures, and provides a computational approach (via a notebook) to handle remaining cases. The results advance our understanding of extremal pseudo-Anosov dynamics and offer a concrete path toward full classification of least-dilatation braids and related mapping class problems.

Abstract

We determine the minimum dilatation among pseudo-Anosov braids with strands, for large enough values of . These are the dilatations attained by the examples of Hironaka-Kin and Venzke, and they satisfy . Together with previous work, this result confirms conjectures by Kin-Takasawa and Venzke, and solves the minimum dilatation problem on the -punctured sphere, for all but values of .

Paper Structure

This paper contains 50 sections, 90 theorems, 68 equations, 21 figures, 1 table.

Table of Contents

  1. Introduction
  2. Previous work
  3. Train tracks, curve complexes, and clique polynomials
  4. Floral train tracks
  5. Braids that attain the minimum dilatation
  6. Pseudo-Anosov maps
  7. General definitions
  8. Lefschetz fixed point theorem
  9. Pseudo-Anosov braids
  10. Clique polynomials and curve complexes
  11. Clique polynomials
  12. Curve complexes
  13. Standardly embedded train tracks
  14. Train track terminology
  15. (Based) standardly embedded train tracks
  16. ...and 35 more sections

Key Result

Theorem 1.1

Let $f$ be a pseudo-Anosov braid with $n$ strands. Then the dilatation of $f$ satisfies

Figures (21)

  • Figure 1: An example of a floral train track. Here the pistil is in red, the anthers are in yellow, the filaments are in green, and the petals are in blue.
  • Figure 2: Left: An unpunctured 3-pronged singular point. Right: A punctured 1-pronged singular point.
  • Figure 3: Summing the vertices $v_1$ and $v_2$ in a weighted graph $(G,w)$ to get a weighted graph $(H,z)$.
  • Figure 4: Top: A train track. Bottom: The tie neighborhood of a train track. The switch ties are bolded.
  • Figure 5: Top: A homeomorphism that sends the tie neighborhood $N$ into the tie neighborhood $N'$. Bottom: The induced map on the train tracks $\tau \to \tau'$. For clarity, we have drawn the image of $\tau$ lying slightly off $\tau'$.
  • ...and 16 more figures

Theorems & Definitions (174)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 2.1: Poincaré-Hopf
  • Proposition 2.2: FLP79
  • Proposition 2.3
  • proof
  • Theorem 2.4: Lefschetz
  • ...and 164 more