Minimal stretch factors of orientation-reversing fully-punctured pseudo-Anosov maps

We show that the stretch factor $λ(f)$ of an orientation-reversing fully-punctured pseudo-Anosov map $f$ on a finite-type orientable surface $S$, with $-χ(S) \geq 4$ and having at least two puncture orbits, satisfies the inequality $λ(f)^{-χ(S)} \geq σ^2$, where $σ=1+\sqrt{2}$ is the silver ratio. We provide examples showing that this bound is asymptotically sharp. This extends previous results of Hironaka and the third author to orientation-reversing maps.

💡 Research Summary

This paper establishes a fundamental lower bound for the stretch factors of orientation-reversing, fully-punctured pseudo-Anosov maps on finite-type orientable surfaces and provides a detailed analysis of the set of such normalized stretch factors.

Core Findings: The authors prove that for an orientation-reversing, fully-punctured pseudo-Anosov map (f) on a surface (S) with (-\chi(S) \geq 4) and at least two puncture orbits, the normalized stretch factor satisfies (\lambda(f)^{-\chi(S)} \geq \sigma^2), where (\sigma = 1+\sqrt{2}) is the silver ratio. They demonstrate that this bound is asymptotically sharp by constructing explicit examples for surfaces with Euler characteristic (\chi(S_k) = -2k) (for integers (k \geq 2)) whose normalized stretch factors converge to (\sigma^2) as (k) increases.

A significant corollary is that for almost all even Euler characteristics (specifically, for (-2k) where (k \geq 1) and (k \neq 3)), the smallest achievable stretch factor among fully-punctured pseudo-Anosov maps with at least two puncture orbits is realized by an orientation-reversing map, not an orientation-preserving one. This contrasts with the intuition from closed surfaces and reveals a nuanced dependency on orientation.

Furthermore, the paper investigates the spectrum of possible normalized stretch factors. It shows that the set for orientation-reversing, fully-punctured maps contains a dense subset of the interval (