PreprintCC BY 4.0arXiv
Entropy and Translation Length in the Ray Graph
Abstract
Let Γ be the mapping class group of the plane minus a Cantor set, acting on the ray graph ℛ, and for γ∈ Γ let τ(γ) denote the translation length of γ on ℛ. We prove that τ(γ) ≤ h(f)/log(2) for every C∞ diffeomorphism f of S2 representing γ, where h denotes topological entropy. The proof falls into two parts: a combinatorial argument bounding distance between two rays in ℛ by the logarithm of the geometric intersection number; and a geometric argument that promotes geometric control (coarse length of iterates of a fixed ray) to combinatorial control (geometric intersection number). We conjecture that the C∞ hypothesis on f can be removed.
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BibTeX
@misc{bavard2026entropy,
title = {{Entropy and Translation Length in the Ray Graph}},
author = {Juliette Bavard and Danny Calegari and Alden Walker},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17879},
eprint = {2609.17879},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17879},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Entropy and Translation Length in the Ray Graph
AU - Juliette Bavard
AU - Danny Calegari
AU - Alden Walker
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17879
UR - https://arxiv.org/abs/2609.17879
AB - Let Γ be the mapping class group of the plane minus a Cantor set, acting on the ray graph ℛ, and for γ∈ Γ let τ(γ) denote the translation length of γ on ℛ. We prove that τ(γ) ≤ h(f)/log(2) for every C^∞ diffeomorphism f of S^2 representing γ, where h denotes topological entropy. The proof falls into two parts: a combinatorial argument bounding distance between two rays in ℛ by the logarithm of the geometric intersection number; and a geometric argument that promotes geometric control (coarse length of iterates of a fixed ray) to combinatorial control (geometric intersection number). We conjecture that the C^∞ hypothesis on f can be removed.
ER -