Abstract
The Decomposition Problem in the class \(LIP({\mathbb{S}^2})\) is to decompose any bi-Lipschitz map \(f:{\mathbb{S}^2} \to {\mathbb{S}^2}\) as a composition of finitely many maps of arbitrarily small isometric distortion. In this paper, we construct a decomposition for certain bi-Lipschitz maps which spiral around every point of a Cantor set X of Assouad dimension strictly smaller than one. These maps are constructed by considering a collection of Dehn twists on the Riemann surface \({\mathbb{S}^2}\backslash X\). The decomposition is then obtained via a bi-Lipschitz path which simultaneously unwinds these Dehn twists. As part of our construction, we also show that \(X \subset {\mathbb{S}^2}\) is uniformly disconnected if and only if the Riemann surface \({\mathbb{S}^2}\backslash X\) has a pants decomposition whose cuffs have hyperbolic length uniformly bounded above, which may be of independent interest.
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Acknowledgements
The authors are indebted to Vladimir Markovic for suggesting this problem and many discussions thereupon. The authors also would like to thank the referee for the helpful comments that improved the exposition of the manuscript.
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A. F. was supported by a grant from the Simons Foundation, #352034.
V. V. was partially supported by the NSF DMS grants 1952510 and 2154918.
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Fletcher, A.N., Vellis, V. Decomposing multitwists. JAMA 152, 421–469 (2024). https://doi.org/10.1007/s11854-023-0301-4
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DOI: https://doi.org/10.1007/s11854-023-0301-4