Constructing pseudo-Anosov maps from integer permutations

John Hubbard; Ahmad Rafiqi; Tom Schang; John Hubbard; Ahmad Rafiqi; Tom Schang

doi:10.3934/math.20261337

AIMS Mathematics

2026, Volume 11, Issue 10: 34280-34302. doi: 10.3934/math.20261337

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Research article

Constructing pseudo-Anosov maps from integer permutations

1.
Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, 14853, New York, USA
2.
Department of Mathematics & Statistics, American University of Sharjah, 26666, Sharjah, UAE
3.
Department of Mathematics, UC Berkeley, 970 Evans Hall, Berkeley, 94720, California, USA

Received: 28 March 2026 Revised: 13 August 2026 Accepted: 12 October 2026 Published: 22 October 2026
MSC : 37E30, 05A05, 37B40

Surface homeomorphisms, particularly pseudo-Anosov maps, are central objects in topology and dynamics. We provide a characterization of orientation-preserving pseudo-Anosov homeomorphisms with orientable foliations that fix all critical trajectories, encoding them as admissible permutations of $ 2g+\nu-1 $ positive integers, where $ g $ is the genus and $ \nu $ the number of singularities. Conversely, from any admissible permutation we construct the surface, measured foliations, and the pseudo-Anosov map. This construction realizes every Abelian differential invariant under some pseudo-Anosov homeomorphism.
- pseudo-Anosov,
- surface homeomorphism,
- permutation,
- entropy,
- algebraic integer
Citation: John Hubbard, Ahmad Rafiqi, Tom Schang. Constructing pseudo-Anosov maps from integer permutations[J]. AIMS Mathematics, 2026, 11(10): 34280-34302. doi: 10.3934/math.20261337

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Abstract

Surface homeomorphisms, particularly pseudo-Anosov maps, are central objects in topology and dynamics. We provide a characterization of orientation-preserving pseudo-Anosov homeomorphisms with orientable foliations that fix all critical trajectories, encoding them as admissible permutations of $ 2g+\nu-1 $ positive integers, where $ g $ is the genus and $ \nu $ the number of singularities. Conversely, from any admissible permutation we construct the surface, measured foliations, and the pseudo-Anosov map. This construction realizes every Abelian differential invariant under some pseudo-Anosov homeomorphism.

References

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