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October 20, 2025Open Access

Pseudo-Anosov surfaces and Dynamics in 3-manifolds

JMJason Fox ManningWest Virginia University CNChristoforos NeofytidisUniversity of Cyprus

Key Points

This research shows a strong connection between self-homeomorphisms and pseudo-Anosov maps on surfaces within 3-manifolds.
Findings indicate that irreducible 3-manifolds admit homeomorphisms isotopic to partially pseudo-Anosov ones, expanding existing geometrical understanding.
Approach relies on analyzing the isotopy classes of self-homeomorphisms when restricting to incompressible subsurfaces.
Results imply new insights into partially hyperbolic dynamics relevant to the broader study of manifold topologies.

Abstract

We determine which closed orientable 3-manifolds M admit a self-homeomorphism restricting to a pseudo-Anosov map on an incompressible subsurface, which we call a pseudo-Anosov surface. When M is irreducible, we show that the self-homeomorphism of M is isotopic rel to a "partially pseudo-Anosov" homeomorphism, a notion that we will introduce. This is motivated by the corresponding results for Anosov tori in irreducible 3-manifolds, and the connection to partially hyperbolic diffeomorphisms, obtained by F. Rodriguez-Hertz, J. Rodriguez-Hertz and R. Ures.

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