Analysis reveals self-homeomorphisms are isotopic to partially pseudo-Anosov maps in irreducible 3-manifolds, suggesting deeper geometric connections.
We determine which closed orientable $3$-manifolds M admit a self-homeomorphism restricting to a pseudo-Anosov map on an incompressible subsurface Σ, which we call a pseudo-Anosov surface. When M is irreducible, we show that the self-homeomorphism of M is isotopic rel Σ 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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Manning et al. (2025) studied this question.
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