Bi-contact plugs and dynamical hyperbolicity

Abstract

We develop a theory of bi-contact plugs in dimension three to construct Anosov flows or, more generally, structurally stable nonsingular flows. For a transversely orientable hyperbolic plug with orientable filling Morse-Smale boundary laminations, we prove that every strongly transverse gluing map can be isotoped through strongly transverse maps to identify a strongly adapted contact form up to sign. The resulting flow is hyperbolic and is Anosov when the quotient is closed. This answers a question of Béguin-Bonatti-Yu under the stated orientability assumptions. The result also extends to gluing nonsingular partially hyperbolic flows. Anosov completion by reflection embeds each such hyperbolic plug in an Anosov flow on its oriented double. Further applications include a classification of structurally generic transversely oriented projectively Anosov flows without saddle periodic orbits, Morse-Smale examples with prescribed saddle count, a generalization of the construction of Bonatti-Bowden-Potrie to give embeddings of (attracting) hyperbolic plugs into (partially hyperbolic) projectively Anosov flows with the same topological entropy, and constructions on doubles and torus bundles with controlled invariant torus dynamics. Finally, every cooriented partially hyperbolic bi-contact plug admits a Liouville structure on its thickening, with Liouville trajectories projecting to positively reparametrized flow lines. Repelling plugs yield four-dimensional Liouville domains, often with chaotic skeletons.