Bi-contact plugs and dynamical hyperbolicity

Abstract

We develop a theory of bi-contact plugs on oriented $3$-manifolds to construct Anosov flows or, more generally, structurally stable nonsingular flows. For a hyperbolic plug with 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 unoriented contact form. The resulting flow is hyperbolic and is Anosov when the quotient is closed. This answers a question of Béguin-Bonatti-Yu. 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 projectively Anosov flows without saddle periodic orbits and Morse-Smale examples with prescribed saddle count. Generalizing the construction of Bonatti-Bowden-Potrie, finite capping embeds any bi-contact plug in a closed projectively Anosov flow with the same topological entropy, preserving partial hyperbolicity for attracting plugs. We also construct flows with controlled invariant torus dynamics on torus bundles and on doubles of plugs with coorientable boundary bi-foliations. 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.

arXiv comment67 pages. 7 figures. All comments are welcomed!