Surgered contact flows, hyperbolicity, and orbit growth

This is a joint project with P. Foulon and A. Vaugon.

• Foulon–Dingв–Geiges–Weinstein–Handel–Thurston surgery is a contact surgery on the unit tangent bundle of a surface.
• This forces orbit complexity.
• It produces several contact 3-flows.
• From the fiber flow a flow with quadratic or exponential complexity.
• From the geodesic flow a contact structure whose every Reeb flow has positive entropy.
• If the surgered geodesic flow is hyperbolic, then it has increased orbit growth.
• There are 3-manifolds with two contact flows, one exponentially complex, one polynomially.