Cocycles over interval exchange transformations and multivalued Hamiltonian flows

TL;DR

This paper constructs ergodic cocycles over special interval exchange transformations and applies these to demonstrate recurrence and ergodicity in certain smooth flows on non-compact manifolds, extending multivalued Hamiltonian flows.

Contribution

It introduces new classes of ergodic cocycles over periodic-type IETs and applies them to analyze recurrence and ergodicity of extended Hamiltonian flows.

Findings

Constructed recurrent ergodic cocycles over IETs of periodic type.

Established recurrence and ergodicity for certain smooth flows on non-compact manifolds.

Extended understanding of multivalued Hamiltonian flows on surfaces.

Abstract

We consider interval exchange transformations of periodic type and construct different classes of recurrent ergodic cocycles of dimension $\geq 1$ over this special class of IETs. Then using Poincar\'e sections we apply this construction to obtain recurrence and ergodicity for some smooth flows on non-compact manifolds which are extensions of multivalued Hamiltonian flows on compact surfaces.