# Flows with Lyapunov one-forms and a generalization of Farber's theorem on homoclinic cycles

Latschev, J (2004). Flows with Lyapunov one-forms and a generalization of Farber's theorem on homoclinic cycles. International Mathematics Research Notices, (5):239-247.

## Abstract

We prove that if a flow on a compact manifold admits a Lyapunov one-form with small zero set Y, then there must exist a generalized homoclinic cycle, that is, a cyclically ordered chain of orbits outside Y such that for every consecutive pair the forward limit set of one and the backward limit set of the next are both contained in the same connected component of Y. The smallness of the zero set is measured in terms of a category-type invariant associated to the cohomology class of the form that was recently introduced by Farber (2002).

We prove that if a flow on a compact manifold admits a Lyapunov one-form with small zero set Y, then there must exist a generalized homoclinic cycle, that is, a cyclically ordered chain of orbits outside Y such that for every consecutive pair the forward limit set of one and the backward limit set of the next are both contained in the same connected component of Y. The smallness of the zero set is measured in terms of a category-type invariant associated to the cohomology class of the form that was recently introduced by Farber (2002).

## Citations

4 citations in Web of Science®
4 citations in Scopus®