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Cherry flow: physical measures and perturbation theory

Published online by Cambridge University Press:  12 May 2016

JIAGANG YANG
JIAGANG YANG*
Affiliation:
Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil email yangjg@impa.br
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Abstract

In this article we consider Cherry flows on the torus which have two singularities, a source and a saddle, and no periodic orbits. We show that every Cherry flow admits a unique physical measure, whose basin has full volume. This proves a conjecture given by Saghin and Vargas [Invariant measures for Cherry flows. Comm. Math. Phys.317(1) (2013), 55–67]. We also show that the perturbation of Cherry flows depends on the divergence at the saddle: when the divergence is negative, this flow admits a neighborhood, such that any flow in this neighborhood belongs to one of the following three cases: it has a saddle connection; it is a Cherry flow; it is a Morse–Smale flow whose non-wandering set consists of two singularities and one periodic sink. In contrast, when the divergence is non-negative, this flow can be approximated by a non-hyperbolic flow with an arbitrarily large number of periodic sinks.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 8 , December 2017 , pp. 2671 - 2688
Copyright
© Cambridge University Press, 2016 

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