TY - JOUR

T1 - Invariant Measures for Cherry Flows

AU - Saghin, Radu

AU - Vargas, Edson

PY - 2013/1

Y1 - 2013/1

N2 - We investigate the invariant probability measures for Cherry flows, i. e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.

AB - We investigate the invariant probability measures for Cherry flows, i. e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.

UR - http://www.scopus.com/inward/record.url?scp=84872423948&partnerID=8YFLogxK

U2 - 10.1007/s00220-012-1611-z

DO - 10.1007/s00220-012-1611-z

M3 - Article

AN - SCOPUS:84872423948

SN - 0010-3616

VL - 317

SP - 55

EP - 67

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -