Invariant Measures for Cherry Flows

Abstract

Abstract. 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.

Cherry flow: physical measures and perturbation theory

“…We show that every Cherry flow admits a unique physical measure, whose basin has full volume. This proves a conjecture given by R. Saghin and E. Vargas in [21]. We also show that the perturbation of Cherry flow 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 the following three cases: (a) has a saddle connection; (b) a Cherry flow; (c) a Morse-Smale flow whose nonwandering set consists two singularities and one periodic sink.…”

“…We denote by D X (σ X ) the divergence at σ X . It was observed by [16,17,21] that the behavior of physical measures for Cherry flows depends on the divergence at the saddle. And the flows in general were assumed to be C ∞ , this is because the C ∞ regularity implies that this flow is C 1+bounded variation linearizable at a neighborhood of σ X (see [15][Appendix]).…”

On physical measures for Cherry flows

“…If λ 1 ≥ −3λ 2 , the ergodic invariant probability measure ν supported on the quasi-minimal set is the physical measure for φ with attraction basin having full Lebesgue measure. Theorem 1.13 together with results of [11] provides a complete description of the physical measures for Cherry flows having rotation number of bounded type (bounded regime). In the unbounded regime the case 1 < λ 1 −λ 2 < 3 remains still open.…”

“…1 To be more precise, let λ 1 > 0 > λ 2 be the two eigenvalues at the saddle point. In the non-positive divergence case when λ 1 ≤ −λ 2 , [11] shows that the Dirac deltas at the singularities are the only ergodic invariant probability measures. Moreover [11] establishes that the Dirac delta at the saddle point is the physical measure for the flow.…”

Cherry flows with non-trivial attractors

“…For the reference we mention that the only studies of non-trivial physical measures concerned the case of inverted Cherry flows, viz. flows with a saddle point and a repulsive point, see [14], [13], [15].…”