Cherry flow: physical measures and perturbation theory

Abstract

In this article we consider Cherry flows on 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 R. Saghin and E. Vargas in~\cite{SV}. 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 non-wandering set consists two singularities and one periodic sink. In contrary, when the divergence is non-negative, this flow can be approximated by non-hyperbolic flow with arbitrarily larger number of periodic sinks

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].…”

Thermodynamical u-formalism I: measures of maximal u-entropy for maps that factor over Anosov

“…A measurable partition of M is said to be -subordinate to the strong-unstable foliation if for -almost every x : and it has uniformly small diameter inside ; contains an open neighborhood of x inside the leaf ; is an increasing partition, meaning that . Then, the u-entropy of is defined by The definition does not depend on the choice of (see [LY85a, Lemma 3.1.2]). These notions go back to Ledrappier and Strelcyn [LS82], who proved that every non-uniformly hyperbolic diffeomorphism with Hölder derivative admits measurable partitions subordinate to the corresponding Pesin unstable lamination. In our present setting, partitions subordinate to the strong-unstable foliation of a partially hyperbolic diffeomorphism were constructed by Yang [Yan17, Lemma 3.2]. Let be a partially hyperbolic diffeomorphism, and be an ergodic measure of f whose center Lyapunov exponents are all negative. Then, . Recall ([Rok67], see also [VO16, Lemma 9.1.12]) that the entropy of relative to a finite partition can be defined as We denote by the partition of M whose elements are the intersections of the elements of with local strong-unstable leaves. Under the hypotheses of Proposition A.4, if is a finite partition with sufficiently small diameter then and, up to zero -measure: ; and is the partition into points. As before, let denote the strong-unstable foliation of f .…”

The hidden unstable orbits of maps with gaps

“…An early example of a discontinuous map arose with the Lorenz semi-flow [1,2], which is an abstraction of Lorenz's famous chaotic flow [3]. Another is the Cherry flow [4][5][6], in which the unstable manifolds from a saddlepoint split apart a flow on a torus. In each case, the return map to a typical section taken through the flow contains a discontinuity, as sketched in figure 1.…”