Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces

Abstract

We consider typical area preserving flows on higher genus surfaces and prove that the flow restricted to mixing minimal components is mixing of all orders, thus answering affimatively to Rohlin's multiple mixing question in this context. The main tool is a variation of the Ratner property originally proved by Ratner for the horocycle flow, i.e. the switchable Ratner property introduced by Fayad and Kanigowski for special flows over rotations. This property, which is of independent interest, provides a quantitative description of the parabolic behaviour of these flows and has implication to joinings classification. The main result is formulated in the language of special flows over interval exchange transformations with asymmetric logarithmic singularities. We also prove a strengthening of one of Fayad and Kanigowski's main results, by showing that Arnold's flows are mixing of all oders for almost every location of the singularities.

Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows

“…Since R ′ δ 1/3 x δ y , m R ′′ max(k, T ) and we are in Case 2, for R ′′ > 1 large enough we have the following lower bound: c ′ 2 q −1 ζ 1/2 k 1/2 mδ x 10 max(C ′ p −1 k 1/2 δ w , q −1 ζ 3/2 C ′ k 3/2 δ x ) and analogously for (z, w), (z, w ′ ). This gives (18). By the cocycle identity and by Lemma 5.5, there exists κ ′′′ ǫ > 0 such that for κ ∈ (0, κ ′′′ ǫ ), for all u, w ∈ [0, 2ζ(M + L)] with |u − w| κM, we have…”

“…Then the authors showed in particular that the SWR-property holds for a full measure set of mixing flows with logarithmic singularities (Arnol'd flows) thereby proving higher order mixing in this class. The result in [5] was strengthened in [18], where the authors showed that the SWR-property holds for a full measure set of Arnol'd flows on surfaces of higher genus.…”

On disjointness properties of some parabolic flows

“…The first estimate (in Lemma 6.4) provides a fine control of f (r) (of order r log r with optimal control on the constants) as long as one assumes that the points in the orbit of x of length r stay sufficiently far from the singularity. Estimates similar to Lemma 6.4 but in the more general context of interval exchange transformations were proved by the third author in [37] (see Proposition 3.4 in [37], as well as its generalization by Ravotti in [32] and Proposition 4.4 in [21]) and inspired in turn by the work of Kochergin on rotations, see e.g. [24].…”

“…The set Z is such that the points y, y satisfy a Ratner-type form of shearing either going forward or backward in time (see (7.4)). This is essentially the set on which the switchable Ratner property (see Section 2.5) holds for the Arnold flow and the definition is indeed the same as the set Z is in [10] or [21]. If y, y display this good form of Ratner-like shearing going forward, we then show that the Forward assumptions (F 1) − (F 4) in F. of Proposition 5.1 hold, while if the Ratner-like form of shearing happens backward, we show that the Backward assumptions (B1) − (B4) in B. of Proposition 5.1 hold.…”

“…The class considered in [10] consists of (some) smooth mixing flows on the torus T 2 with one fixed point (these are sometimes known as Arnol'd and Kochergin flows). This triggered further investigations and the first and third authors and J. Kułaga-Przymus in [21] proved that the same variant of the Ratner property holds for smooth Arnol'd flows on every surface of genus g > 1. The variation of the Ratner property considered in [10] and [21] is the so called switchable Ratner property (or SRproperty for short).…”