Abstract
Arnold and Kochergin mixing conservative flows on surfaces stand as the main and almost only natural class of mixing transformations for which higher order mixing has not been established, nor disproved. Under suitable arithmetic conditions on their unique rotation vector, of full Lebesgue measure in the first case and of full Hausdorff dimension in the second, we show that these flows are mixing of any order. For this, we show that they display a generalization of the so called Ratner property on slow divergence of nearby orbits, that implies strong restrictions on their joinings, which in turn yields higher order mixing. This is the first case in which the Ratner property is used to prove multiple mixing outside its original context of horocycle flows and we expect our approach will have further applications.
Highlights
A major open problem in ergodic theory is Rokhlin’s question on whether mixing implies mixing of all orders, called multiple mixing [26]
The most noteworthy are K-systems where multiple mixing always holds [4], horocycle flows [23], mixing systems with singular spectrum that display multiple mixing by a celebrated theorem of Host [13], and finite rank systems since Kalikow showed that rank one and mixing implies multiple mixing [14], a result that was extended to finite rank mixing systems by Ryzhikov [30]
The possibility of mixing for these flows was studied in two different cases: first, by Kochergin who obtained mixing when ω has higher order zeros and the flow has degenerate saddles [17], and by Arnold [1] who suggested that mixing is possible on a minimal component even in the case where ω is Morse but the saddles on the minimal component appear in asymmetric configurations, for example because of a saddle loop
Summary
A major open problem in ergodic theory is Rokhlin’s question on whether mixing implies mixing of all orders, called multiple mixing [26]. Whether natural classes of special flows (not necessarily mixing) over irrational rotations may have the R-property remained open until Fraczek and Lemanczyk [9,10] showed that a generalized R-property holds in some classes of special flows with roof functions of bounded variation (which, by [16], are not mixing) They have introduced a weaker notion than the R-property, called weak Ratner or WR-property that still implies the FEJ-property (see Definition 2.1 and the comment after it). This line of thought can be extended to show that horocycle flows are never isomorphic to special flows above an irrational rotation and under a roof function that is convex and C2 except at one point For the latter result, one needs to introduce the concept of strong Ratner property, which is an isomorphism invariant, that specifies the occurrence of slow divergence of nearby orbits to the first time when the orbits do split apart. The statement of the Lemma follows by induction and because a single a1 is always badly approximable by α
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