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


Kanigowski, Adam; Kułaga-Przymus, Joanna; Ulcigrai, Corinna (2019). Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces. Journal of the European Mathematical Society, 21(12):3797-3855.

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 affirmatively Rokhlin’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 behavior of these flows and has implications for joining 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 orders for almost every location of the singularities.

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 affirmatively Rokhlin’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 behavior of these flows and has implications for joining 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 orders for almost every location of the singularities.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:340 Law
610 Medicine & health
510 Mathematics
Uncontrolled Keywords:Applied Mathematics, General Mathematics
Language:English
Date:30 August 2019
Deposited On:09 Jan 2020 13:20
Last Modified:06 Feb 2020 09:35
Publisher:European Mathematical Society
ISSN:1435-9855
OA Status:Green
Publisher DOI:https://doi.org/10.4171/jems/914

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