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ON TYPICALITY OF TRANSLATION FLOWS WHICH ARE DISJOINT WITH THEIR INVERSE

Part of: Smooth dynamical systems: general theory Low-dimensional dynamical systems Ergodic theory

Published online by Cambridge University Press:  11 December 2018

Przemysław Berk ,
Krzysztof Frączek  and
Thierry de la Rue
Przemysław Berk
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland (zimowy@mat.umk.pl; fraczek@mat.umk.pl)
Krzysztof Frączek
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland (zimowy@mat.umk.pl; fraczek@mat.umk.pl)
Thierry de la Rue
Affiliation:
Laboratoire de Mathématiques Raphaël Salem, Normandie Université, Université de Rouen, CNRS, Avenue de l’Université, 76801 Saint Etienne du Rouvray, France (Thierry.de-la-Rue@univ-rouen.fr)
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Abstract

In this paper we prove that the set of translation structures for which the corresponding vertical translation flows are disjoint with its inverse contains a $G_{\unicode[STIX]{x1D6FF}}$-dense subset in every non-hyperelliptic connected component of the moduli space ${\mathcal{M}}$. This is in contrast to hyperelliptic case, where for every translation structure the associated vertical flow is reversible, i.e., it is isomorphic to its inverse by an involution. To prove the main result, we study limits of the off-diagonal 3-joinings of special representations of vertical translation flows. Moreover, we construct a locally defined continuous embedding of the moduli space into the space of measure-preserving flows to obtain the $G_{\unicode[STIX]{x1D6FF}}$-condition. Moreover, as a by-product we get that in every non-hyperelliptic connected component of the moduli space there is a dense subset of translation structures whose vertical flow is reversible.

Keywords

measure-preserving flowstranslation surfacesreversibility of dynamical systemsjoinings methods in ergodic theory

MSC classification

Primary: 37A10: One-parameter continuous families of measure-preserving transformations 37E35: Flows on surfaces
Secondary: 37C80: Symmetries, equivariant dynamical systems
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 5 , September 2020 , pp. 1677 - 1737
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Copyright
© Cambridge University Press 2018

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