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Journal of Modern Dynamics (JMD)
 

Minimality of interval exchange transformations with restrictions

Pages: 219 - 248, Volume 11, 2017      doi:10.3934/jmd.2017010

 
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Ivan Dynnikov - Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Str., Moscow 119991, Russian Federation (email)
Alexandra Skripchenko - Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Str., Moscow 119991, Russian Federation (email)

Abstract: It is known since a 40-year-old paper by M.Keane that minimality is a generic (i.e., holding with probability one) property of an irreducible interval exchange transformation. If one puts some integral linear restrictions on the parameters of the interval exchange transformation, then minimality may become an ``exotic'' property. We conjecture in this paper that this occurs if and only if the linear restrictions contain a Lagrangian subspace of the first homology of the suspension surface. We partially prove it in the `only if' direction and provide a series of examples to support the converse one. We show that the unique ergodicity remains a generic property if the restrictions on the parameters do not contain a Lagrangian subspace (this result is due to Barak Weiss).

Keywords:  Interval exchange transformation, minimality, unique ergodicity, measurable foliation.
Mathematics Subject Classification:  Primary: 37E05; Secondary: 37E35.

Received: October 2015;      Revised: December 2016;      Available Online: March 2017.

 References