Minimality of interval exchange transformations with restrictions

Ivan Dynnikov, Alexandra Skripchenko

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

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).

Original languageEnglish
Pages (from-to)219-248
Number of pages30
JournalJournal of Modern Dynamics
Volume11
DOIs
Publication statusPublished - 2017
Externally publishedYes

Keywords

  • Interval exchange transformation
  • Measurable foliation
  • Minimality
  • Unique ergodicity

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