Symmetric interval identification systems of order three
Alexandra Skripchenko
Discrete & Continuous Dynamical Systems - A 2012, 32(2): 643-656 doi: 10.3934/dcds.2012.32.643
In the present paper we study symmetric interval identification systems of order three. We prove that the Rauzy induction preserves symmetry: for any symmetric interval identification system of order 3 after finitely many iterations of the Rauzy induction we always obtain a symmetric system. We also provide an example of symmetric interval identification system of thin type.
keywords: Interval identification systems Rauzy induction $\mathbb{R}$-tree.
Minimality of interval exchange transformations with restrictions
Ivan Dynnikov Alexandra Skripchenko
Journal of Modern Dynamics 2017, 11(1): 219-248 doi: 10.3934/jmd.2017010

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

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