Minimality of interval exchange transformations with restrictions

Ivan Dynnikov; Alexandra Skripchenko

doi:10.3934/jmd.2017010

Article Contents

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

This volume Previous Article Effective equidistribution of circles in the limit sets of Kleinian groups Next Article Most interval exchanges have no roots

Minimality of interval exchange transformations with restrictions

Ivan Dynnikov and
Alexandra Skripchenko

Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Str., Moscow 119991, Russia

Author Bio: Ivan Dynnikov <dynnikov@mech.math.msu.su>; Alexandra Skripchenko <sashaskrip@gmail.com>

Received: October 12, 2015

Revised: December 12, 2016

Published: March 2017

The work is supported by the Russian Science Foundation under grant 14-50-00005 and performed at Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia.

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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).

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Mathematics Subject Classification: Primary: 37E05; Secondary: 37E35.

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Figure 1. The subset $M(\pi, \mathscr U)\subset\Delta^n\cap\mathscr U$ in Examples 1.7 (left) and 1.8 (right). Only points with $\dim_{\mathbb Q}\langle a_1, a_2, 1\rangle=3$ are considered

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Figure 2. Singularities of $\mathscr F_{\pi, \mathbf a}$

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Figure 3. A transversal representing a separating cycle (bold line) and the restriction cycle (dashed line) in Example 2.18

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Figure 4. Double suspension surface. Each vertical straight line segment in $\partial D$ is collapsed to a point, the bottom side of the square is identified with the top one

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