Minimality of interval exchange transformations with restrictions

TL;DR

This paper investigates how linear restrictions on interval exchange transformations affect their minimality, proposing a conjecture linking minimality to the presence of Lagrangian subspaces in the restrictions.

Contribution

It formulates a conjecture relating minimality under restrictions to Lagrangian subspaces and provides partial proofs and examples supporting this relationship.

Findings

Minimality can become non-generic under certain linear restrictions.

Unique ergodicity remains generic if restrictions lack a Lagrangian subspace.

Partial proof of the conjecture's 'only if' direction.

Abstract

It is known since 40 years 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).