Local rigidity for periodic generalised interval exchange transformations

AbstractWe investigate the properties of absolutely continuous invariant probability measures (ACIPs), especially those measures with bounded variation densities, for piecewise area preserving maps (PAPs) on ℝd. This class of maps unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where Lebesgue measure is locally preserved. Using a functional analytic approach, we first explore the relationship between topological transitivity and uniqueness of ACIPs, and then give an approach to construct invariant measures with bounded variation densities for PWIs. Our results ‘partially’ answer one of the fundamental questions posed in [13]—to determine all invariant non-atomic probability Borel measures in piecewise rotations. When restricting PAPs to interval exchange transformations (IETs), our results imply that for non-uniquely ergodic IETs with two or more ACIPs, these ACIPs have very irregular densities, i.e. they have unbounded variation.

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Dynamics and geometry of the Rauzy–Veech induction for quadratic differentials

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385708080565 ◽

2009 ◽

Vol 29 (3) ◽

pp. 767-816 ◽

Cited By ~ 16

Author(s):

CORENTIN BOISSY ◽

ERWAN LANNEAU

Keyword(s):

Geodesic Flow ◽

Natural Generalization ◽

Interval Exchange ◽

Connected Components ◽

Quadratic Differentials ◽

Flat Surfaces ◽

Interval Exchange Transformations ◽

Transverse Segment ◽

First Return Maps ◽

Interval Exchange Maps

AbstractInterval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy–Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmüller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely linear involutions. These maps are related to general measured foliations on surfaces (whether orientable or not). In this paper we are interested by such maps related to geodesic flow on (orientable) flat surfaces with ℤ/2ℤ linear holonomy. We relate geometry and dynamics of such maps to the combinatorics of generalized permutations. We study an analogue of the Rauzy–Veech induction and give an efficient combinatorial characterization of its attractors. We establish a natural bijection between the extended Rauzy classes of generalized permutations and connected components of the strata of meromorphic quadratic differentials with at most simple poles, which allows us, in particular, to classify the connected components of all exceptional strata.

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Mild mixing of certain interval-exchange transformations

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2017.31 ◽

2017 ◽

Vol 39 (1) ◽

pp. 248-256

Author(s):

DONALD ROBERTSON

Keyword(s):

Hausdorff Dimension ◽

Interval Exchange ◽

Interval Exchange Transformations ◽

Recurrent Type ◽

Recurrent Interval

We prove that irreducible, linearly recurrent, type W interval-exchange transformations are always mild mixing. For every irreducible permutation, the set of linearly recurrent interval-exchange transformations has full Hausdorff dimension.

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