Dynamics of self-similar interval exchange transformations on three intervals

2016 ◽  
pp. 297-316
Author(s):  
Kelly Yancey
Keyword(s):  
Interval Exchange ◽  
Interval Exchange Transformations ◽  
Self Similar

scholarly journals Persistence of wandering intervals in self-similar affine interval exchange transformations

2009 ◽  
Vol 30 (3) ◽  
pp. 665-686 ◽  
Author(s):  
XAVIER BRESSAUD ◽  
PASCAL HUBERT ◽  
ALEJANDRO MAASS
Keyword(s):  
Interval Exchange ◽  
Interval Exchange Transformation ◽  
Algebraic Condition ◽  
Exchange Transformation ◽  
Interval Exchange Transformations ◽  
Affine Maps ◽  
Interval Exchanges ◽  
Self Similar ◽  
Associated Matrix

AbstractIn this article we prove that given a self-similar interval exchange transformation T(λ,π), whose associated matrix verifies a quite general algebraic condition, there exists an affine interval exchange transformation with wandering intervals that is semi-conjugated to it. That is, in this context the existence of Denjoy counterexamples occurs very often, generalizing the result of Cobo [Piece-wise affine maps conjugate to interval exchanges. Ergod. Th. & Dynam. Sys.22 (2002), 375–407].

scholarly journals Invariant measures with bounded variation densities for piecewise area preserving maps

2012 ◽  
Vol 33 (2) ◽  
pp. 624-642 ◽  
Author(s):  
YIWEI ZHANG ◽  
CONGPING LIN
Keyword(s):  
Bounded Variation ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Topological Transitivity ◽  
Borel Measures ◽  
Interval Exchange Transformations ◽  
Area Preserving Maps ◽  
Piecewise Isometries ◽  
Area Preserving ◽  
Absolutely Continuous Invariant

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.

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

2009 ◽  
Vol 29 (3) ◽  
pp. 767-816 ◽  
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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