scholarly journals Interval exchanges, admissibility and branching Rauzy induction

2017 ◽  
Vol 51 (3) ◽  
pp. 135-139
Author(s):  
Francesco Dolce ◽  
Dominique Perrin
Keyword(s):  
Interval Exchanges ◽  
Rauzy Induction

scholarly journals Dynamics of non-classical interval exchanges

2011 ◽  
Vol 32 (6) ◽  
pp. 1930-1971 ◽  
Author(s):  
VAIBHAV S. GADRE
Keyword(s):  
Dynamical Property ◽  
Natural Generalization ◽  
Interval Exchange ◽  
Flat Surfaces ◽  
Interval Exchanges ◽  
Refinement Process ◽  
Train Tracks ◽  
Rauzy Induction ◽  
Interval Exchange Maps ◽  
Uniquely Ergodic

AbstractA natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira [Measured foliations on non-orientable surfaces.Ann. Sci. Éc. Norm. Supér.(4)26(6) (1993), 645–664]. Recurrent train tracks with a single switch provide a subclass of linear involutions. We call such linear involutions non-classical interval exchanges. They are related to measured foliations on orientable flat surfaces. Non-classical interval exchanges can be studied as a dynamical system by considering Rauzy induction in this context. This gives a refinement process on the parameter space similar to Kerckhoff’s simplicial systems. We show that the refinement process gives an expansion that has a key dynamical property calleduniform distortion. We use uniform distortion to prove normality of the expansion. Consequently, we prove an analog of Keane’s conjecture: almost every non-classical interval exchange is uniquely ergodic. Uniform distortion has been independently shown in [A. Avila and M. Resende. Exponential mixing for the Teichmüller flow in the space of quadratic differentials, http://arxiv.org/abs/0908.1102].

scholarly journals Bifix codes and interval exchanges

2015 ◽  
Vol 219 (7) ◽  
pp. 2781-2798 ◽  
Author(s):  
Valérie Berthé ◽  
Clelia De Felice ◽  
Francesco Dolce ◽  
Julien Leroy ◽  
Dominique Perrin ◽  
...  
Keyword(s):  
Interval Exchanges

Generalized interval exchanges and the 2–3 conjecture

2005 ◽  
Vol 3 (3) ◽  
pp. 412-429
Author(s):  
Shmuel Friedland ◽  
Benjamin Weiss
Keyword(s):  
Interval Exchanges

scholarly journals On Sarnak’s conjecture and Veech’s question for interval exchanges

2018 ◽  
Vol 134 (2) ◽  
pp. 545-573 ◽  
Author(s):  
Sébastien Ferenczi ◽  
Christian Mauduit
Keyword(s):  
Interval Exchanges

scholarly journals Rigidity and Substitutive Dendric Words

2018 ◽  
Vol 29 (05) ◽  
pp. 705-720 ◽  
Author(s):  
V. Berthé ◽  
F. Dolce ◽  
F. Durand ◽  
J. Leroy ◽  
D. Perrin
Keyword(s):  
Bipartite Graphs ◽  
Constant Length ◽  
Infinite Words ◽  
Algebraic Properties ◽  
Interval Exchanges ◽  
Left And Right ◽  
Sturmian Words

Dendric words are infinite words that are defined in terms of extension graphs. These are bipartite graphs that describe the left and right extensions of factors. Dendric words are such that all their extension graphs are trees. They are also called tree words. This class of words includes classical families of words such as Sturmian words, codings of interval exchanges, or else, Arnoux–Rauzy words. We investigate here the properties of substitutive dendric words and prove some rigidity properties, that is, algebraic properties on the set of substitutions that fix a dendric word. We also prove that aperiodic minimal dendric subshifts (generated by dendric words) cannot have rational topological eigenvalues, and thus, cannot be generated by constant length substitutions.

scholarly journals Interval exchanges that do not occur in free groups

2012 ◽  
Vol 6 (4) ◽  
pp. 755-763 ◽  
Author(s):  
Christopher Novak
Keyword(s):  
Free Groups ◽  
Interval Exchanges

Cutting and stacking, interval exchanges and geometric models

1985 ◽  
Vol 50 (1-2) ◽  
pp. 160-168 ◽  
Author(s):  
Pierre Arnoux ◽  
Donald S. Ornstein ◽  
Benjamin Weiss
Keyword(s):  
Geometric Models ◽  
Interval Exchanges ◽  
Cutting And Stacking

scholarly journals An alternative approach to the ergodic theory of measured foliations on surfaces

1981 ◽  
Vol 1 (4) ◽  
pp. 461-488 ◽  
Author(s):  
Mary Rees
Keyword(s):  
Projective Space ◽  
Lebesgue Measure ◽  
Ergodic Theory ◽  
Diffeomorphism Group ◽  
Alternative Approach ◽  
Interval Exchanges ◽  
Almost All ◽  
Uniquely Ergodic

AbstractWe consider measured foliations on surfaces, and interval exchanges. We give alternative proofs of the following theorems first proved by Masur and (independently) Veech. The action of the diffeomorphism group of the surface on the projective space of measured foliations (with respect to a natural ‘Lebesgue’ measure) is ergodic. Almost all measured foliations are uniquely ergodic. Almost all interval exchanges (again, with respect to a natural ‘Lebesgue’ measure) are uniquely ergodic.

scholarly journals Cascades in the dynamics of affine interval exchange transformations

2019 ◽  
Vol 40 (8) ◽  
pp. 2073-2097
Author(s):  
ADRIEN BOULANGER ◽  
CHARLES FOUGERON ◽  
SELIM GHAZOUANI
Keyword(s):  
Cantor Set ◽  
Parameter Family ◽  
Interval Exchange ◽  
Link Type ◽  
Interval Exchange Transformations ◽  
Veech Group ◽  
Veech Groups ◽  
Rauzy Induction

We describe in this article the dynamics of a one-parameter family of affine interval exchange transformations. This amounts to studying the directional foliations of a particular dilatation surface introduced in Duryev et al [Affine surfaces and their Veech groups. Preprint, 2016, arXiv:1609.02130], the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. This study is achieved through analysis of the dynamics of the Veech group of this surface combined with a modified version of Rauzy induction in the context of affine interval exchange transformations.

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