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 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.

scholarly journals Simplicial systems for interval exchange maps and measured foliations

1985 ◽  
Vol 5 (2) ◽  
pp. 257-271 ◽  
Author(s):  
S. P. Kerckhoff
Keyword(s):  
General Theorem ◽  
Interval Exchange ◽  
Alternative Proof ◽  
Simple Assumption ◽  
Refinement Process ◽  
Almost All ◽  
Interval Exchange Maps ◽  
Uniquely Ergodic

AbstractThe spaces of interval exchange maps and measured foliations are considered and an alternative proof that almost all interval exchange maps and measured foliations are uniquely ergodic is given. These spaces are endowed with a refinement process, called a simplicial system, which is studied abstractly and is shown to be normal under a simple assumption. The results follow and thus are a corollary of a more general theorem in a broader setting.

On the dual of Rauzy induction

2016 ◽  
Vol 37 (5) ◽  
pp. 1492-1536 ◽  
Author(s):  
KAE INOUE ◽  
HITOSHI NAKADA
Keyword(s):  
Continued Fractions ◽  
Natural Extension ◽  
Gauss Map ◽  
Interval Exchange ◽  
Dual Relationship ◽  
Rauzy Induction ◽  
Interval Exchange Maps

We investigate a certain dual relationship between piecewise rotations of a circle and interval exchange maps. In 2005, Cruz and da Rocha [A generalization of the Gauss map and some classical theorems on continued fractions. Nonlinearity18 (2005), 505–525]  introduced a notion of ‘castles’ arising from piecewise rotations of a circle. We extend their idea and introduce a continuum version of castles, which we show to be equivalent to Veech’s zippered rectangles [Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), 201–242]. We show that a fairly natural map defined on castles represents the inverse of the natural extension of the Rauzy map.

Minimal interval exchange transformations with flips

2017 ◽  
Vol 38 (8) ◽  
pp. 3101-3144 ◽  
Author(s):  
ANTONIO LINERO BAS ◽  
GABRIEL SOLER LÓPEZ
Keyword(s):  
Linear Algebra ◽  
Geometric Approach ◽  
Interval Exchange ◽  
Positive Power ◽  
Direct Construction ◽  
Interval Exchange Transformations ◽  
Associated Matrix ◽  
Rauzy Induction ◽  
Positive Integers ◽  
Uniquely Ergodic

We consider interval exchange transformations of$n$intervals with$k$flips, or$(n,k)$-IETs for short, for positive integers$k,n$with$k\leq n$. Our main result establishes the existence of minimal uniquely ergodic$(n,k)$-IETs when$n\geq 4$; moreover, these IETs are self-induced for$2\leq k\leq n-1$. This result extends the work on transitivity in Gutierrezet al[Transitive circle exchange transformations with flips.Discrete Contin. Dyn. Syst. 26(1) (2010), 251–263]. In order to achieve our objective we make a direct construction; in particular, we use the Rauzy induction to build a periodic Rauzy graph whose associated matrix has a positive power. Then we use a result in the Perron–Frobenius theory [Pullman, A geometric approach to the theory of non-negative matrices.Linear Algebra Appl. 4(1971) 297–312] which allows us to ensure the existence of these minimal self-induced and uniquely ergodic$(n,k)$-IETs,$2\leq k\leq n-1$. We then find other permutations in the same Rauzy class generating minimal uniquely ergodic$(n,1)$- and$(n,n)$-IETs.

scholarly journals Rauzy induction of polygon partitions and toral $ \mathbb{Z}^2 $-rotations

2021 ◽  
Vol 17 (0) ◽  
pp. 481
Author(s):  
Sébastien Labbé
Keyword(s):  
Dynamical System ◽  
Periodic Sequence ◽  
Markov Partition ◽  
Interval Exchange ◽  
Irrational Rotations ◽  
Interval Exchange Transformations ◽  
Symbolic Dynamical System ◽  
Rauzy Induction ◽  
Sturmian Sequences ◽  
Uniquely Ergodic

<p style='text-indent:20px;'>We extend the notion of Rauzy induction of interval exchange transformations to the case of toral <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation, i.e., <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action defined by rotations on a 2-torus. If <inline-formula><tex-math id="M3">\begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document}</tex-math></inline-formula> denotes the symbolic dynamical system corresponding to a partition <inline-formula><tex-math id="M4">\begin{document}$ \mathscr{P} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M7">\begin{document}$ R $\end{document}</tex-math></inline-formula> is Cartesian on a sub-domain <inline-formula><tex-math id="M8">\begin{document}$ W $\end{document}</tex-math></inline-formula>, we express the 2-dimensional configurations in <inline-formula><tex-math id="M9">\begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document}</tex-math></inline-formula> as the image under a <inline-formula><tex-math id="M10">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-dimensional morphism (up to a shift) of a configuration in <inline-formula><tex-math id="M11">\begin{document}$ \mathscr{X}_{\widehat{\mathscr{P}}|_W, \widehat{R}|_W} $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M12">\begin{document}$ \widehat{\mathscr{P}}|_W $\end{document}</tex-math></inline-formula> is the induced partition and <inline-formula><tex-math id="M13">\begin{document}$ \widehat{R}|_W $\end{document}</tex-math></inline-formula> is the induced <inline-formula><tex-math id="M14">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action on <inline-formula><tex-math id="M15">\begin{document}$ W $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We focus on one example, <inline-formula><tex-math id="M16">\begin{document}$ \mathscr{X}_{\mathscr{P}_0, R_0} $\end{document}</tex-math></inline-formula>, for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift <inline-formula><tex-math id="M17">\begin{document}$ X_0 $\end{document}</tex-math></inline-formula> of the Jeandel–Rao Wang shift computed in an earlier work by the author. As a consequence, <inline-formula><tex-math id="M18">\begin{document}$ {\mathscr{P}}_0 $\end{document}</tex-math></inline-formula> is a Markov partition for the associated toral <inline-formula><tex-math id="M19">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation <inline-formula><tex-math id="M20">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula>. It also implies that the subshift <inline-formula><tex-math id="M21">\begin{document}$ X_0 $\end{document}</tex-math></inline-formula> is uniquely ergodic and is isomorphic to the toral <inline-formula><tex-math id="M22">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation <inline-formula><tex-math id="M23">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> which can be seen as a generalization for 2-dimensional subshifts of the relation between Sturmian sequences and irrational rotations on a circle. Batteries included: the algorithms and code to reproduce the proofs are provided.</p>

scholarly journals A Sufficient Condition for Absolute Continuity of Conjugations between Interval Exchange Maps

2019 ◽  
pp. 276-284
Author(s):  
Abdumajid S. Begmatov
Keyword(s):  
Absolute Continuity ◽  
Interval Exchange ◽  
Sufficient Condition ◽  
Genus One ◽  
Interval Exchange Maps

A class of topological equivalent generalized interval exchange maps of genus one and of the same bounded combinatorics is considered in the paper. A sufficient condition for absolute continuity of the conjugation between two maps from this class is provided

The recurrence time for interval exchange maps

Nonlinearity ◽  
2008 ◽  
Vol 21 (9) ◽  
pp. 2201-2210 ◽  
Author(s):  
Dong Han Kim ◽  
Stefano Marmi
Keyword(s):  
Interval Exchange ◽  
Recurrence Time ◽  
Interval Exchange Maps
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