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.

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.

Exactness of the Euclidean algorithm and of the Rauzy induction on the space of interval exchange transformations

2011 ◽  
Vol 33 (1) ◽  
pp. 221-246 ◽  
Author(s):  
TOMASZ MIERNOWSKI ◽  
ARNALDO NOGUEIRA
Keyword(s):  
Lebesgue Measure ◽  
Continued Fraction ◽  
Euclidean Algorithm ◽  
Interval Exchange ◽  
Two Dimensional ◽  
Interval Exchange Transformations ◽  
Multidimensional Continued Fraction ◽  
Rauzy Induction ◽  
Definition Of

AbstractThe two-dimensional homogeneous Euclidean algorithm is the central motivation for the definition of the classical multidimensional continued fraction algorithms, such as Jacobi–Perron, Poincaré, Brun and Selmer algorithms. The Rauzy induction, a generalization of the Euclidean algorithm, is a key tool in the study of interval exchange transformations. Both maps are known to be dissipative and ergodic with respect to Lebesgue measure. Here we prove that they are exact.

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>

Affine interval exchange transformations with wandering intervals

1997 ◽  
Vol 17 (6) ◽  
pp. 1315-1338 ◽  
Author(s):  
RICARDO CAMELIER ◽  
CARLOS GUTIERREZ
Keyword(s):  
Invariant Measure ◽  
Cantor Set ◽  
Interval Exchange ◽  
Interval Exchange Transformations ◽  
Uniquely Ergodic

There exist uniquely ergodic bijective affine interval exchange transformations of $[0,1)$ having wandering intervals and such that the support of the invariant measure is a Cantor set.

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