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.

EXPONENTIALLY STRONG CONVERGENCE OF NON-CLASSICAL MULTIDIMENSIONAL CONTINUED FRACTION ALGORITHMS

2002 ◽  
Vol 02 (04) ◽  
pp. 563-586
Author(s):  
KENTARO NAKAISHI
Keyword(s):  
Strong Convergence ◽  
Continued Fraction ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Deterministic Case ◽  
Convergence Properties ◽  
Almost Everywhere ◽  
Multidimensional Continued Fraction

Convergence properties of multidimensional continued fraction algorithms introduced by V. Baladi and A. Nogueira are studied. The paper contains an arithmetic proof of almost everywhere exponentially strong convergence of some two-dimensional Markovian random algorithms and dynamically defined ones. A special three-dimensional deterministic case is also discussed.

A NEW SYMMETRIC TWO-DIMENSIONAL ALGORITHM

2006 ◽  
Vol 02 (04) ◽  
pp. 489-498
Author(s):  
PEDRO FORTUNY AYUSO ◽  
FRITZ SCHWEIGER
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Dual Graph ◽  
Plane Curve ◽  
Singularity Theory ◽  
Main Topic ◽  
Two Dimensional ◽  
Dimension 2 ◽  
Multidimensional Continued Fraction ◽  
Fraction Algorithm

Continued fractions are deeply related to Singularity Theory, as the computation of the Puiseux exponents of a plane curve from its dual graph clearly shows. Another closely related topic is Euclid's Algorithm for computing the gcd of two integers (see [2] for a detailed overview). In the first section, we describe a subtractive algorithm for computing the gcd of n integers, related to singularities of curves in affine n-space. This gives rise to a multidimensional continued fraction algorithm whose version in dimension 2 is the main topic of the paper.

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.

scholarly journals Actions of finite rank: weak rational ergodicity and partial rigidity

2015 ◽  
Vol 36 (7) ◽  
pp. 2138-2171 ◽  
Author(s):  
ALEXANDRE I. DANILENKO
Keyword(s):  
Simple Proof ◽  
Finite Rank ◽  
Abelian Groups ◽  
Interval Exchange ◽  
Infinite Measure ◽  
Interval Exchange Transformations ◽  
Bratteli Diagrams ◽  
Rank One ◽  
Partial Rigidity ◽  
Definition Of

A simple proof of the fact that each rank-one infinite measure preserving (i.m.p.) transformation is subsequence weakly rationally ergodic is found. Some classes of funny rank-one i.m.p. actions of Abelian groups are shown to be subsequence weakly rationally ergodic. A constructive definition of finite funny rank for actions of arbitrary infinite countable groups is given. It is shown that the ergodic i.m.p. transformations of balanced finite funny rank are subsequence weakly rationally ergodic. It is shown that the ergodic probability preserving transformations of exact finite rank, the ergodic Bratteli–Vershik maps corresponding to the ‘consecutively ordered’ Bratteli diagrams of finite rank, some their generalizations and the ergodic interval exchange transformations are partially rigid.

scholarly journals On some symmetric multidimensional continued fraction algorithms

2017 ◽  
Vol 38 (5) ◽  
pp. 1601-1626 ◽  
Author(s):  
PIERRE ARNOUX ◽  
SÉBASTIEN LABBÉ
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Positive Measure ◽  
Continued Fraction ◽  
Natural Extension ◽  
Fractal Boundary ◽  
Absolutely Continuous ◽  
Invariant Domain ◽  
Multidimensional Continued Fraction ◽  
Unique Invariant Measure

We compute explicitly the density of the invariant measure for the reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the unsorted version of the Brun algorithm and Cassaigne algorithm. We illustrate some experimentations on the domain of the natural extension of those algorithms. For some other algorithms, which are known to have a unique invariant measure absolutely continuous with respect to Lebesgue measure, the invariant domain found by this method seems to have a fractal boundary, and it is unclear whether it is of positive measure.

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 Diophantine type of interval exchange maps

2013 ◽  
Vol 34 (6) ◽  
pp. 1990-2017
Author(s):  
DONG HAN KIM
Keyword(s):  
Continued Fraction ◽  
Interval Exchange ◽  
Gauss Measure ◽  
Return Times ◽  
Interval Exchange Transformations ◽  
Cohomological Equation ◽  
Rotation Numbers ◽  
Interval Exchange Maps ◽  
Fraction Algorithm ◽  
Asymptotic Scaling

AbstractRoth type irrational rotation numbers have several equivalent arithmetical characterizations as well as several equivalent characterizations in terms of the dynamics of the corresponding circle rotations. In this paper we investigate how to generalize Roth-like Diophantine conditions to interval exchange maps. If one considers the dynamics in parameter space one can introduce two non-equivalent Roth type conditions, the first (condition (Z)) by means of the Zorich cocycle [Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier 46(2) (1996), 325–370], the second (condition (A)) by means of a further acceleration of the continued fraction algorithm by Marmi–Moussa–Yoccoz introduced in [The cohomological equation for Roth type interval exchange maps, J. Amer. Math. Soc. 18 (2005), 823–872]. A third very natural condition (condition (D)) arises by considering the distance between the discontinuity points of the iterates of the map. If one considers the dynamics of an interval exchange map in phase space then one can introduce the notion of Diophantine type by considering the asymptotic scaling of return times pointwise or with respect to uniform convergence (respectively conditions (R) and (U)). In the case of circle rotations all the above conditions are equivalent. For interval exchange maps of three intervals we show that (D) and (A) are equivalent and imply (Z), (U) and (R), which are equivalent among them. For maps of four intervals or more we prove several results; the only relation that we cannot decide is whether (Z) implies (R) or not.

Introductory paper

1966 ◽  
Vol 24 ◽  
pp. 3-5
Author(s):  
W. W. Morgan
Keyword(s):  
Line Intensity ◽  
Classification Scheme ◽  
Two Dimensional ◽  
Spectral Classification ◽  
Dimensional Array ◽  
Rr Lyrae ◽  
Metallic Line ◽  
Introductory Paper ◽  
Parameter Varying ◽  
Definition Of

1. The definition of “normal” stars in spectral classification changes with time; at the time of the publication of theYerkes Spectral Atlasthe term “normal” was applied to stars whose spectra could be fitted smoothly into a two-dimensional array. Thus, at that time, weak-lined spectra (RR Lyrae and HD 140283) would have been considered peculiar. At the present time we would tend to classify such spectra as “normal”—in a more complicated classification scheme which would have a parameter varying with metallic-line intensity within a specific spectral subdivision.

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