The set of uniquely ergodic interval exchange transformations is path-connected

2017 ◽  
Vol 39 (2) ◽  
pp. 311-356
Author(s):  
JON CHAIKA ◽  
SEBASTIAN HENSEL
Keyword(s):  
Interval Exchange ◽  
Interval Exchange Transformations ◽  
Path Connected ◽  
Uniquely Ergodic

Let $\unicode[STIX]{x1D70B}$ be a non-degenerate permutation on at least four symbols. We show that the set of uniquely ergodic interval exchange transformations with permutation $\unicode[STIX]{x1D70B}$ is path-connected.

Deviation for interval exchange transformations

1997 ◽  
Vol 17 (6) ◽  
pp. 1477-1499 ◽  
Author(s):  
ANTON ZORICH
Keyword(s):  
Ergodic Theorem ◽  
Gauss Map ◽  
Interval Exchange ◽  
Actual Number ◽  
Interval Exchange Transformation ◽  
Exchange Transformation ◽  
Interval Exchange Transformations ◽  
Starting Point ◽  
Almost All ◽  
Uniquely Ergodic

Consider a long piece of a trajectory $x, T(x), T(T(x)), \ldots, T^{n-1}(x)$ of an interval exchange transformation $T$. A generic interval exchange transformation is uniquely ergodic. Hence, the ergodic theorem predicts that the number $\chi_i(x,n)$ of visits of our trajectory to the $i$th subinterval would be approximately $\lambda_i n$. Here $\lambda_i$ is the length of the corresponding subinterval of our unit interval $X$. In this paper we give an estimate for the deviation of the actual number of visits to the $i$th subinterval $X_i$ from one predicted by the ergodic theorem.We prove that for almost all interval exchange transformations the following bound is valid: $$ \max_{\ssty x\in X \atop \ssty 1\le i\le m} \limsup_{n\to +\infty} \frac {\log | \chi_i(x,n) -\lambda_in|}{\log n} = \frac{\theta_2}{\theta_1} < 1. $$ Roughly speaking the error term is bounded by $n^{\theta_2/\theta_1}$. The numbers $0\le \theta_2 < \theta_1$ depend only on the permutation $\pi$ corresponding to the interval exchange transformation (actually, only on the Rauzy class of the permutation). In the case of interval exchange of two intervals we obviously have $\theta_2=0$. In the case of exchange of three and more intervals the numbers $\theta_1, \theta_2$ are the two top Lyapunov exponents related to the corresponding generalized Gauss map on the space of interval exchange transformations.The limit above ‘converges to the bound’ uniformly for all $x\in X$ in the following sense. For any $\varepsilon >0$ the ratio of logarithms would be less than $\theta_2(\pi)/\theta_1(\pi)+\varepsilon $ for all $n\ge N(\varepsilon)$, where $N(\varepsilon)$ does not depend on the starting point $x\in X$.

scholarly journals Discrepancy properties and conjugacy classes of interval exchange transformations

2021 ◽  
Author(s):  
Christian Weiß
Keyword(s):  
Conjugacy Class ◽  
Exceptional Case ◽  
Conjugacy Classes ◽  
Interval Exchange ◽  
Class Invariant ◽  
Interval Exchange Transformations ◽  
Star Discrepancy ◽  
Uniquely Ergodic

AbstractInterval exchange transformations are typically uniquely ergodic maps and therefore have uniformly distributed orbits. Their degree of uniformity can be measured in terms of the star-discrepancy. Few examples of interval exchange transformations with low-discrepancy orbits are known so far and only for $$n=2,3$$ n = 2 , 3 intervals, there are criteria to completely characterize those interval exchange transformations. In this paper, it is shown that having low-discrepancy orbits is a conjugacy class invariant under composition of maps. To a certain extent, this approach allows us to distinguish interval exchange transformations with low-discrepancy orbits from those without. For $$n=4$$ n = 4 intervals, the classification is almost complete with the only exceptional case having monodromy invariant $$\rho = (4,3,2,1)$$ ρ = ( 4 , 3 , 2 , 1 ) . This particular monodromy invariant is discussed in detail.

scholarly journals A switched server system semiconjugate to a minimal interval exchange

2019 ◽  
Vol 31 (4) ◽  
pp. 682-708
Author(s):  
FILIPE FERNANDES ◽  
BENITO PIRES
Keyword(s):  
Limit Cycles ◽  
Queueing Systems ◽  
Topological Dynamics ◽  
Interval Exchange ◽  
Interval Exchange Transformations ◽  
Server Systems ◽  
Asymptotically Periodic ◽  
Ultimate Result ◽  
Uniquely Ergodic ◽  
Server System

Switched server systems are mathematical models of manufacturing, traffic and queueing systems that have being studied since the early 1990s. In particular, it is known that typically the dynamics of such systems is asymptotically periodic: each orbit of the system converges to one of its finitely many limit cycles. In this article, we provide an explicit example of a switched server system with exotic behaviour: each orbit of the system converges to the same Cantor attractor. To accomplish this goal, we bring together recent advances in the understanding of the topological dynamics of piecewise contractions and interval exchange transformations (IETs) with flips. The ultimate result is a switched server system whose Poincaré map is semiconjugate to a minimal and uniquely ergodic IET with flips.

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>

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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