scholarly journals Persistence of wandering intervals in self-similar affine interval exchange transformations

2009 ◽  
Vol 30 (3) ◽  
pp. 665-686 ◽  
Author(s):  
XAVIER BRESSAUD ◽  
PASCAL HUBERT ◽  
ALEJANDRO MAASS
Keyword(s):  
Interval Exchange ◽  
Interval Exchange Transformation ◽  
Algebraic Condition ◽  
Exchange Transformation ◽  
Interval Exchange Transformations ◽  
Affine Maps ◽  
Interval Exchanges ◽  
Self Similar ◽  
Associated Matrix

AbstractIn this article we prove that given a self-similar interval exchange transformation T(λ,π), whose associated matrix verifies a quite general algebraic condition, there exists an affine interval exchange transformation with wandering intervals that is semi-conjugated to it. That is, in this context the existence of Denjoy counterexamples occurs very often, generalizing the result of Cobo [Piece-wise affine maps conjugate to interval exchanges. Ergod. Th. & Dynam. Sys.22 (2002), 375–407].

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 Infinite interval exchange transformations from shifts

2016 ◽  
Vol 37 (6) ◽  
pp. 1935-1965 ◽  
Author(s):  
LUIS-MIGUEL LOPEZ ◽  
PHILIPPE NARBEL
Keyword(s):  
Topological Entropy ◽  
Interval Exchange ◽  
Infinite Interval ◽  
Linear Factor ◽  
Interval Exchange Transformations ◽  
Interval Exchanges ◽  
Finiteness Properties ◽  
Block Growth ◽  
Linear Block

We show that minimal shifts with zero topological entropy are topologically conjugate to interval exchange transformations, which are generally infinite. When these shifts have linear factor complexity (linear block growth), the conjugate interval exchanges are proved to satisfy strong finiteness properties.

scholarly journals There exists a topologically mixing interval exchange transformation

2011 ◽  
Vol 32 (3) ◽  
pp. 869-875 ◽  
Author(s):  
JON CHAIKA
Keyword(s):  
Interval Exchange ◽  
Interval Exchange Transformation ◽  
Exchange Transformation ◽  
Topologically Mixing

AbstractWe prove the existence of a topologically mixing interval exchange transformation and prove that no interval exchange is topologically mixing of all orders.

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