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 Extensive amenability and an application to interval exchanges

2016 ◽  
Vol 38 (1) ◽  
pp. 195-219 ◽  
Author(s):  
KATE JUSCHENKO ◽  
NICOLÁS MATTE BON ◽  
NICOLAS MONOD ◽  
MIKAEL DE LA SALLE
Keyword(s):  
Random Walks ◽  
Group Actions ◽  
Interval Exchange ◽  
General Construction ◽  
Poisson Boundary ◽  
Semidirect Products ◽  
Interval Exchange Transformations ◽  
Interval Exchanges ◽  
Probabilistic Proof ◽  
Rational Rank

Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group$\text{IET}$of interval exchange transformations that have angular components of rational rank less than or equal to two. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on$\text{IET}$and show that there are subgroups$G<\text{IET}$admitting no finitely supported measure with trivial boundary.

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].

scholarly journals On the number of ergodic measures for minimal shifts with eventually constant complexity growth

2016 ◽  
Vol 37 (7) ◽  
pp. 2099-2130
Author(s):  
MICHAEL DAMRON ◽  
JON FICKENSCHER
Keyword(s):  
Symbolic Dynamics ◽  
Explicit Description ◽  
Probability Measures ◽  
Bounded Number ◽  
Ergodic Measures ◽  
The One ◽  
Constant Growth ◽  
Block Growth ◽  
Linear Block

In 1985, Boshernitzan showed that a minimal (sub)shift satisfying a linear block growth condition must have a bounded number of ergodic probability measures. Recently, this bound was shown to be sharp through examples constructed by Cyr and Kra. In this paper, we show that under the stronger assumption of eventually constant growth, an improved bound exists. To this end, we introduce special Rauzy graphs. Variants of the well-known Rauzy graphs from symbolic dynamics, these graphs provide an explicit description of how a Rauzy graph for words of length $n$ relates to the one for words of length $n+1$ for each $n=1,2,3,\ldots \,$.

Sign in / Sign up

Export Citation Format

Share Document