scholarly journals Hausdorff dimension for ergodic measures of interval exchange transformations

2008 ◽  
Vol 2 (3) ◽  
pp. 457-464 ◽  
Author(s):  
Jon Chaika ◽  
Keyword(s):  
Hausdorff Dimension ◽  
Interval Exchange ◽  
Interval Exchange Transformations ◽  
Ergodic Measures

scholarly journals Mild mixing of certain interval-exchange transformations

2017 ◽  
Vol 39 (1) ◽  
pp. 248-256
Author(s):  
DONALD ROBERTSON
Keyword(s):  
Hausdorff Dimension ◽  
Interval Exchange ◽  
Interval Exchange Transformations ◽  
Recurrent Type ◽  
Recurrent Interval

We prove that irreducible, linearly recurrent, type W interval-exchange transformations are always mild mixing. For every irreducible permutation, the set of linearly recurrent interval-exchange transformations has full Hausdorff dimension.

scholarly journals The number of ergodic measures for transitive subshifts under the regular bispecial condition

2020 ◽  
pp. 1-55
Author(s):  
MICHAEL DAMRON ◽  
JON FICKENSCHER
Keyword(s):  
Dynamical System ◽  
Shift Operator ◽  
Interval Exchange ◽  
Combinatorial Proof ◽  
Interval Exchange Transformations ◽  
Left Shift ◽  
Ergodic Measures ◽  
Finite Set ◽  
Limiting Value ◽  
Shift Dynamical System

Abstract If $\mathcal {A}$ is a finite set (alphabet), the shift dynamical system consists of the space $\mathcal {A}^{\mathbb {N}}$ of sequences with entries in $\mathcal {A}$ , along with the left shift operator S. Closed S-invariant subsets are called subshifts and arise naturally as encodings of other systems. In this paper, we study the number of ergodic measures for transitive subshifts under a condition (‘regular bispecial condition’) on the possible extensions of words in the associated language. Our main result shows that under this condition, the subshift can support at most $({K+1})/{2}$ ergodic measures, where K is the limiting value of $p(n+1)-p(n)$ , and p is the complexity function of the language. As a consequence, we answer a question of Boshernitzan from 1984, providing a combinatorial proof for the bound on the number of ergodic measures for interval exchange transformations.

scholarly journals SELF-INVERSES, LAGRANGIAN PERMUTATIONS AND MINIMAL INTERVAL EXCHANGE TRANSFORMATIONS WITH MANY ERGODIC MEASURES

2014 ◽  
Vol 16 (01) ◽  
pp. 1350019 ◽  
Author(s):  
JONATHAN FICKENSCHER
Keyword(s):  
Sufficient Conditions ◽  
Explicit Construction ◽  
Interval Exchange ◽  
Probability Measures ◽  
Exchange Transformation ◽  
Interval Exchange Transformations ◽  
Lagrangian Subspace ◽  
Ergodic Measures ◽  
Rauzy Classes

Thanks to works by Kontsevich and Zorich followed by Boissy, we have a classification of all Rauzy Classes of any given genus. It follows from these works that Rauzy Classes are closed under the operation of inverting the permutation. In this paper, we shall prove the existence of self-inverse permutations in every Rauzy Class by giving an explicit construction of such an element satisfying the sufficient conditions. We will also show that self-inverse permutations are Lagrangian, meaning any suspension has its vertical cycles span a Lagrangian subspace in homology. This will simplify the proof of a lemma in a work by Forni. Veech proved a bound on the number of distinct ergodic probability measures for a given minimal interval exchange transformation. We verify that this bound is sharp by constructing examples in each Rauzy Class.

scholarly journals Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Author(s):  
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh
Keyword(s):  
Risk Management ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Management Model ◽  
Closing Lemma ◽  
Symbolic Model ◽  
Metric Properties ◽  
Ergodic Measures ◽  
Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

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