scholarly journals Embeddings of interval exchange transformations into planar piecewise isometries

2018 ◽  
Vol 40 (5) ◽  
pp. 1153-1179 ◽  
Author(s):  
PETER ASHWIN ◽  
AREK GOETZ ◽  
PEDRO PERES ◽  
ANA RODRIGUES
Keyword(s):  
Necessary Conditions ◽  
Interval Exchange ◽  
One Dimensional ◽  
Interval Exchange Transformations ◽  
Higher Dimensional ◽  
Piecewise Isometries ◽  
Dynamical Properties ◽  
Similarities And Differences

Althoughpiecewise isometries(PWIs) are higher-dimensional generalizations of one-dimensionalinterval exchange transformations(IETs), their generic dynamical properties seem to be quite different. In this paper, we consider embeddings of IET dynamics into PWI with a view to better understanding their similarities and differences. We derive some necessary conditions for existence of such embeddings using combinatorial, topological and measure-theoretic properties of IETs. In particular, we prove that continuous embeddings of minimal 2-IETs into orientation-preserving PWIs are necessarily trivial and that any 3-PWI has at most one non-trivially continuously embedded minimal 3-IET with the same underlying permutation. Finally, we introduce a family of 4-PWIs, with an apparent abundance of invariant non-smooth fractal curves supporting IETs, that limit to a trivial embedding of an IET.

scholarly journals Invariant measures with bounded variation densities for piecewise area preserving maps

2012 ◽  
Vol 33 (2) ◽  
pp. 624-642 ◽  
Author(s):  
YIWEI ZHANG ◽  
CONGPING LIN
Keyword(s):  
Bounded Variation ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Topological Transitivity ◽  
Borel Measures ◽  
Interval Exchange Transformations ◽  
Area Preserving Maps ◽  
Piecewise Isometries ◽  
Area Preserving ◽  
Absolutely Continuous Invariant

AbstractWe investigate the properties of absolutely continuous invariant probability measures (ACIPs), especially those measures with bounded variation densities, for piecewise area preserving maps (PAPs) on ℝd. This class of maps unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where Lebesgue measure is locally preserved. Using a functional analytic approach, we first explore the relationship between topological transitivity and uniqueness of ACIPs, and then give an approach to construct invariant measures with bounded variation densities for PWIs. Our results ‘partially’ answer one of the fundamental questions posed in [13]—to determine all invariant non-atomic probability Borel measures in piecewise rotations. When restricting PAPs to interval exchange transformations (IETs), our results imply that for non-uniquely ergodic IETs with two or more ACIPs, these ACIPs have very irregular densities, i.e. they have unbounded variation.

Interval translation mappings

1995 ◽  
Vol 15 (5) ◽  
pp. 821-832 ◽  
Author(s):  
Michael Boshernitzan ◽  
Isaac Kornfeld
Keyword(s):  
Invariant Measures ◽  
Cantor Sets ◽  
The Other ◽  
Interval Exchange ◽  
Interval Exchange Transformations ◽  
Other Hand ◽  
Dynamical Properties

AbstractA class of locally isometric, but not necessarily invertible mappings of an interval is considered. We show that under some conditions the study of the dynamical properties of these mappings can be reduced to interval exchange transformations. On the other hand, there are examples of mappings in this class with ergodic invariant measures supported by Cantor sets. The so-called μβ -sets studied by Y. Katznelson appear naturally in such examples.

scholarly journals Cone exchange transformations and boundedness of orbits

2009 ◽  
Vol 30 (5) ◽  
pp. 1311-1330 ◽  
Author(s):  
PETER ASHWIN ◽  
AREK GOETZ
Keyword(s):  
Sufficient Conditions ◽  
Point Of View ◽  
Unique Ergodicity ◽  
Interval Exchange ◽  
Ergodic Properties ◽  
Interval Exchange Transformations ◽  
Piecewise Isometries ◽  
Almost All ◽  
Necessary And Sufficient ◽  
Negative Flux

AbstractWe introduce a class of two-dimensional piecewise isometries on the plane that we refer to as cone exchange transformations (CETs). These are generalizations of interval exchange transformations (IETs) to 2D unbounded domains. We show for a typical CET that boundedness of orbits is determined by ergodic properties of an associated IET and a quantity we refer to as the ‘flux at infinity’. In particular we show, under an assumption of unique ergodicity of the associated IET, that a positive flux at infinity implies unboundedness of almost all orbits outside some bounded region, while a negative flux at infinity implies boundedness of all orbits. We also discuss some examples of CETs for which the flux is zero and/or we do not have unique ergodicity of the associated IET; in these cases (which are of great interest from the point of view of applications such as dual billiards) it remains an outstanding problem to find computable necessary and sufficient conditions for boundedness of orbits.

scholarly journals CUTTING AND SHUFFLING A LINE SEGMENT: MIXING BY INTERVAL EXCHANGE TRANSFORMATIONS

2012 ◽  
Vol 22 (12) ◽  
pp. 1230041 ◽  
Author(s):  
MARISSA K. KROTTER ◽  
IVAN C. CHRISTOV ◽  
JULIO M. OTTINO ◽  
RICHARD M. LUEPTOW
Keyword(s):  
Line Segment ◽  
Computational Study ◽  
Interval Exchange ◽  
Usual Sense ◽  
Ergodic Theorems ◽  
Mixing Processes ◽  
One Dimensional ◽  
Interval Exchange Transformations ◽  
Number Of Iterations ◽  
Finite Number Of Iterations

We present a computational study of finite-time mixing of a line segment by cutting and shuffling. A family of one-dimensional interval exchange transformations is constructed as a model system in which to study these types of mixing processes. Illustrative examples of the mixing behaviors, including pathological cases that violate the assumptions of the known governing theorems and lead to poor mixing, are shown. Since the mathematical theory applies as the number of iterations of the map goes to infinity, we introduce practical measures of mixing (the percent unmixed and the number of intermaterial interfaces) that can be computed over given (finite) numbers of iterations. We find that good mixing can be achieved after a finite number of iterations of a one-dimensional cutting and shuffling map, even though such a map cannot be considered chaotic in the usual sense and/or it may not fulfill the conditions of the ergodic theorems for interval exchange transformations. Specifically, good shuffling can occur with only six or seven intervals of roughly the same length, as long as the rearrangement order is an irreducible permutation. This study has implications for a number of mixing processes in which discontinuities arise either by construction or due to the underlying physics.

Approximations of Dynamical Systems and Their Applications to Cryptography

2003 ◽  
Vol 13 (07) ◽  
pp. 1937-1948 ◽  
Author(s):  
J. M. Amigó ◽  
J. Szczepański
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Block Structure ◽  
Point Of View ◽  
Interval Exchange ◽  
Linear Cryptanalysis ◽  
New Approach ◽  
Interval Exchange Transformations ◽  
Measure Spaces ◽  
Dynamical Properties

During the last years a new approach to construct safe block and stream ciphers has been developed using the theory of dynamical systems. Since a block cryptosystem is generally, from the mathematical point of view, a family (parametrized by the keys) of permutations of n-bit numbers, one of the main problems of this approach is to adapt the dynamics defined by a map f to the block structure of the cryptosystem. In this paper we propose a method based on the approximation of f by periodic maps Tn (v.g. some interval exchange transformations). The approximation of automorphisms of measure spaces by periodic automorphisms was introduced by Halmos and Rohlin. One important aspect studied in our paper is the relation between the dynamical properties of the map f (say, ergodicity or mixing) and the immunity of the resulting cipher to cryptolinear attacks, which is currently one of the standard benchmarks for cryptosystems to be considered secure. Linear cryptanalysis, first proposed by M. Matsui, exploits some statistical inhomogeneities of expressions called linear approximations for a given cipher. Our paper quantifies immunity to cryptolinear attacks in terms of the approximation speed of the map f by the periodic Tn. We show that the most resistant block ciphers are expected when the approximated dynamical system is mixing.

scholarly journals Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds

2020 ◽  
Vol 8 (1) ◽  
pp. 68-91
Author(s):  
Gianmarco Giovannardi
Keyword(s):  
Characteristic Direction ◽  
Fixed Degree ◽  
One Dimensional ◽  
First Order ◽  
Natural Choice ◽  
Higher Dimensional ◽  
Graded Manifold ◽  
The One ◽  
Graded Manifolds ◽  
Holonomy Map

AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

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