Dynamical Spectrum and Diffraction

2014 ◽  
Vol 70 (a1) ◽  
pp. C526-C526
Author(s):  
Lorenzo Sadun
Keyword(s):  
Dynamical System ◽  
Measure Theory ◽  
Point Pattern ◽  
Topological Conjugacy ◽  
Diffraction Spectrum ◽  
Dynamical Spectrum ◽  
Famous Conjecture ◽  
And Topology ◽  
Local Complexity ◽  
Shape Deformations

The diffraction spectrum of a point pattern is very closely related to the dynamical spectrum of an associated dynamical system. This dynamical spectrum is invariant under topological conjugacy and measurable conjugacy, and in particular under a large class of shape deformations. Using measure theory and topology, we construct a pure-point diffractive set, with finite local complexity, that is not a Meyer set. This provides a counterexample to a famous conjecture of Lagarias.

scholarly journals Dynamical versus diffraction spectrum for structures with finite local complexity

2014 ◽  
Vol 35 (7) ◽  
pp. 2017-2043 ◽  
Author(s):  
MICHAEL BAAKE ◽  
DANIEL LENZ ◽  
AERNOUT VAN ENTER
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Symbolic Dynamics ◽  
Ergodic Measure ◽  
Pure Point ◽  
Diffraction Spectrum ◽  
Dynamical Spectrum ◽  
Typical Element ◽  
Local Complexity ◽  
Uniquely Ergodic

It is well known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as ergodic Delone dynamical systems, both via suitable embeddings. The connection is rather well understood when the spectrum is pure point, where the two spectral notions are essentially equivalent. In general, however, the dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of finite local complexity and establish the equivalence of the dynamical spectrum with a collection of diffraction spectra of the system and certain factors. This equivalence gives access to the dynamical spectrum via these diffraction spectra. It is particularly useful as the diffraction spectra are often simpler to determine and, in many cases, only very few of them need to be calculated.

scholarly journals Vladimir Abramovich Rokhlin (23 August 1919 to 3 December 1984): Introductory note

1989 ◽  
Vol 9 (4) ◽  
pp. 605-608
Author(s):  
A. Katok
Keyword(s):  
Ergodic Theory ◽  
Measure Theory ◽  
The Other ◽  
Natural Evolution ◽  
And Topology ◽  
Introductory Note ◽  
Time Periods ◽  
The Right ◽  
Short Time ◽  
The Impact

The Ergodic Theory and Dynamical Systems journal pays tribute to V. A. Rokhlin, one of the founders of ergodic theory, a world-renowned topologist and geometer, and a man of tragic fate and exceptional courage. Rokhlin's mathematical heritage splits rather sharply into the ergodic theory – measure theory and topology – geometry parts. This fact has to do with a natural evolution of his interests but also with the keen sense of style in mathematics that Rokhlin possessed to an unusual degree. Naturally, we will concentrate on Rokhlin's contributions to ergodic theory and measure theory, his influence on other mathematicians working in those fields, and the development of some of his ideas. Fortunately, the topology part of Rokhlin's heritage has been superbly presented in Part I of the book ‘A la recherche de la topologie perdue’ published by Birkhaüser in Progress in Mathematics series (v. 62, 1986). The same cannot be said about the work on real algebraic geometry, Rokhlin's last big achievement. The impact of that work, carried out by his students, however, is very much felt now, and is very unlikely to be forgotten or neglected. On the other hand, his work in ergodic theory and measure theory, was restricted primarily to two relatively short time periods, 1947–1950 and 1959–1964, and for the most part was not followed by Rokhlin's immediate students. Hence, it runs a certain risk of being underestimated. We hope that the articles by A. Vershik; S. Yuzvinsky and B: Weiss, published in this issue, will put Rokhlin's work and his influence in ergodic theory into the right perspective.

scholarly journals Unfoldings of discrete dynamical systems

1984 ◽  
Vol 4 (3) ◽  
pp. 421-486 ◽  
Author(s):  
Joel W. Robbin
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Conjugacy Class ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Topological Conjugacy ◽  
Universal Unfolding ◽  
Moser Iteration

AbstractA universal unfolding of a discrete dynamical system f0 is a manifold F of dynamical systems such that each system g sufficiently near f0 is topologically conjugate to an element f of F with the conjugacy φ and the element f depending continuously on f0. An infinitesimally universal unfolding of f0 is (roughly speaking) a manifold F transversal to the topological conjugacy class of f0. Using Nash-Moser iteration we show infinitesimally universal unfoldings are universal and (in part II) give a class of examples relating to moduli of stability introduced by Palis and De Melo.

THE DISTAL AND ASYMPTOTIC SETS FOR CONTINUED FRACTIONS

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950139
Author(s):  
WEIBIN LIU ◽  
SHUAILING WANG
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Underlying Space ◽  
And Topology

For continued fraction dynamical system [Formula: see text], we give a classification of the underlying space [Formula: see text] according to the orbit of a given point [Formula: see text]. The sizes of all classes are determined from the viewpoints of measure, Hausdorff dimension and topology. For instance, the Hausdorff dimension of the distal set of [Formula: see text] is one and the Hausdorff dimension of the asymptotic set is either zero or [Formula: see text] according to [Formula: see text] is rational or not.

Topological Conjugacy Classification of Elementary Cellular Automata with Majority Memory

2017 ◽  
Vol 27 (14) ◽  
pp. 1750217 ◽  
Author(s):  
Haiyun Xu ◽  
Fangyue Chen ◽  
Weifeng Jin
Keyword(s):  
Cellular Automata ◽  
Vector Space ◽  
Symbolic Dynamics ◽  
Continuous Functions ◽  
Equivalence Classes ◽  
Topological Conjugacy ◽  
Symbolic Sequence ◽  
And Topology ◽  
Elementary Cellular Automata

The topological conjugacy classification of elementary cellular automata with majority memory (ECAMs) is studied under the framework of symbolic dynamics. In the light of the conventional symbolic sequence space, the compact symbolic vector space is identified with a feasible metric and topology. A slight change is introduced to present that all global maps of ECAMs are continuous functions, thereafter generating the compact dynamical systems. By exploiting two fundamental homeomorphisms in symbolic vector space, all ECAMs are furthermore grouped into 88 equivalence classes in the sense that different mappings in the same global equivalence are mutually topologically conjugate.

Chaos to multiple mappings from a set-valued view

2020 ◽  
Vol 34 (11) ◽  
pp. 2050108
Author(s):  
Hongbo Zeng ◽  
Lidong Wang ◽  
Tao Sun
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Topological Conjugacy ◽  
Compact Metric Space ◽  
Distributional Chaos ◽  
Continuous Maps ◽  
Weakly Mixing

Let [Formula: see text] be a compact metric space and [Formula: see text] be an [Formula: see text]-tuple of continuous maps from [Formula: see text] to itself. In this paper, we investigate the multiple mappings dynamical system [Formula: see text] with Hausdorff metric Li–Yorke chaos, distributional chaos and distributional chaos in a sequence properties from a set-valued view. On the basis of this research, we draw main conclusions as follows: (i) two topological conjugacy dynamical systems to multiple mappings have simultaneously Hausdorff metric Li–Yorke chaos or distributional chaos. (ii) Hausdorff metric Li–Yorke [Formula: see text]-chaos is equivalent to Hausdorff metric distributional [Formula: see text]-chaos in a sequence. (iii) By giving two examples, we show that there is non-mutual implication between that the multiple mappings [Formula: see text] is Hausdorff metric Li–Yorke chaos and that each element [Formula: see text] [Formula: see text] in [Formula: see text] is Li–Yorke chaos. (iv) For the multiple mappings, weakly mixing implies the Hausdorff metric strongly Li–Yorke chaos and Hausdorff metric distributional chaos in a sequence.

scholarly journals Squirals and beyond: substitution tilings with singular continuous spectrum

2013 ◽  
Vol 34 (4) ◽  
pp. 1077-1102 ◽  
Author(s):  
MICHAEL BAAKE ◽  
UWE GRIMM
Keyword(s):  
Dynamical System ◽  
Mixed Type ◽  
Constructive Proof ◽  
Singular Continuous Spectrum ◽  
Dynamical Spectrum ◽  
Substitution Rule ◽  
Continuous Components ◽  
Lattice Dynamical System ◽  
Substitution Tilings ◽  
Inflation Rule

AbstractThe squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of${ \mathbb{Z} }^{2} $. In particular, its balanced version has purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive proof that admits a generalization to bijective block substitutions of trivial height on${ \mathbb{Z} }^{d} $.

scholarly journals A characterization of the Morse minimal set up to topological conjugacy

2008 ◽  
Vol 28 (5) ◽  
pp. 1443-1451 ◽  
Author(s):  
ETHAN M. COVEN ◽  
MICHAEL KEANE ◽  
MICHELLE LEMASURIER
Keyword(s):  
Dynamical System ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Conjugacy ◽  
Orbit Closure ◽  
Minimal Set ◽  
Set Up ◽  
Necessary And Sufficient

AbstractWe establish necessary and sufficient conditions for a dynamical system to be topologically conjugate to the Morse minimal set, the shift orbit closure of the Morse sequence. Conditions for topological conjugacy to the closely related Toeplitz minimal set are also derived.

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