Methods from Multiscale Theory and Wavelets Applied to Nonlinear Dynamics

Dutkay, Dorin Ervin; Jørgensen, Palle E. T.

doi:10.1007/3-7643-7588-4_4

Cited by 18 publications

(18 citation statements)

References 51 publications

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“…In particular, this means that one can apply to the Sierpinski set S A all the techniques for constructions of wavelets on fractals from representations of Cuntz algebras developed, for instance, in [3,4,[12][13][14][16][17][18][19][26][27][28], etc.…”

Section: Sierpinski Fractalsmentioning

confidence: 99%

Cuntz–Krieger Algebras and Wavelets on Fractals

Marcolli

Paolucci

2009

Complex Anal. Oper. Theory

114

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Section: Sierpinski Fractalsmentioning

confidence: 99%

Cuntz–Krieger Algebras and Wavelets on Fractals

Marcolli

Paolucci

2009

Complex Anal. Oper. Theory

114

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“…In the recent years, substantial efforts are made in establishing broad interplay between C * -algebras and non-invertible dynamical systems, actions of semigroups, equivalence relations, (semi-)groupoids, correspondences (see for example works by Exel [21,22], Exel and Vershik [23] Arzumanian and Vershik [4,5], Deaconu [13], Renault [39][40][41], Adji, Laca, Nielsen and Raeburn [1], an Huef and Raeburn [2], Bratteli, Jorgensen and Evans [6], Bratteli and Jorgensen [7,8], Dutkay and Jorgensen [14][15][16][17], Jorgensen [25], Dai and Larson [11], Kajiwara and Watatani [26], Kawamura [29], Watatani [48], Ostrovskyȋ and Samoȋlenko [36], Cuntz and Krieger [10], Matsumoto [35], Eilers [19], Carlsen and Silvestrov [9], and references therein).…”

Section: Theoremmentioning

confidence: 99%

On the Exel Crossed Product of Topological Covering Maps

Carlsen

Silvestrov

2008

Acta Appl Math

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For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C * -algebras C(X) α,ᏸ N introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X) α,ᏸ N is a maximal abelian C * -subalgebra of C(X) α,ᏸ N; any nontrivial two sided ideal of C(X) α,ᏸ N has non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X) α,ᏸ N is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C * -algebras of homeomorphism dynamical systems.Keywords Crossed product algebra · Covering map · Topologically free dynamical system · Maximal abelian subalgebra · Ideals Mathematics Subject Classification (2000) 47L65 · 46L55 · 47L40 · 54H20 · 37B05 · 54H15

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“…The aim of this paper is to revisit the Fourier asymptotics of the measures μ (A,B) in light of recent results on IFS involving dynamics and representation theory, see e.g., [10,11,13,14]. The measures μ (A,B) are equilibrium measures for the corresponding affine system.…”

Section: Introductionmentioning

confidence: 99%

Affine Systems: Asymptotics at Infinity for Fractal Measures

2007

Self Cite

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We study measures on R d which are induced by a class of infinite and recursive iterations in symbolic dynamics. Beginning with a finite set of data, we analyze prescribed recursive iteration systems, each involving subdivisions. The construction includes measures arising from affine and contractive iterated function systems with and without overlap (IFSs), i.e., limit measures μ induced by a finite family of affine mappings in R d (the focus of our paper), as well as equilibrium measures in complex dynamics.By a systematic analysis of the Fourier transform of the measure μ at hand (frequency domain), we identify asymptotic laws, spectral types, dichotomy, and chaos laws. In particular we show that the cases when μ is singular carry a gradation, ranging from Cantor-like fractal measures to measures exhibiting chaos, i.e., a situation when small changes in the initial data produce large fluctuations in the outcome, or rather, the iteration limit (in this case the measures). Our method depends on asymptotic estimates on the Fourier transform of μ for paths at infinity in R d . We show how properties of μ depend on perturbations of the initial data, e.g., variations in a prescribed finite set of affine mappings in R d , in parameters of a rational function in one complex variable (Julia sets and equilibrium measures), or in the entries of a given infinite positive definite matrix.

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Methods from Multiscale Theory and Wavelets Applied to Nonlinear Dynamics

Cited by 18 publications

References 51 publications

Cuntz–Krieger Algebras and Wavelets on Fractals

Cuntz–Krieger Algebras and Wavelets on Fractals

On the Exel Crossed Product of Topological Covering Maps

Affine Systems: Asymptotics at Infinity for Fractal Measures

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