Book chapter
The Role of Transfer Operators and Shifts in the Study of Fractals: Encoding-Models, Analysis and Geometry, Commutative and Non-commutative
Geometry and Analysis of Fractals, pp.65-95
Springer Proceedings in Mathematics & Statistics, Springer Berlin Heidelberg
08/02/2014
DOI: 10.1007/978-3-662-43920-3_3
Abstract
We study a class of dynamical systems in L2\documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document} spaces of infinite products X\documentclass[12pt]{minimal}
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\begin{document}$$X$$\end{document}. Fix a compact Hausdorff space B\documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document}. Our setting encompasses such cases when the dynamics on X=BN\documentclass[12pt]{minimal}
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\begin{document}$$X = B^\mathbb {N}$$\end{document} is determined by the one-sided shift in X\documentclass[12pt]{minimal}
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\begin{document}$$X$$\end{document}, and by a given transition-operator R\documentclass[12pt]{minimal}
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\begin{document}$$R$$\end{document}. Our results apply to any positive operator R\documentclass[12pt]{minimal}
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\begin{document}$$R$$\end{document} in C(B)\documentclass[12pt]{minimal}
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\begin{document}$$C(B)$$\end{document} such that R1=1\documentclass[12pt]{minimal}
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\begin{document}$$R1 = 1$$\end{document}. From this we obtain induced measures Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} on X\documentclass[12pt]{minimal}
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\begin{document}$$X$$\end{document}, and we study spectral theory in the associated L2(X,Σ)\documentclass[12pt]{minimal}
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\begin{document}$$L^2(X,\Sigma )$$\end{document}. For the second class of dynamics, we introduce a fixed endomorphism r\documentclass[12pt]{minimal}
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\begin{document}$$r$$\end{document} in the base space B\documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document}, and specialize to the induced solenoid Sol(r)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{Sol }(r)$$\end{document}. The solenoid Sol(r)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{Sol }(r)$$\end{document} is then naturally embedded in X=BN\documentclass[12pt]{minimal}
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\begin{document}$$X = B^\mathbb {N}$$\end{document}, and r\documentclass[12pt]{minimal}
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\begin{document}$$r$$\end{document} induces an automorphism in Sol(r)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{Sol }(r)$$\end{document}. The induced systems will then live in L2(Sol(r),Σ)\documentclass[12pt]{minimal}
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\begin{document}$$L^2(\mathrm{Sol }(r), \Sigma )$$\end{document}. The applications include wavelet analysis, both in the classical setting of Rn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^n$$\end{document}, and Cantor-wavelets in the setting of fractals induced by affine iterated function systems (IFS). But our solenoid analysis includes such hyperbolic systems as the Smale-Williams attractor, with the endomorphism r\documentclass[12pt]{minimal}
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\begin{document}$$r$$\end{document} there prescribed to preserve a foliation by meridional disks. And our setting includes the study of Julia set-attractors in complex dynamics.
Details
- Title: Subtitle
- The Role of Transfer Operators and Shifts in the Study of Fractals: Encoding-Models, Analysis and Geometry, Commutative and Non-commutative
- Creators
- Dorin Ervin Dutkay - University of Central FloridaPalle E. T Jorgensen - University of Iowa
- Resource Type
- Book chapter
- Publication Details
- Geometry and Analysis of Fractals, pp.65-95
- Series
- Springer Proceedings in Mathematics & Statistics
- DOI
- 10.1007/978-3-662-43920-3_3
- eISSN
- 2194-1017
- ISSN
- 2194-1009
- Publisher
- Springer Berlin Heidelberg; Berlin, Heidelberg
- Language
- English
- Date published
- 08/02/2014
- Academic Unit
- Mathematics
- Record Identifier
- 9984241050002771
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