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The Role of Transfer Operators and Shifts in the Study of Fractals: Encoding-Models, Analysis and Geometry, Commutative and Non-commutative
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The Role of Transfer Operators and Shifts in the Study of Fractals: Encoding-Models, Analysis and Geometry, Commutative and Non-commutative

Dorin Ervin Dutkay and Palle E. T Jorgensen
Geometry and Analysis of Fractals, pp.65-95
Springer Proceedings in Mathematics & Statistics, v. 88, Springer Berlin Heidelberg
2014
DOI: 10.1007/978-3-662-43920-3_3

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Abstract

We study a class of dynamical systems in L2 spaces of infinite products X. Fix a compact Hausdorff space B. Our setting encompasses such cases when the dynamics on X = Bℕ is determined by the one-sided shift in X, and by a given transition-operator R. Our results apply to any positive operator R in C(B) such that R1 = 1. From this we obtain induced measures Σ on X, and we study spectral theory in the associated L2(X,Σ). For the second class of dynamics, we introduce a fixed endomorphism r in the base space B, and specialize to the induced solenoid Sol(r). The solenoid Sol(r) is then naturally embedded in X = Bℕ, and r induces an automorphism in Sol(r). The induced systems will then live in L2(Sol(r),Σ). The applications include wavelet analysis, both in the classical setting of ℝn, 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 there prescribed to preserve a foliation by meridional disks. And our setting includes the study of Julia set-attractors in complex dynamics.
28A80 37A30 37C30 37C85 46G25 47B65 Attractors Cantor Fractal Infinite-product measures Ruelle-operators Shift-spaces Solenoid Transition operators Wavelet representations Wavelets

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