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

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. © Springer-Verlag Berlin Heidelberg 2014