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

Abstract

We study a class of dynamical systems in $L^2$ spaces of infinite products $X$. Fix a compact Hausdorff space $B$. Our setting encompasses such cases when the dynamics on $X = B^\mathbb {N}$ 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 $\Sigma$ on $X$, and we study spectral theory in the associated $L^2(X,\Sigma )$. For the second class of dynamics, we introduce a fixed endomorphism $r$ in the base space $B$, and specialize to the induced solenoid $\mathrm{Sol }(r)$. The solenoid $\mathrm{Sol }(r)$ is then naturally embedded in $X = B^\mathbb {N}$, and $r$ induces an automorphism in $\mathrm{Sol }(r)$. The induced systems will then live in $L^2(\mathrm{Sol }(r), \Sigma )$. The applications include wavelet analysis, both in the classical setting of $\mathbb {R}^n$, 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.