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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 whenthe dynamics on $X = B^\bn$ is determined by the one-sided shift in $X$, and bya 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 $r$ inthe base space $B$, and specialize to the induced solenoid $\Sol(r)$. Thesolenoid $\Sol(r)$ is then naturally embedded in $X = B^\bn$, and $r$ inducesan automorphism in $\Sol(r)$. The induced systems will then live in$L^2(\Sol(r), \Sigma)$.  The applications include wavelet analysis, both in the classical setting of$\br^n$, and Cantor-wavelets in the setting of fractals induced by affineiterated function systems (IFS). But our solenoid analysis includes suchhyperbolic systems as the Smale-Williams attractor, with the endomorphism $r$there prescribed to preserve a foliation by meridional disks. And our settingincludes the study of Julia set-attractors in complex dynamics.

The role of transfer operators and shifts in the study of fractals: encoding-models, analysis and geometry, commutative and non-commutative