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We develop a Hilbert space framework for a number of general multi-scaleproblems from dynamics. The aim is to identify a spectral theory for a class ofsystems based on iterations of a non-invertible endomorphism.  We are motivated by the more familiar approach to wavelet theory which startswith the two-to-one endomorphism $r: z \mapsto z^2$ in the one-torus $\bt$, awavelet filter, and an associated transfer operator. This leads to a scalingfunction and a corresponding closed subspace $V_0$ in the Hilbert space$L^2(\br)$. Using the dyadic scaling on the line $\br$, one has a nested familyof closed subspaces $V_n$, $n \in \bz$, with trivial intersection, and withdense union in $L^2(\br)$. More generally, we achieve the same outcome, but indifferent Hilbert spaces, for a class of non-linear problems. In fact, we seethat the geometry of scales of subspaces in Hilbert space is ubiquitous in theanalysis of multiscale problems, e.g., martingales, complex iteration dynamicalsystems, graph-iterated function systems of affine type, and subshifts insymbolic dynamics. We develop a general framework for these examples whichstarts with a fixed endomorphism $r$ (i.e., generalizing $r(z) = z^2$) in acompact metric space $X$. It is assumed that $r : X\to X$ is onto, andfinite-to-one.Comment: v3, minor addition

Hilbert spaces built on a similarity and on dynamical renormalization