We develop a Hilbert space framework for a number of general multi-scale
problems from dynamics. The aim is to identify a spectral theory for a class of
systems based on iterations of a non-invertible endomorphism.
We are motivated by the more familiar approach to wavelet theory which starts
with the two-to-one endomorphism $r: z \mapsto z^2$ in the one-torus $\bt$, a
wavelet filter, and an associated transfer operator. This leads to a scaling
function 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 family
of closed subspaces $V_n$, $n \in \bz$, with trivial intersection, and with
dense union in $L^2(\br)$. More generally, we achieve the same outcome, but in
different Hilbert spaces, for a class of non-linear problems. In fact, we see
that the geometry of scales of subspaces in Hilbert space is ubiquitous in the
analysis of multiscale problems, e.g., martingales, complex iteration dynamical
systems, graph-iterated function systems of affine type, and subshifts in
symbolic dynamics. We develop a general framework for these examples which
starts with a fixed endomorphism $r$ (i.e., generalizing $r(z) = z^2$) in a
compact metric space $X$. It is assumed that $r : X\to X$ is onto, and
finite-to-one.