Hilbert spaces built on a similarity and on dynamical renormalization

Abstract

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.

v3, minor additions

References in corpus (4)

• Iterated function systems, Ruelle operators, and invariant projective measures
• Martingales, endomorphisms, and covariant systems of operators in Hilbert space
• Construction of Parseval wavelets from redundant filter systems
• Projective multiresolution analyses for dilations in higher dimensions

Cited by in corpus (7)

• Harmonic analysis of iterated function systems with overlap
• The Measure of a Measurement
• Unitary Representations of Wavelet Groups and Encoding of Iterated Function Systems in Solenoids
• Irreducible wavelet representations and ergodic automorphisms on solenoids
• $C^{\ast }$-algebras associated with algebraic correspondences on the Riemann sphere
• Affine systems: asymptotics at infinity for fractal measures
• Probability measures on solenoids corresponding to fractal wavelets