Conjugacies provided by fractal transformations I : Conjugate measures, Hilbert spaces, orthogonal expansions, and flows, on self-referential spaces

TL;DR

This paper explores how fractal transformations can create continuous, measure-preserving maps between self-similar sets, enabling spectral analysis, Fourier analysis with fractal basis functions, and the study of fractal flows on complex geometries.

Contribution

It introduces new methods for using fractal transformations to establish unitary maps between Hilbert spaces on self-similar sets, with applications to spectral analysis and fractal flows.

Findings

Fractal Fourier basis functions are discontinuous at dense points but still effective for approximation.

Fractal transformations can produce measure-preserving, almost everywhere continuous maps between self-similar sets.

Applications include spectral analysis and ergodic flows on fractal and polygonal regions.

Abstract

Theorems and explicit examples are used to show how transformations between self-similar sets (general sense) may be continuous almost everywhere with respect to stationary measures on the sets and may be used to carry well known flows and spectral analysis over from familiar settings to new ones. The focus of this work is on a number of surprising applications including (i) what we call fractal Fourier analysis, in which the graphs of the basis functions are Cantor sets, being discontinuous at a countable dense set of points, yet have very good approximation properties; (ii) Lebesgue measure-preserving flows, on polygonal laminas, whose wave-fronts are fractals. The key idea is to exploit fractal transformations to provide unitary transformations between Hilbert spaces defined on attractors of iterated function systems. Some of the examples relate to work of Oxtoby and Ulam concerning ergodic flows on regions bounded by polygons.