IFS measures on generalized Bratteli diagrams

Abstract

The purpose of or paper is a general analysis of path space measures. Our focus is a certain path space analysis on generalized Bratteli diagrams. We use this in a systematic study of systems of self-similar measures (the term “IFS measures” is used in the paper) for both types of such diagrams, discrete and continuous. In special cases, such measures arise in the study of iterated function systems (IFS). In the literature, similarity may be defined by, e.g., systems of affine maps (Sierpinski), or systems of conformal maps (Julia). We study new classes of semi-branching function systems related to stationary Bratteli diagrams. The latter play a big role in our understanding of new forms of harmonic analysis on fractals. The measures considered here arise in classes of discrete-time, multi-level dynamical systems where a similarity is specified between levels. These structures are made precise by prescribed systems of functions which in turn serve to define self-similarity, i.e., similarity of large scales, and small scales. For path space systems, in our main result, we give a necessary and sufficient condition for existence of such generalized IFS measures. For the corresponding semi-branching function systems, we further identify the measures which are also shift-invariant.