IFS measures on generalized Bratteli diagrams

TL;DR

This paper provides a comprehensive analysis of path space measures on generalized Bratteli diagrams, introducing new classes of semi-branching function systems and establishing conditions for the existence and invariance of IFS measures.

Contribution

It introduces a systematic study of IFS measures on generalized Bratteli diagrams, including new semi-branching function systems and criteria for measure existence and invariance.

Findings

Necessary and sufficient conditions for the existence of generalized IFS measures.

Identification of shift-invariant measures in semi-branching function systems.

Application to harmonic analysis on fractals and multi-level dynamical systems.

Abstract

The purpose of the 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 plays 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 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., the similarity of large scales, and small scales. For path space systems, in our main result, we give a necessary and sufficient condition for the existence of such generalized IFS measures. For the corresponding semi-branching function systems, we further identify the measures which are also shift-invariant.