The Eigenvalue Problem for Linear and Affine Iterated Function Systems

TL;DR

This paper extends the eigenvalue problem to linear and affine iterated function systems, establishing the existence and uniqueness of an eigenvalue equal to the joint spectral radius and characterizing the eigenset.

Contribution

It generalizes the eigenvalue problem to iterated function systems and links the eigenvalue to the joint spectral radius, providing geometric properties of the eigenset.

Findings

Eigenvalue equals the joint spectral radius for irreducible systems

Unique eigenset is centrally symmetric and star-shaped

Results encompass previous theorems as corollaries

Abstract

The eigenvalue problem for a linear function L centers on solving the eigen-equation Lx = rx. This paper generalizes the eigenvalue problem from a single linear function to an iterated function system F consisting of possibly an infinite number of linear or affine functions. The eigen-equation becomes F(X) = rX, where r>0 is real, X is a compact set, and F(X)is the union of f(X), for f in F. The main result is that an irreducible, linear iterated function system F has a unique eigenvalue r equal to the joint spectral radius of the functions in F and a corresponding eigenset S that is centrally symmetric, star-shaped, and full dimensional. Results of Barabanov and of Dranishnikov-Konyagin-Protasov on the joint spectral radius follow as corollaries.