The Eigenvalue Problem for Linear and Affine Iterated Function Systems

Details The Eigenvalue Problem for Linear and Affine Iterated Function Systems The Eigenvalue Problem for Linear and Affine Iterated Function Systems

2010-04-27

arxiv.org/abs/1004.5040v1

Mathematics

Metric Geometry

18 pages, 3 figures

Scientific paper

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.

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