The Eigenvalue Problem for Linear and Affine Iterated Function Systems

Michael Barnsley; Andrew Vince

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arXiv · 1004.5040

The Eigenvalue Problem for Linear and Affine Iterated Function Systems

Michael Barnsley·Andrew Vince

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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.

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BibTeX RIS

Michael Barnsley, Andrew Vince. 2010-04-27. The Eigenvalue Problem for Linear and Affine Iterated Function Systems. https://arxiv.org/abs/1004.5040

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