The eigenvalue problem for linear and affine iterated function systems |
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Authors: | Michael Barnsley |
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Institution: | a Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia b University of Florida, Department of Mathematics, 358 Little Hall, P.O. Box 118105, Gainesville, FL 32611-8105, USA |
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Abstract: | The eigenvalue problem for a linear function L centers on solving the eigen-equation . 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)=λX, where λ>0 is real, X is a compact set, and F(X)=?f∈Ff(X). The main result is that an irreducible, linear iterated function system F has a unique eigenvalue λ 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 Dranisnikov-Konyagin-Protasov on the joint spectral radius follow as corollaries. |
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Keywords: | 15A18 28A80 |
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