2010/04/27 by Michael F. Barnsley, Michael Barnsley, Barnsley, Michael +2
Mathematics · Physics and Astronomy · #15A18 #28A80 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Metric Geometry (math.MG) #Quantum chaos and dynamical systems #advanced mathematical theories #math.MG #msc:15A18 #msc:28A80
paper · pdf · doi:10.48550/arxiv.1004.5040
18 pages, 3 figures
arxiv created 2010/04/27 · openalex publication_date 2010/04/27 · arxiv updated 2010/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.