The eigenvalue problem for linear and affine iterated function systems

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.