TY - JOUR
T1 - The eigenvalue problem for linear and affine iterated function systems
AU - Barnsley, Michael
AU - Vince, Andrew
PY - 2011/12/15
Y1 - 2011/12/15
N2 - The eigenvalue problem for a linear function L centers on solving the eigen-equation Lx=λx. 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.
AB - The eigenvalue problem for a linear function L centers on solving the eigen-equation Lx=λx. 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.
KW - Eigenvalue problem
KW - Iterated function system
KW - Joint spectral radius
UR - http://www.scopus.com/inward/record.url?scp=80051584357&partnerID=8YFLogxK
U2 - 10.1016/j.laa.2011.05.011
DO - 10.1016/j.laa.2011.05.011
M3 - Article
SN - 0024-3795
VL - 435
SP - 3124
EP - 3138
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - 12
ER -