2011/07/01 by Xiongping Dai, Yu Huang, Dai, Xiongping +3
Mathematics · #15B52 #Dynamical Systems (math.DS) #FOS: Mathematics #Graph theory and applications #Mathematical Dynamics and Fractals #Rings and Algebras (math.RA) #Spectral Theory in Mathematical Physics #math.DS #math.RA #msc:15B52
paper · pdf · doi:10.48550/arxiv.1107.0123
9 pages; accepted by Proceedings of the AMS
arxiv created 2011/07/01 · openalex publication_date 2011/07/01 · arxiv updated 2011/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \bS=\S1,...,SK\ be a finite set of complex d× d matrices and \varSigmaK+ the compact space of all one-sided infinite sequences i\bcdot\colonℕ→\1,...,K\. An ergodic probability μ_* of the Markov shift θ\colon\varSigmaK+→\varSigmaK+; i\bcdot↦ i\bcdot+1, is called "extremal" for \bS, if ρ(\bS)=limn→∞√[n]\normSi1...Sin holds for μ_*-a.e. i\bcdot∈\varSigmaK+, where ρ(\bS) denotes the generalized/joint spectral radius of \bS. Using extremal norm and Kingman subadditive ergodic theorem, it is shown that \bS has the spectral finiteness property (i.e. ρ(\bS)=√[n]ρ(Si1...Sin) for some finite-length word (i1,...,in)) if and only if for some extremal measure μ_* of \bS, it has at least one periodic density point i\bcdot∈\varSigmaK+.