EXTREMAL ERGODIC MEASURES AND THE FINITENESS PROPERTY OF MATRIX SEMIGROUPS

Let S = {S-1, ... , S-K} be a finite set of complex d x d matrices and Sigma(+)(K) be the compact space of all one-sided infinite sequences i. : N -> {1, ... , K}. An ergodic probability mu(*) of the Markov shift theta : Sigma(+)(K) -> Sigma(+)(K) ;i, -> i. + 1, is called " extremal" for S if rho(S) = lim(n ->infinity) root parallel to S-i1 ... S-in parallel to holds for mu(*)-a.e. i. is an element of Sigma(+)(K), where rho(S) denotes the generalized/joint spectral radius of S. Using the extremal norm and the Kingman subadditive ergodic theorem, it is shown that S has the spectral finiteness property (i. e. rho(S) = (n)root rho((Si1) ... S-in) for some finite-length word (i(1), ... , i(n))) if and only if for some extremal measure mu* of S, it has at least one periodic density point i. is an element of Sigma(+)(K)