2017/02/23 by Feng, De-Jun, Lo, Chiu-Hong, Shen, Shuang
#28A78 #28A80 #37A60 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Primary 15A60 #Secondary 37D35
paper · doi:10.48550/arxiv.1702.07251
Let \bf M=(M1,…, Mk) be a tuple of real d× d matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether \bf M possesses the following property: there exist two constants λ∈ \Bbb R and C>0 such that for any n∈ \Bbb N and any i1, …, in ∈ \1,…, k\, either Mi1 ⋯ Min=\bf 0 or C-1 eλn ≤ ‖ Mi1 ⋯ Min ‖ ≤ C eλn, where ‖⋅‖ is a matrix norm. The proof is based on symbolic dynamics and the thermodynamic formalism for matrix products. As applications, we are able to check the absolute continuity of a class of overlapping self-similar measures on \Bbb R, the absolute continuity of certain self-affine measures in \Bbb Rd and the dimensional regularity of a class of sofic affine-invariant sets in the plane.