Piecewise linear maps, Liapunov exponents and entropy

Abstract

Let L A = { f A , x : x is a partition of [ 0 , 1 ] } be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if f A , x ∈ L A , then the Liapunov exponent λ ( x ) of f A , x is equal to a measure theoretic entropy h m A , x of f A , x , where m A , x is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that max x λ ( x ) = max x h m A , x = log ( λ 1 ) , where λ 1 is the maximal eigenvalue of A.