Piecewise linear maps, Liapunov exponents and entropy

Let LA = {fA,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 fA,x ∈ LA, then the Liapunov exponent λ(x) of fA,x is equal to a measure theoretic entropy hmA,x of fA,x, where mA,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 maxx λ(x) = maxx hmA,x = log(λ1), where λ1 is the maximal eigenvalue of A. © 2007 Elsevier Inc. All rights reserved