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) is an element of L-A, then the Liapunov exponent lambda(x) of f(A,x) is equal to a measure theoretic entropy h(mA,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) lambda(x) = max(x)h(mA,x) = log(lambda(1)), where lambda(1)is the maximal eigenvalue of A. (c) 2007 Elsevier Inc. All rights reserved.