Piecewise linear maps, Liapunov exponents and entropy

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_{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 λ (x) = max_x h_{mA, x} = log (λ₁), where λ₁ is the maximal eigenvalue of A.