Convexity of the Lyapunov exponent

Let gamma(p) be the maximal Lyapunov exponent for an independently and iden tically distributed (i.i.d). random product of matrices where each factor e quals A with probability p and B with probability 1 - p. Counterexamples sh ow that the two conjectures of E. Key on the convexity and concavity of the function gamma in the interval (0, 1) are too general. Under suitable assu mptions on A,B convexity and concavity of gamma are proved. (C) 1999 Elsevi er Science Inc. All rights reserved.