On Growth Rates of Subadditive Functions for Semiflows

Let ,: X_T+ X be a semiflow on a compact metric space X. A function F: X_T+ X is subadditive with respect to , if F(x, t+s) F(x, t)+F(,(x, t), s). We define the maximal growth rate of F to be supx # X lim supt (1 t) F(x, t). This growth rate is shown to equal the maximal growth rate of the subadditive function restricted to the minimal center of attraction of the semiflow. Applications to Birkhoff sums, characteristic exponents of linear skew-product semiflows on Banach bundles, and average Lyapunov functions are developed. In particular, a relationship between the dynamical spectrum and the measurable spectrum of a linear skew-product flow established by R. A. Johnson, K. J. Palmer, and G. R. Sell (SIAM J. Math. Anal. 18, 1987, 1 33) is extended to semiflows in an infinite dimensional setting. 1998 Academic Press