On the existence of higher order Lyapunov exponents

We prove the existence of an exponential rate of growth for the second derivative of a one-dimensional dynamical system with positive Lyapunov exponent, and prove that the second order Lyapunov exponent thus defined is equal to precisely twice the first order derivative. Similar techniques shed some light on the multidimensional case. This work extends results of Dressier and Farmer, and verifies a conjecture of Farmer and Sidorowich related to a priori estimates for time series forecasting.