Conditional exponents, entropies and a measure of dynamical self-organization

In dynamical systems composed of interacting parts, conditional exponents, conditional exponent entropies and cylindrical entropies are shown to be well defined ergodic invariants which characterize the dynamical selforganization and statitical independence of the constituent parts. An example of interacting Bernoulli units is used to illustrate the nature of these invariants. 1 Conditional exponents The notion of conditional Lyapunov exponents (originally called sub-Lyapunov exponents) was introduced by Pecora and Carroll in their study of synchronization of chaotic systems[1] [2]. It turns out, as I will show below, that, like the full Lyapunov exponent, the conditional exponents are well defined ergodic invariants. Therefore they are reliable quantities to quantify the relation of a global dynamical system to its constituent parts and to characterize dynamical selforganization. Given a dynamical system defined by a map f : M → M , with M ⊂ R the conditional exponents associated to the splitting R × R are the eigenvalues of the limit lim n→∞ (Dkf (x)Dkf (x)) 1 2n (1)