Finite-time Lyapunov exponents for products of random transformations (Q1397943)

It is shown how continuous products of random transformations constrained by a generic group structure can be studied by using Iwasawa's decompositions into ``angular'', ``diagonal'', and ``shear'' degrees of freedom. The paper contains seven sections. In Section 2 a random process \(X(t)\) is defined by assigning its covariance in terms of intrinsic geometric features of the corresponding algebra. The multiplicative process \(g(t)\) is defined as a time ordered exponential through the appropriate regularization. In Sections 3-5 new variables are defined through the following steps: (i) performimg the Iwasawa decomposition of the process \(g(t);\) (ii) introducing intermediate variables, adapted to the Iwasawa decomposition; (iii) rewriting the Gibbs weight in terms of new variables; (iv) computing the Jacobian factor of the variable transformation; (v) performing a Lie algebra transformation that leads to new, normally distributed variables. In Section 6 the new variables are used to compute the statistics of the finite time Lyapunov spectrum of a generic linear representation of the process \(g(t)\). In Section 7 the cases of \(\text{sl}(n,{\mathbb R})\) and \(\text{sp}(n,{\mathbb R})\) algebras are worked out in detail.