RDS on Homogeneous Spaces of the General Linear Group

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SummaryFollowing the pioneering work of Furstenberg [153], the study of the asymptotic behavior (law of large numbers, positivity of top Lyapunov exponent alias Furstenberg constant, simplicity of the Lyapunov spectrum, central limit theorem, large deviations principle, etc.) of products of i.i.d. random matrices (more generally, products of i.i.d. random variables in a Lie group) became an important research area. This development was highlighted in 1985 by Guivarc’h and Raugi’s profound paper [163] and the book [77] by Bougerol and Lacroix.Furstenberg made the following basic observation: In order to determine the asymptotic behavior of the product Φ n = A n · · · A l of group-valued i.i.d. random variables (A n ), one has to let the group act on certain manifolds M (Furstenberg boundaries — in the matrix case Grassmannian and flag manifolds), (g, x) ↦ gx, and study e. g. the stationary measures of the transition probability P(x, B) = ℙ{g: gx ∈ B} of the Markov chain x n =Φ n x 0 on M. Furstenberg’s approach proved to be extremely fertile and practically every author investigating products of random matrices has made use of it.One of the lasting outcomes was Furstenberg’s formula for the top Lyapunov exponent as an integral over projective space [153: Formula (7.5)]. This formula was independently found by Khasminskii [205, 206] for linear SDE.As explained in the Preface, we decided to omit the subject — except for the study of the action of linear cocycles on certain homogeneous spaces, thus permitting us in particular to derive Furstenberg-Khasminskii formulas for all Lyapunov exponents.More specifically, in this chapter we will study some nonlinear RDS which are induced by a Gl(d, ℝ)-valued cocycle Φ on homogeneous spaces of Gl (d, ℝ).The most interesting of these homogeneous spaces for us are: the unit sphere S d−1, the projective space P d−1, the Grassmannian manifold G k (d) of k-planes, the Grassmannian manifold G k + (d) of oriented k-planes, the Stiefel manifold St k (d) of orthonormal k-frames (the action of Gl (d, ℝ) is followed by orthonormalization) and the flag manifolds F τ (d), where τ = (d 1, ..., d r ), 1 ≤ d l < ... < d r ≤ d, is a multi-index of dimensions d i, of subspaces V i and f = (V1 ⊂ V2 ⊂ ... ⊂ V r ) is a generic element of F τ (d).We also need to study RDS on a principal bundle and investigate under which conditions it induces an RDS on the base manifold. In case that the original RDS has a generator, we determine the generator of the induced system. The cases of particular interest for us are the principal bundles \(S{t_2}(d)\mathop \to \limits^{\mathbb{Z}_2^2} {F_{1,2}}(d),S{t_k}(d)\mathop \to \limits^{O(k,\mathbb{R})} {G_K}(d)\;and\;S{t_k}(d)\mathop \to \limits^{SO(k,\mathbb{R})} G_k^ + (d)\) which are the basis of our theory of rotation numbers in Sect. 6.5.In Sect. 6.1 we collect some “abstract nonsense” about group-valued cocycles and cocycles on principal bundles.Sect. 6.2 is devoted to the dynamics and ergodic theory of the RDS induced on S d−1 and P d−1 by a linear RDS Φ. We determine all invariant measures (Theorem 6.2.3) and their spectrum and splitting (Theorem 6.2.20). On this basis we obtain Furstenberg-Khasminskii formulas (i. e. phase averages over S d−1) for all Lyapunov exponents of Φ (Theorem 6.2.8 for the real noise case and Theorem 6.2.14 for the white noise case). The white noise formulas are derived by anticipative calculus.In order to obtain Furstenberg-Khasminskii formulas for sums of Lyapunov exponents (Theorem 6.3.3) we have to study the RDS induced by Φ on Grassmannian manifolds (see Sect. 6.3).Manifold versions of the above are treated in Sect. 6.4.Finally, Sect. 6.5 is devoted to our concept of rotation numbers, a simultaneous generalization of the two-dimensional concept and the notion of “imaginary part of eigenvalue of a matrix” to d dimensions and to nonlinear RDS. We prove an MET for rotation numbers which states that if the Lyapunov spectrum is simple, then every two- plane has a rotation number ρ, and this ρ is taken from a finite collection of basic rotation numbers ρ ij realized in the canonical planes E i Λ E j (Theorem 6.5.14 for the real noise case and Theorem 6.5.16 for the white noise case).