Linear Cocycles Research Articles

Linear cocycles with simple Lyapunov spectrum are dense in $L^\infty$

We prove that the set of linear cocycles with simple Lyapunov spectrum is dense in the space of all linear cocycles equipped with the uniform topology.

Jordan normal form for linear cocycles

Jordan normal form for linear cocycles

The giant spiral

Cocycles ofZm actions on compact metric spaces provide a means for constructingRm actions or flows, called suspension flows. It is known that allRm flows with a free dense orbit have an almost one-to-one extension which is a suspension flow. Whenm=1, examples of cocycles are easy to construct; there is a one-to-one correspondence between cocycles and real valued continuous functions. However, whenm>1 the construction of examples of cocycles becomes more problematic. The only existing class of examples, the close to linear cocycles, have strong linearity properties and are well understood. In fact, when theZm action is uniquely ergodic, all cocycles are close to linear. We will show that in general this need not be the case. We present a method, suggested to us by Hillel Furstenberg, for constructing examples of cocycles whenm>1 and use this method to construct a non close to linear cocycle on a minimalZ2 action.

On the simplicity of the Lyapunov spectrum of products of random matrices

Assuming that the underlying probability space is non-atomic, we prove that products of random matrices (linear cocycles) with simple Lyapunov spectrum form an $L^p$-dense set ($1 \leq p < \infty$) in the space of all cocycles satisfying the integrability conditions of the multiplicative ergodic theorem. However, the linear cocycles with one-point spectrum are also $L^p$-dense. Further, in any $L^\infty$-neighborhood of an orthogonal cocycle there is a diagonalizable cocycle.For products of independent identically distributed random matrices (with distribution $\mu$), simplicity of the Lyapunov spectrum holds on a set of $\mu$'s which is open and dense in both the topology of total variation and the topology of weak convergence, hence is generic in both topologies. For products of matrices which form a Markov chain, the spectrum is simple on a set of transition functions dense in the topology of weak convergence.

Topological classification of linear hyperbolic cocycles

In this paper linear hyperbolic cocycles are classified by the relation of topological conjugacy. Roughly speaking, two linear cocycles are conjugate if there exists a homeomorphism which maps their trajectories into each other. The problem of classification of discrete-time deterministic hyperbolic dynamical systems was investigated by Robbin (1972). He proved that there exist 4d classes ofd-dimensional deterministic discrete hyperbolic dynamical systems. We obtain a criterion for topological conjugacy of two linear hyperbolic cocycles and show that the number of classes depends crucially on the ergodic properties of the metric dynamical system over which they are defined. Our result is a generalization of the deterministic theorem of Robbin.

On close to linear cocycles

On close to linear cocycles

Furstenberg-khasminskii formulas for lyapunov exponents via anticipative calculus

According to a classical formula due to Furstenberg and Khasminskii the top Lyapunov exponent of a random linear cocycle can be expressed as a phase average of a function over projective space. Using a description of the Oseledets spaces as intersections of two flags of random subspaces of the phase space, one forward and one backward in time, we derive similar formulas for the remaining Lyapunov exponents of the cocycle generated by a linear stochastic differential equation. This approach makes elements of anticipative calculus enter the scene quite naturally. Indeed, in the correction terms appearing in the analogues of the Furstenberg-Khasminskii formula, averages of the Malliavin gradient of the orthogonal projectors on the Oseledets spaces come into play explicitly

On the Characteristic Classes of Actions of Lattices in Higher Rank Lie Groups

We show that under certain assumptions, the measurable cohomology class of the linear holonomy cocycle of a foliation yields information about the characteristic classes of the foliation. Combined with the results of a previous paper, this yields vanishing theorems for characteristic classes of certain actions of lattices in higher rank semisimple Lie groups.

On the characteristic classes of actions of lattices in higher rank Lie groups

We show that under certain assumptions, the measurable cohomology class of the linear holonomy cocycle of a foliation yields information about the characteristic classes of the foliation. Combined with the results of a previous paper, this yields vanishing theorems for characteristic classes of certain actions of lattices in higher rank semisimple Lie groups. Let F be a discrete group acting by diffeomorphisms on a smooth compact manifold M. Associated to this action are certain characteristic classes in H (F, R), which are constructed as characteristic classes of a natural foliation associated to the group action. The action of F on M induces an action of F on the principal frame bundle P(M) of M and the characteristic classes of the action can be interpreted as obstructions to the existence of invariant geometric structures on M, i.e., principal subbundles of P(M) invariant by the F-action. If F is a lattice in a higher rank semisimple Lie group, then F has strong rigidity properties (see, e.g., [M, Z1]). In an earlier paper [S], we showed, using techniques from ergodic theory, that for a certain class of F-actions there is always an invariant measurable reductive geometric structure, i.e., a measurable principal subbundle with reductive structure group, which is invariant by the F-action. Moreover, the noncompact semisimple part of this reductive group is locally isomorphic to a semisimple factor of the ambient Lie group of F [Zi]. Zimmer [Z3] recently proved this result for a large class of actions (which does not a priori include the class of actions considered in [S]). A natural question is whether these results remain true in the smooth category. The purpose of this paper is to show that the characteristic classes, which obstruct a smooth geometric structure, vanish in the presence of a measurable geometric structure. Explicitly, we have Main Theorem. Let (M, Y) be a codimension n, C 2-foliated manifold and suppose that the linear holonomy cocycle is measurably equivalent to a locally tempered cocycle taking values in a subgroup H c GL(n, R) which is stable under transpose. Then the Weil homomorphism X: H'(g[(n), 0(n)) -Hc(M, Yi) Received by the editors February 24, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R32; Secondary 57S20. Research partially supported by an Alfred P. Sloan Dissertation Fellowship. ? 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

The lyapunov spectrum and stable manifolds for stochastic linear delay equations

A class of linear stochastic retarded functional differential equations is considered. These equations have diffusion coefficients that do not look into the past. It is shown that the trajectories of such equations form a continuous linear cocycle on the underlying state space. At times greater than the delay the cocycle is almost surely compact. Consequently, using an infinite-dimensional Oseledec multiplicative ergodic theorem of Ruelle, the existence of a countable non-random Lyapunov spectrum is proved. In the hyperbolic case it is shown that the state space admits an almost sure saddle-point splitting which is cocycle-invariant and corresponds to an exponential dichotomy for the stochastic flow