### The Multiplicative Ergodic Theorem in Hilbert Space

Alex Blumenthal

Abstract.
For a smooth, invertible dynamical system on a compact manifold equipped
with an invariant probability measure, the Multiplicative Ergodic Theorem
of Oseledets tells us the existence of certain numbers, Lyapunov exponents,
associated with a splitting of the tangent space at almost all points in
the system. These reflect the asymptotic exponential rate at which small
changes in initial condition will blow up or contract over time, according
to whether the exponent associated with the perturbative direction is
positive or negative.
Following Ruelle [1], we will prove a Multiplicative Ergodic Theorem for
smooth, injective dynamical systems on a hilbert space with compact,
injective derivative, equipped with an invariant probability measure
supported on a compact set. This result has potential applications in the
theory of certain parabolic partial differential equations, where we view
the PDE as giving rise to a dynamical system on an infinite-dimensional
function space.
[1] David Ruelle. Characteristic Exponents and Invariant Manifolds in
Hilbert Space. Annals of Mathematics, Second Series, Vol. 115, No. 2 (Mar.,
1982), pp. 243-290