On the Oseledets Regularity Functions

We investigate regularity functions corresponding to a fixed Lyapunov bundle E in the Oseledets multiplicative ergodic theorem for an ergodic flow or diffeomorphism. These functions measure the deviation of the growth in norm of the linearized system from constant exponential growth. In general, Oseledets regularity functions were only known to be Borel measurable. Our main result is that if E is continuous on the entire manifold, then the corresponding regularity functions are locally bounded on an open set of full measure. If the system is uniformly mixing, then they are in L, for all 1 ≤ p <∞. In this note we investigate the regularity of “regularity functions” (cf., [9, 1]) associated with a Lyapunov exponent of a smooth dynamical system on a compact manifold. We focus on flows; the treatment of diffeomorphisms is completely analogous. Throughout, Φ = {ft} will denote a C1 flow on a closed (compact and without boundary) Riemannian manifold M . For x ∈M and v ∈ TxM \ {0}, the Lyapunov exponent of v is defined by χ = lim |t|→∞ 1 t log ‖Txft(v)‖ , (∗) where Txft denotes the tangent map of ft : M → M . This means that ‖Txft(v)‖ ∼ eχt ‖v‖, as |t| → ∞. If this limit exists, the set of vectors in TxM (including zero) with the same Lyapunov exponent χ is a linear subspace of TxM , called the Lyapunov space of χ and denoted by Eχ(x). Our goal is to understand the deviation of ‖Txft Eχ‖ from e(χ±ε)t as a function of x. The fundamental properties of Lyapunov exponents and their Lyapunov spaces are described by the Oseledets Multiplicative Ergodic Theorem, which guarantees [1, 11, 7] the existence of a Φ-invariant set R ⊂ M of full measure with respect to any Φ-invariant Borel probability measure μ, such that every point x ∈ R is Lyapunov regular. Recall that this means Date: May 21, 2007. 2000 Mathematics Subject Classification. 37C40, 37A25.