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We also show that both ergodic and geometric properties of such a measure are very close to the corresponding properties of the Lebesgue measure with respect to the linear action $\a_0$.

We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism $f$ of a compact manifold $X$ preserving a hyperbolic ergodic probability measure $μ$. The cocycle $\mathcal{A}$ over $f$ is Hölder continuous and takes values in $GL(d, \mathbb{R})$ or, more generally, in the group of invertible bounded linear operators on a Banach space. For a $GL(d, \mathbb{R})$-valued cocycle $\mathcal{A}$ we prove that the Lyapunov exponents of $\mathcal{A}$ with respect to $μ$ can be approximated by the Lyapunov exponents of $\mathcal{A}$ with respect to measures on hyperbolic periodic orbits of $f$. In the infinite-dimensional setting one can define the upper and lower Lyapunov exponents of $\mathcal{A}$ with respect to $μ$, but they cannot always be approximated by the exponents of $\mathcal{A}$ on periodic orbits. We prove that they can be approximated in terms of the norms of the return values of $\mathcal{A}$ on hyperbolic periodic orbits of $f$.

Let $f$ be a measure-preserving transformation of a Lebesgue space $(X,\mu)$ and let ${\mathscr{F}}$ be its extension to a bundle $\mathscr{E} = X \times {\mathbb{R}}^m$ by smooth fiber maps ${\mathscr{F}}_x : {\mathscr{E}}_x \to {\mathscr{E}}_{fx}$ so that the derivative of ${\mathscr{F}}$ at the zero section has negative Lyapunov exponents. We construct a measurable system of smooth coordinate changes ${\mathscr{H}}_x$ on ${\mathscr{E}}_x$ for $\mu$-a.e. $x$ so that the maps ${\mathscr{P}}_x ={\mathscr{H}}_{fx} \circ {\mathscr{F}}_x \circ {\mathscr{H}}_x^{-1}$ are sub-resonance polynomials in a finite dimensional Lie group. Our construction shows that such ${\mathscr{H}}_x$ and ${\mathscr{P}}_x$ are unique up to a sub-resonance polynomial. As a consequence, we obtain the centralizer theorem that the coordinate change $\mathscr{H}$ also conjugates any commuting extension to a polynomial extension of the same type. We apply our results to a measure-preserving diffeomorphism $f$ with a non-uniformly contracting invariant foliation $W$. We construct a measurable system of smooth coordinate changes ${\mathscr{H}}_x: W_x \to T_xW$ such that the maps ${\mathscr{H}}_{fx} \circ f \circ {\mathscr{H}}_x^{-1}$ are polynomials of sub-resonance type. Moreover, we show that for almost every leaf the coordinate changes exist at each point on the leaf and give a coherent atlas with transition maps in a finite dimensional Lie group.

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