Random Dynamical Systems pp 111-162 | Cite as

# The Multiplicative Ergodic Theorem in Euclidean Space

## Summary

This chapter is devoted to the presentation of Oseledets’s multiplicative ergodic theorem, the most fundamental theorem of the book. It provides a spectral theory for linear cocycles, and one can say without exaggeration that all what follows are just applications of this one theorem.

We hence have tried hard to give complete and precise formulations and clearly structured proofs of the various versions of the multiplicative ergodic theorem (Theorem 3.4.1 for one-sided time T = ℕ and T = ℝ^{+}, Theorem 3.4.11 for two-sided time T = ℤ and T = ℝ). In particular, we carefully describe the invariance properties of the W sets on which the statements hold.

As the proofs of the various versions of the multiplicative ergodic theorem are based on corresponding versions of the Furstenberg-Kesten theorem (Theorem 3.3.3 for one-sided time T = ℕ and T = ℝ^{+}, Theorem 3.3.10 for two-sided time T = ℤ and T = ℝ), the same care has to be taken for the formulations and proofs of the latter.

A certain pedantry of our style is motivated by the aim of making this chapter a source of reference.

### Keywords

Filtration Manifold Assure Mane Sinal## Preview

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