5 - Stationary measures
Published online by Cambridge University Press: 05 July 2014
Summary
Further development of the theory requires that we exploit in more depth the connection between the Lyapunov exponents of a linear cocycle and the invariant measures of the corresponding projective cocycle, of which we had a brief glimpse in the proof of the multiplicative ergodic theorem. In this chapter we introduce a general formalism that will be very useful towards that end.
Linearity is not relevant at this stage, so we formulate the results for a class of systems more general than linear and projective cocycles, that we call random transformations. The definition and fundamental properties of such systems are discussed in Section 5.1.
In Section 5.2 we introduce the key notion of stationary measure for a random transformation. We will see in the next chapter that the measures stationary under the projective cocycle completely determine the Lyapunov exponents of the linear cocycle. The properties of stationary measures of general (possibly non-invertible) random transformations are studied in Sections 5.2 and 5.3.
The invertible case is treated in more detail in Section 5.4 and leads to the important concepts of u-state and s-state, which are invariant probability measures whose disintegrations along the fibers have special invariance properties. These disintegrations are revisited, from a different angle, in Section 5.5.
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- Lectures on Lyapunov Exponents , pp. 67 - 95Publisher: Cambridge University PressPrint publication year: 2014