Deterministic version lemmas in ergodic theory of random dynamical systems (Q912230)

Let S and M be measurable spaces (with \(\mu\) a measure on M). For each \(s\in S\) let \(f_ s:\quad M\to M\), and let \(\{\xi_ n\}\) be a stationary stochastic process (in this paper an i.i.d. sequence) on a probability space \((\Omega,{\mathcal F},P)\) with values in S. Then a random dynamical system starting at \(x\in M\) is given by \(X_ n(\omega)x=f_{\xi_ n(\omega)}\circ...\circ f_{\xi_ 1(\omega)}(x).\) A transformation \(T:\quad M\times \Omega \to M\times \Omega\) is defined by \(T(x,\omega)=(f_{\xi_ 1(\omega)}(x),\sigma (\omega)),\) where \(\sigma\) is a measure-preserving transformation such that \(\xi_{n+1}=\xi_ n\circ \sigma.\) It is shown that the \(L_ 1(\mu \times P)\) eigenfunctions \(\Phi\) (with eigenvalue of unit modulus) of T are deterministic in the sense that there exists \(\phi \in L_ 1(\mu)\) such that, for almost all \((x,\omega),\) \(\Phi (x,\omega)=\phi (x).\) Several applications to more specific random dynamical systems are given.