##
**Ergodic theory of random transformations.**
*(English)*
Zbl 0604.28014

Progress in Probability and Statistics, Vol. 10, Boston-Basel-Stuttgart: BirkhĂ¤user. X, 210 p. DM 82.00 (1986).

This book is a monograph on those aspects of ergodic theory which have been and are currently being extended from iterations of a single measure-preserving transformation to compositions of independent and identically distributed random transformations. There are two relevant definitions of ”measure-preserving” for random transformations. Let \({\mathcal F}\) be a measure space of transformations acting on a measurable space \({\mathcal M}\), and let the measure \(m\) on \({\mathcal F}\) be the distribution of a random transformation \(f\). The notion of ”measure- preserving” used in Kakutani’s (1951) pioneering work on random ergodic theory is that all transformations f in the support of \(m\) preserve the \(same\) measure \(\eta\) on \({\mathcal M}\). The more general notion, used throughout most of the present book, is \(''P^*\)-invariance”: \(\eta\) is \(P^*\)-invariant if the measure \(P^*_{\eta}\) defined by
\[
P^*\eta (G)\equiv \iint \chi_ G(f(x))dm(f)d\eta (x)
\]
coincides with \(\eta\).

The coverage of the book is as follows. First, a general introduction describes the various notions of invariant measures for random transformations and relates them to ergodicity and other properties of the Markov sequence \(\{f_ n\circ f_{n-1}\circ...\circ f_ 1(x)\}\) on \({\mathcal M}\), where \(\{\) \(f_ n\}\) is an \(m\)-distributed independent sequence of transformations in \({\mathcal F}\). Chapter II recapitulates and extends the standard theory of measure-theoretic and topological entropy to the random-transformation case. Chapter III is an exposition of new work of the author extending Oseledec’s multiplicative ergodic theorem to the setting of the compositions of independent transformations of a measurable vector bundle which act linearly on the fibers. Chapter IV is concerned with continuity of invariant sub-bundles and stability of characteristic exponents under random perturbations. Finally, Chapter V treats ergodic theory of some smooth random transformations: random diffeomorphisms and stochastic flows. The book has an index of notations, but no index of subjects or definitions.

Under each of the abstract headings mentioned above, the author provides very clear exposition and gives new insight into methods of proof by showing how the results generalize to random transformations. One drawback of the very technical approach of the book, which will limit its usefulness for non-specialists, is that very few concrete examples of the theory are worked out. The main motivating examples for the author seem to have been the ”random perturbations” \(f_ n\) of a fixed nonrandom transformation f. These appear explicitly in connection with the theory of stability of characteristic exponents in Chapter IV, but they are not concretely motivated here, as they often have been elsewhere, by considerations of dynamical systems observed with some experimental error. For all physical motivations and most quasi-physical models, the reader must look elsewhere.

In summary, this is a good if very dry book on the theory of ergodicity, entropy, and characteristic exponents for compositions of independent random transformations. What the author has done clearly and concisely, primarily for specialists in ergodic theory, is to unify a sizeable body of recent research in these areas.

The coverage of the book is as follows. First, a general introduction describes the various notions of invariant measures for random transformations and relates them to ergodicity and other properties of the Markov sequence \(\{f_ n\circ f_{n-1}\circ...\circ f_ 1(x)\}\) on \({\mathcal M}\), where \(\{\) \(f_ n\}\) is an \(m\)-distributed independent sequence of transformations in \({\mathcal F}\). Chapter II recapitulates and extends the standard theory of measure-theoretic and topological entropy to the random-transformation case. Chapter III is an exposition of new work of the author extending Oseledec’s multiplicative ergodic theorem to the setting of the compositions of independent transformations of a measurable vector bundle which act linearly on the fibers. Chapter IV is concerned with continuity of invariant sub-bundles and stability of characteristic exponents under random perturbations. Finally, Chapter V treats ergodic theory of some smooth random transformations: random diffeomorphisms and stochastic flows. The book has an index of notations, but no index of subjects or definitions.

Under each of the abstract headings mentioned above, the author provides very clear exposition and gives new insight into methods of proof by showing how the results generalize to random transformations. One drawback of the very technical approach of the book, which will limit its usefulness for non-specialists, is that very few concrete examples of the theory are worked out. The main motivating examples for the author seem to have been the ”random perturbations” \(f_ n\) of a fixed nonrandom transformation f. These appear explicitly in connection with the theory of stability of characteristic exponents in Chapter IV, but they are not concretely motivated here, as they often have been elsewhere, by considerations of dynamical systems observed with some experimental error. For all physical motivations and most quasi-physical models, the reader must look elsewhere.

In summary, this is a good if very dry book on the theory of ergodicity, entropy, and characteristic exponents for compositions of independent random transformations. What the author has done clearly and concisely, primarily for specialists in ergodic theory, is to unify a sizeable body of recent research in these areas.

Reviewer: E.Slud

### MSC:

28D05 | Measure-preserving transformations |

28D20 | Entropy and other invariants |

60J05 | Discrete-time Markov processes on general state spaces |