##
**Probabilistic properties of deterministic systems.**
*(English)*
Zbl 0606.58002

Cambridge etc.: Cambridge University Press. X, 358 p. £40.00, $ 49.50 (1985).

This book is about densities. The message of the authors is that many systems arising in physics, biology and economics which show ”chaotic” properties can be profitably investigated by using densities. The authors aim at giving a unified treatment of a variety of mathematical systems generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro- partial differential equations.

Starting with a simple quadratic map S defined on the unit interval, the authors first show how the Frobenius-Perron operator corresponding to S can be introduced in order to study the evolution of densities. In the third chapter, Markov and Frobenius-Perron operators are introduced in a general fashion. In the fourth chapter, ergodic, mixing and exact transformations are defined, and it is shown how the irregularity of their behaviour can be studied by using Frobenius-Perron and Koopman operators. The next two chapters deal with the asymptotic properties of densities, and with transformations defined on intervals and manifolds. In the remainder of their book, the authors study continuous time systems. Dynamical and semidynamical systems as well as semigroups of Frobenius-Perron operators and their infinitesimal operators are introduced, and, finally, the Hille-Yosida theorem is derived.

In the eighth chapter it is shown how a discrete time process can be embedded into a continuous time system via a suitable embedding of its Frobenius-Perron operator, the result of which being an abstract form of the Boltzmann equation. The next chapter deals with entropy. Stochastic perturbations of discrete time as well as of continuous time systems are studied in the two final chapters (in particular, the last chapter contains an elementary introduction into Itô’s stochastic integral).

In our opinion, the authors have succeeded in writing a stimulating and very readable book which is abundant with examples. It should turn out to be useful for graduate students as well as specialists working in stochastic processes, ergodic theory, physics, and biomathematics.

Starting with a simple quadratic map S defined on the unit interval, the authors first show how the Frobenius-Perron operator corresponding to S can be introduced in order to study the evolution of densities. In the third chapter, Markov and Frobenius-Perron operators are introduced in a general fashion. In the fourth chapter, ergodic, mixing and exact transformations are defined, and it is shown how the irregularity of their behaviour can be studied by using Frobenius-Perron and Koopman operators. The next two chapters deal with the asymptotic properties of densities, and with transformations defined on intervals and manifolds. In the remainder of their book, the authors study continuous time systems. Dynamical and semidynamical systems as well as semigroups of Frobenius-Perron operators and their infinitesimal operators are introduced, and, finally, the Hille-Yosida theorem is derived.

In the eighth chapter it is shown how a discrete time process can be embedded into a continuous time system via a suitable embedding of its Frobenius-Perron operator, the result of which being an abstract form of the Boltzmann equation. The next chapter deals with entropy. Stochastic perturbations of discrete time as well as of continuous time systems are studied in the two final chapters (in particular, the last chapter contains an elementary introduction into Itô’s stochastic integral).

In our opinion, the authors have succeeded in writing a stimulating and very readable book which is abundant with examples. It should turn out to be useful for graduate students as well as specialists working in stochastic processes, ergodic theory, physics, and biomathematics.

Reviewer: K.Schürger

### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37A99 | Ergodic theory |

37D15 | Morse-Smale systems |

60Hxx | Stochastic analysis |

37N99 | Applications of dynamical systems |