Topics in ergodic theory.

*(English)*Zbl 0805.58005
Princeton Mathematical Series 44. Princeton, NJ: University Press (ISBN 0-691-03277-7). 218 p. (1994).

The author describes his book as a continuation of his earlier work: ‘Introduction to Ergodic Theory’, Princeton Univ. Press (1976; Zbl 0375.28011). In that respect, he assumes the reader has some sophistication in the areas of Ergodic Theory and Dynamical Systems, if not complete familiarity with the earlier book. There is some overlap in these two books, mainly in Part I and II.

The book is divided into 5 parts, each part being split into several lectures. The different parts are fairly independent of each other, as are many of the lectures. Each lecture ends with some references and comments.

Part I is concerned with the basis of Ergodic Theory. These include measure preserving transformations, Lebesgue spaces and partitions, interval exchange transformations and their spectra (a topic also treated in detail in the earlier book). Isomorphism of dynamical systems is discussed (including Meshalkin’s example) and also the discrete spectrum theorem (where it is shown that the example connected with Feigenbaum’s universality in the theory of one-dimensional maps, has pure point spectrum).

Part II gives lectures on the entropy theory of dynamical systems, paying particular attention to multi-dimensional time and cellular automata. Although much of the material in Parts I and II is standard, new proofs are often presented, giving new insights into well known results. Ornstein’s isomorphism theory for Bernoulli shifts is only briefly mentioned.

In Part III, one-dimensional dynamics are studied. The lectures start with some applications to continued and Farey fractions, then go on to a study of homeomorphisms and diffeomorphisms of the circle. Lecture 11 gives a proof of Sharkovskij’s Theorem and the main ideas of Feigenbaum’s universality, and finally a lecture on expanding maps of the circle, including a brief mention of the Ruelle-Perron-Frobenius equation.

In Part IV a study of two-dimensional dynamics is given, starting with a lecture on the Aubrey-Mather theory of twist maps (the standard or Chirilov map in this theory is \(T: C\to C\), where \(C\) is the cylinder, defined by \(T(z,\phi) = (z',\phi')\), \(z' = z + k \sin 2\pi\phi\); \(\phi' = \phi+z\)). For such maps, this is followed by a study of their periodic hyperbolic points, stable manifolds, homoclinic and heteroclinic points and stochastic layers.

The last part, Part V, presents some aspects of the theory of hyperbolic dynamical systems. In particular, lectures 16, 17 and 18 are devoted to geodesic flows and their generalizations. A number of examples are given, including the Lozi map (as a simplification of the Hénon map), Lorenz systems and geodesic flows on compact manifolds of negative curvature. The importance of local unstable manifolds, unstable manifolds and local stable manifolds is emphasized. Their existence is discussed in lecture 17, where the notion of Gibbs measure is introduced. Finally, in lecture 18, the theory of Markov partitions and thermodynamic formalism is applied to the analysis of stochastic properties of deterministic dynamical systems.

The author, being one of the principal developers of much of the theory in this book, gives many interesting asides about where certain ideas originated. The book gives a well written and readable account of some difficult topics of current interest in dynamical systems, some of which are not available in other books. Much of the presentation is new, and will be invaluable to workers in these fields as the source of ideas. It will be particularly useful to the graduate student wishing to choose a thesis topic in dynamical systems, and also as his or her first readings in those directions.

The book is divided into 5 parts, each part being split into several lectures. The different parts are fairly independent of each other, as are many of the lectures. Each lecture ends with some references and comments.

Part I is concerned with the basis of Ergodic Theory. These include measure preserving transformations, Lebesgue spaces and partitions, interval exchange transformations and their spectra (a topic also treated in detail in the earlier book). Isomorphism of dynamical systems is discussed (including Meshalkin’s example) and also the discrete spectrum theorem (where it is shown that the example connected with Feigenbaum’s universality in the theory of one-dimensional maps, has pure point spectrum).

Part II gives lectures on the entropy theory of dynamical systems, paying particular attention to multi-dimensional time and cellular automata. Although much of the material in Parts I and II is standard, new proofs are often presented, giving new insights into well known results. Ornstein’s isomorphism theory for Bernoulli shifts is only briefly mentioned.

In Part III, one-dimensional dynamics are studied. The lectures start with some applications to continued and Farey fractions, then go on to a study of homeomorphisms and diffeomorphisms of the circle. Lecture 11 gives a proof of Sharkovskij’s Theorem and the main ideas of Feigenbaum’s universality, and finally a lecture on expanding maps of the circle, including a brief mention of the Ruelle-Perron-Frobenius equation.

In Part IV a study of two-dimensional dynamics is given, starting with a lecture on the Aubrey-Mather theory of twist maps (the standard or Chirilov map in this theory is \(T: C\to C\), where \(C\) is the cylinder, defined by \(T(z,\phi) = (z',\phi')\), \(z' = z + k \sin 2\pi\phi\); \(\phi' = \phi+z\)). For such maps, this is followed by a study of their periodic hyperbolic points, stable manifolds, homoclinic and heteroclinic points and stochastic layers.

The last part, Part V, presents some aspects of the theory of hyperbolic dynamical systems. In particular, lectures 16, 17 and 18 are devoted to geodesic flows and their generalizations. A number of examples are given, including the Lozi map (as a simplification of the Hénon map), Lorenz systems and geodesic flows on compact manifolds of negative curvature. The importance of local unstable manifolds, unstable manifolds and local stable manifolds is emphasized. Their existence is discussed in lecture 17, where the notion of Gibbs measure is introduced. Finally, in lecture 18, the theory of Markov partitions and thermodynamic formalism is applied to the analysis of stochastic properties of deterministic dynamical systems.

The author, being one of the principal developers of much of the theory in this book, gives many interesting asides about where certain ideas originated. The book gives a well written and readable account of some difficult topics of current interest in dynamical systems, some of which are not available in other books. Much of the presentation is new, and will be invaluable to workers in these fields as the source of ideas. It will be particularly useful to the graduate student wishing to choose a thesis topic in dynamical systems, and also as his or her first readings in those directions.

Reviewer: G.R.Goodson (Towson)

##### MSC:

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

37E99 | Low-dimensional dynamical systems |

37D99 | Dynamical systems with hyperbolic behavior |

28D05 | Measure-preserving transformations |

37A99 | Ergodic theory |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

53D25 | Geodesic flows in symplectic geometry and contact geometry |