##
**Dynamical systems of algebraic origin.**
*(English)*
Zbl 0833.28001

Progress in Mathematics (Boston, Mass.). 128. Basel: Birkhäuser. xviii, 310 p. (1995).

This beautifully written monograph is the most recent recipient of the Ferran Sunyer i Balaguer Prize, awarded annually by the Institute d’Estudies Catalans for an outstanding mathematical exposition in an area which has had major contributions from the author.

The aim of this book is to introduce and study a wide collection of \(\mathbb{Z}^d\)-actions from Ergodic Theory that exhibit many new and interesting phenomena not present for \(\mathbb{Z}\)-actions, and which can nevertheless be studied in a systematic way. These are the \(\mathbb{Z}^d\)- actions by automorphisms \(\alpha\) of a compact abelian group \(X\). Much of the material in the book is motivated by the now famous example of Ledrappier: Consider the shift \(\sigma\) of \(\mathbb{Z}^2\) on the closed shift invariant subgroup \[ X = \Bigl\{ {\mathbf x} = (x_m) \in (\mathbb{Z}_2)^{ \mathbb{Z}^2} : x_{(m_1, m_2)} + x_{(m_1 + 1,m_2)} + x_{(m_1,m_2 + 1)} = 0 \bmod 2 \;\forall (m_1,m_2) \in \mathbb{Z}^2 \Bigr\}, \] of the full (two-dimensional) two shift. \(\sigma\) has zero entropy, is mixing, but not mixing of order 3. However, if the group \(\mathbb{Z}_2\) is replaced by the circle group \(\mathbb{T}\), then the corresponding shift \(\sigma^\mathbb{T}\) turns out to be mixing of all orders. These are examples of actions of a countable group \(\Gamma\) by an automorphism \(\alpha\) of a compact group \(X\). To answer the mixing question, in Chapter VIII it is shown that every mixing \(\mathbb{Z}^d\)- action by automorphisms of a compact connected abelian group \(X\) is mixing of all orders, and that \(\mathbb{Z}^d\)-actiofs by automorphisms of compact, zero-dimensional abelian groups are mixing of all orders if and only if they have completely positive entropy.

Chapter I gives general background material, including the generalizations of Halmos’ results on the ergodic properties of a single automorphism of a compact abelian group, to the nonabelian and multi- dimensional situation. It is shown that an action of a countable group \(\Gamma\) by an automorphism \(\alpha\) of a compact group \(X\) is ergodic if and only if it is topologically transitive. The finiteness conditions: expansiveness and d.c.c. (descending chain condition) are introduced for such actions, and the connection to subshifts of finite type is studied.

Chapter II concentrates on \(\mathbb{Z}^d\)-actions by automorphisms of compact abelian groups and their connections with commutative algebra.

Chapter III reviews the classical theory of automorphisms of compact (nonabelian) groups, and uses entropy to obtain a classification for topological conjugacy of expansive and ergodic automorphisms of compact zero-dimensional groups.

The next chapters deal with the entropy of \(\mathbb{Z}^d\)-actions on compact groups This includes the introduction of the Mahler measure of a Laurent polynomial as a tool for calculating such entropies, a characterization of completely positive entropy, a discussion of measures of maximal entropy, the distribution of periodic points and Bernoullicity, and finally \(\mathbb{Z}^d\)-actions having zero entropy.

The higher order mixing properties of \(\mathbb{Z}^d\)-actions and the results mentioned earlier are given in Chapter VIII. A direct connection between mixing properties and solutions to certain arithmetical equations is given. Finally, Chapter IX is devoted to rigidity properties of zero entropy \(\mathbb{Z}^d\)-actions. The first cohomology of \(\mathbb{Z}^d\)- actions is studied, and striking differences for the cases \(d = 1\) and \(d > 1\) are demonstrated.

The prerequisites for the reading of this book are fairly substantial, and include a knowledge of the character and representation theory of locally compact groups, some commutative algebra and arithmetical algebraic geometry. A good knowledge of ergodic theory and topological dynamics for a single automorphism is really essential in order for the material to be fully meaningful, especially as the technicalities involved in the very general results can be substantial.

This book is a very important addition to the literature, giving the first systematic account of the ergodic theory of algebraic \(\mathbb{Z}^d\)- actions. It will be of immense value to any researchers and graduate students interested in such multi-dimensional actions.

The aim of this book is to introduce and study a wide collection of \(\mathbb{Z}^d\)-actions from Ergodic Theory that exhibit many new and interesting phenomena not present for \(\mathbb{Z}\)-actions, and which can nevertheless be studied in a systematic way. These are the \(\mathbb{Z}^d\)- actions by automorphisms \(\alpha\) of a compact abelian group \(X\). Much of the material in the book is motivated by the now famous example of Ledrappier: Consider the shift \(\sigma\) of \(\mathbb{Z}^2\) on the closed shift invariant subgroup \[ X = \Bigl\{ {\mathbf x} = (x_m) \in (\mathbb{Z}_2)^{ \mathbb{Z}^2} : x_{(m_1, m_2)} + x_{(m_1 + 1,m_2)} + x_{(m_1,m_2 + 1)} = 0 \bmod 2 \;\forall (m_1,m_2) \in \mathbb{Z}^2 \Bigr\}, \] of the full (two-dimensional) two shift. \(\sigma\) has zero entropy, is mixing, but not mixing of order 3. However, if the group \(\mathbb{Z}_2\) is replaced by the circle group \(\mathbb{T}\), then the corresponding shift \(\sigma^\mathbb{T}\) turns out to be mixing of all orders. These are examples of actions of a countable group \(\Gamma\) by an automorphism \(\alpha\) of a compact group \(X\). To answer the mixing question, in Chapter VIII it is shown that every mixing \(\mathbb{Z}^d\)- action by automorphisms of a compact connected abelian group \(X\) is mixing of all orders, and that \(\mathbb{Z}^d\)-actiofs by automorphisms of compact, zero-dimensional abelian groups are mixing of all orders if and only if they have completely positive entropy.

Chapter I gives general background material, including the generalizations of Halmos’ results on the ergodic properties of a single automorphism of a compact abelian group, to the nonabelian and multi- dimensional situation. It is shown that an action of a countable group \(\Gamma\) by an automorphism \(\alpha\) of a compact group \(X\) is ergodic if and only if it is topologically transitive. The finiteness conditions: expansiveness and d.c.c. (descending chain condition) are introduced for such actions, and the connection to subshifts of finite type is studied.

Chapter II concentrates on \(\mathbb{Z}^d\)-actions by automorphisms of compact abelian groups and their connections with commutative algebra.

Chapter III reviews the classical theory of automorphisms of compact (nonabelian) groups, and uses entropy to obtain a classification for topological conjugacy of expansive and ergodic automorphisms of compact zero-dimensional groups.

The next chapters deal with the entropy of \(\mathbb{Z}^d\)-actions on compact groups This includes the introduction of the Mahler measure of a Laurent polynomial as a tool for calculating such entropies, a characterization of completely positive entropy, a discussion of measures of maximal entropy, the distribution of periodic points and Bernoullicity, and finally \(\mathbb{Z}^d\)-actions having zero entropy.

The higher order mixing properties of \(\mathbb{Z}^d\)-actions and the results mentioned earlier are given in Chapter VIII. A direct connection between mixing properties and solutions to certain arithmetical equations is given. Finally, Chapter IX is devoted to rigidity properties of zero entropy \(\mathbb{Z}^d\)-actions. The first cohomology of \(\mathbb{Z}^d\)- actions is studied, and striking differences for the cases \(d = 1\) and \(d > 1\) are demonstrated.

The prerequisites for the reading of this book are fairly substantial, and include a knowledge of the character and representation theory of locally compact groups, some commutative algebra and arithmetical algebraic geometry. A good knowledge of ergodic theory and topological dynamics for a single automorphism is really essential in order for the material to be fully meaningful, especially as the technicalities involved in the very general results can be substantial.

This book is a very important addition to the literature, giving the first systematic account of the ergodic theory of algebraic \(\mathbb{Z}^d\)- actions. It will be of immense value to any researchers and graduate students interested in such multi-dimensional actions.

Reviewer: G.R.Goodson (Towson)

### MSC:

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

28D15 | General groups of measure-preserving transformations |

28D20 | Entropy and other invariants |