##
**Ergodic theorems. With a supplement by Antoine Brunel.**
*(English)*
Zbl 0575.28009

De Gruyter Studies in Mathematics, 6. Berlin-New York: Walter de Gruyter. VIII, 357 p. DM 128.00; $ 49.95 (1985).

The aim of the book under review is the study of ergodic properties of operators which has its origin in statistical mechanics. Namely, for an operator \(T\) convergence properties of the averages \(\frac{1}{n}\sum^{n- 1}_{k=0}T^ k\) are investigated. The first results on ergodic properties of operators go back to 1931 when J. von Neumann proved the mean ergodic theorem for contractions on Hilbert space and G. D. Birkhoff proved the pointwise ergodic theorem for measure preserving transformations. The author gives an up-to-date presentation of the theory in question which covers a larger spectrum of ergodic theorems.

Chapter I is devoted to the study of measure preserving and null preserving mappings. Namely, let \(\tau\) be a measure preserving endomorphism of a measure space (\(\Omega\),\({\mathcal A},\mu)\), i.e. \(\tau^{-1}{\mathcal A}\subset {\mathcal A}\) and \(\mu (\tau^{-1}A)=\mu (A)\) for any \(A\in {\mathcal A}\). For \(f\in L^ 1(\Omega,{\mathcal A},\mu)\) consider the averages \(A_ nf=\frac{1}{n}\sum^{n-1}_{k=0}f\circ \tau^ k\). The problem is to investigate convergence properties of \(A_ n\). For \(f\in L^ 2\) the norm converges of \(A_ nf\) follows from von Neumann’s mean ergodic theorem which claims that for a contraction \(T\) (i.e. \(\| T\| \leq 1)\) on a Hilbert space the averages \(\frac{1}{n}\sum^{n- 1}_{k=0}T^ k\) converge in the strong operator topology. A deeper result on the behaviour of \(A_ nf\) is Birkhoff’s pointwise ergodic theorem which claims that \(A_ nf\) converges a.e. for \(f\in L^ 1\). The limit of \(A_ nf\) is a \(\tau\)-invariant function. If \(\tau\) is ergodic (i.e. \(\tau A=A\) only in case \(\mu (A)=0\) or \(\mu (\Omega \setminus A)=0)\) and \(\mu (\Omega)<\infty\) then \(\lim_{n\to \infty}A_ nf=\frac{1}{\mu (\Omega)}\int_{\Omega}f\).

Similar results hold in the case of continuous time, i.e. for flows \(\{\tau_ t:\) \(t\in {\mathbb R}\}\) or semiflows \(\{\tau_ t:\) \(t\in {\mathbb R}_+\}\) of measure preserving transformations. In this case there is another problem concerning the behaviour of \((A_{\epsilon}f)(\omega)=\epsilon^{- 1}\int^{\epsilon}_{0}f(\tau_ t\omega)dt\) as \(\epsilon\) \(\to 0\). Wiener’s local ergodic theorem claims that \((A_{\epsilon}f)(\omega)\to_{\epsilon \to 0}f(\omega)\) a.e.

Next, recurrence properties of null preserving mappings are investigated. The classical Poincaré theorem claims that if \(\tau\) is a measure preserving transformation then \(\tau\) is infinitely recurrent, i.e. for any \(A\in {\mathcal A}\) for \(\mu\)-almost all \(\omega\in A\) the sequence \(\tau^{-k}\omega\) falls to \(A\) for infinitely many \(k\). For an arbitrary null preserving mapping there is a result due to Hopf which claims that \(\Omega\) can be decomposed into two disjoint subsets \(C, D\) such that \(C\subset \tau^{-1}C\), \(\tau\) \(| C\) is recurrent and \(D\) is at most countable union of wandering sets.

In § 1.4 examples of measure preserving mappings arising in the theory of stationary processes are given. In § 1.5 Kingman’s ergodic theorem for subadditive processes is proved which leads to numerous applications including the multiplicative ergodic theorem of Oseledets.

The rest of Chapter 1 concerns some dominated estimates for more general operators (operators which are contractions in \(L^ 1\) and \(L^{\infty})\) and related topics.

Chapter 2 is devoted to mean ergodic properties. It is proved that for a power bounded operator (i.e. \(\| T^ n\| \leq const,\quad n\geq 0\)) on a reflexive Banach space the averages \(A_ nx=\frac{1}{n}\sum^{n- 1}_{k=0}T^ kx\) converge for any \(x\in X\) in norm to a \(T\)-invariant limit.

Some more general classes of operators are considered (e.g. Cesàro bounded operators, i.e. operators \(T\) with \(\sup_{n}\| A_ n\| <\infty)\). For such operators the problem of limit behaviour of \(A_ nx\) is investigated, too. More general semigroups of operators acting on locally convex spaces are also considered.

In § 2.2 conditions are found under which \(A_ n\) converges uniformly. In § 2.3 weak mixing properties for contractions on Hilbert space are studied. This study is based on Wiener’s theorem on convergence in density of Fourier coefficients of measures. The result is applied to the investigation of mixing properties of endomorphisms of probability spaces. In § 2.4 weakly almost periodic semigroups acting on a Banach space are investigated. For such semigroups a theorem of Jacobs-DeLeeuw-Glicksberg is proved which splits the Banach space into a direct sum of the subspace spanned by the eigenvectors with unimodular eigenvalues and the space of the so-called flight vectors having 0 as a weak limit point.

Chapter 3 concerns positive contractions on \(L^ 1\), i.e. operators \(T\) on \(L^ 1\) such that \(\| T\| \leq 1\) and \(Tf\geq 0\) for \(f\geq 0\). For such operators an analogue of the Hopf decomposition is established. For an arbitrary positive contraction \(T\) on \(L^ 1\) the Cesáro means \(A_ nf=\frac{1}{n}\sum^{n-1}_{k=0}T^ kf\) can be divergent. But the remarkable Chacon-Ornstein theorem claims that for any positive \(g\) in \(L^ 1\) and for any \(f\) in \(L^ 1\) the sequence \(A_ nf/A_ ng\) converges a.e. on the set \(\{x:\sum_{n\geq 0}(T^ ng)(x)\geq 0\}\). This result yields the a.e. -convergence of \(A_ nf\) on the set \(\{x: \sum_{n\geq 0}(T^ ng)(x)>0\}\). The limit \(S_ nf/S_ ng\) was evaluated by Brunel. The corresponding result is presented in § 3.3. The rest of the chapter is devoted to the problem of the existence of invariant measures to the subadditive ergodic theorem for positive contractions on \(L^ 1\) and to the construction of Chacon’s example of divergence of Cesàro means for a positive contraction.

Chapter 4 deals with some more general objects than Chapter 3. First the case of arbitrary contractions on \(L^ 1\) is considered. It is proved that each such contraction \(T\) has a contractive modulus \(| T|\). For such operators \(T\) Chacon’s theorem is proved on the convergence of the ratios \(\sum^{n-1}_{k=0}T^ kf/\sum^{n-1}_{k=0}p_ k\) on \(\{\) x:\(\sum^{\infty}_{k=0}p_ k>0\}\), where \(\{p_ k\}\) is \(T\)-admissible, i.e. \(| f| \leq p_ k\) implies \(| Tf| \leq p_{k+1}\). Next, the vector-valued case and the case of power bounded operators are considered.

Chapter 5 concerns operators on \(C(K)\) and \(L^ p\), \(1<p<\infty\). The main results for operators on \(C(K)\) are Jamison’s theorem, which claims that for an irreducible Markov operator \(S\) on \(C(K)\) the pointwise convergence \((S^ nf)(\omega)\to 0\) implies \(\| S^ nf\| \to 0\), and Breiman’s strong law for Markov processes.

In the case of \(L^ p\) operators the main result of the chapter is the pointwise Akcoglu theorem for positive contractions on \(L^ p\) which claims that for any positive contraction \(T\) on \(L^ p\) the limit \(\lim_{n\to \infty}\frac{1}{n}\sum^{n-1}_{k=0}T^ kf\) exist a.e. An example due to Burkholder is given of a contraction on \(L^ 2\) for which the convergence fails. Another result given here is the dominated Stein estimate for the powers \(T^ kf\) where \(T\) is a self-adjoint positive contraction on \(L^ 2.\)

The other chapters concern some more general settings. Chapter 6 deals with multiparameter and amenable semigroups, Chapter 7 is devoted to local ergodic theorems. In Chapter 8 more general averages are investigated. Chapter 9 concerns ergodic theorems in von Neumann algebras, entropy and information, nonlinear nonexpansive mappings and other miscellaneous topics.

The book is completed by the supplement, written by A. Brunel, on Harris processes which is an important class of Markov processes. The book gives thorough and self-contained presentation of the theory. Different parts of the book can be read independently of each other.

Chapter I is devoted to the study of measure preserving and null preserving mappings. Namely, let \(\tau\) be a measure preserving endomorphism of a measure space (\(\Omega\),\({\mathcal A},\mu)\), i.e. \(\tau^{-1}{\mathcal A}\subset {\mathcal A}\) and \(\mu (\tau^{-1}A)=\mu (A)\) for any \(A\in {\mathcal A}\). For \(f\in L^ 1(\Omega,{\mathcal A},\mu)\) consider the averages \(A_ nf=\frac{1}{n}\sum^{n-1}_{k=0}f\circ \tau^ k\). The problem is to investigate convergence properties of \(A_ n\). For \(f\in L^ 2\) the norm converges of \(A_ nf\) follows from von Neumann’s mean ergodic theorem which claims that for a contraction \(T\) (i.e. \(\| T\| \leq 1)\) on a Hilbert space the averages \(\frac{1}{n}\sum^{n- 1}_{k=0}T^ k\) converge in the strong operator topology. A deeper result on the behaviour of \(A_ nf\) is Birkhoff’s pointwise ergodic theorem which claims that \(A_ nf\) converges a.e. for \(f\in L^ 1\). The limit of \(A_ nf\) is a \(\tau\)-invariant function. If \(\tau\) is ergodic (i.e. \(\tau A=A\) only in case \(\mu (A)=0\) or \(\mu (\Omega \setminus A)=0)\) and \(\mu (\Omega)<\infty\) then \(\lim_{n\to \infty}A_ nf=\frac{1}{\mu (\Omega)}\int_{\Omega}f\).

Similar results hold in the case of continuous time, i.e. for flows \(\{\tau_ t:\) \(t\in {\mathbb R}\}\) or semiflows \(\{\tau_ t:\) \(t\in {\mathbb R}_+\}\) of measure preserving transformations. In this case there is another problem concerning the behaviour of \((A_{\epsilon}f)(\omega)=\epsilon^{- 1}\int^{\epsilon}_{0}f(\tau_ t\omega)dt\) as \(\epsilon\) \(\to 0\). Wiener’s local ergodic theorem claims that \((A_{\epsilon}f)(\omega)\to_{\epsilon \to 0}f(\omega)\) a.e.

Next, recurrence properties of null preserving mappings are investigated. The classical Poincaré theorem claims that if \(\tau\) is a measure preserving transformation then \(\tau\) is infinitely recurrent, i.e. for any \(A\in {\mathcal A}\) for \(\mu\)-almost all \(\omega\in A\) the sequence \(\tau^{-k}\omega\) falls to \(A\) for infinitely many \(k\). For an arbitrary null preserving mapping there is a result due to Hopf which claims that \(\Omega\) can be decomposed into two disjoint subsets \(C, D\) such that \(C\subset \tau^{-1}C\), \(\tau\) \(| C\) is recurrent and \(D\) is at most countable union of wandering sets.

In § 1.4 examples of measure preserving mappings arising in the theory of stationary processes are given. In § 1.5 Kingman’s ergodic theorem for subadditive processes is proved which leads to numerous applications including the multiplicative ergodic theorem of Oseledets.

The rest of Chapter 1 concerns some dominated estimates for more general operators (operators which are contractions in \(L^ 1\) and \(L^{\infty})\) and related topics.

Chapter 2 is devoted to mean ergodic properties. It is proved that for a power bounded operator (i.e. \(\| T^ n\| \leq const,\quad n\geq 0\)) on a reflexive Banach space the averages \(A_ nx=\frac{1}{n}\sum^{n- 1}_{k=0}T^ kx\) converge for any \(x\in X\) in norm to a \(T\)-invariant limit.

Some more general classes of operators are considered (e.g. Cesàro bounded operators, i.e. operators \(T\) with \(\sup_{n}\| A_ n\| <\infty)\). For such operators the problem of limit behaviour of \(A_ nx\) is investigated, too. More general semigroups of operators acting on locally convex spaces are also considered.

In § 2.2 conditions are found under which \(A_ n\) converges uniformly. In § 2.3 weak mixing properties for contractions on Hilbert space are studied. This study is based on Wiener’s theorem on convergence in density of Fourier coefficients of measures. The result is applied to the investigation of mixing properties of endomorphisms of probability spaces. In § 2.4 weakly almost periodic semigroups acting on a Banach space are investigated. For such semigroups a theorem of Jacobs-DeLeeuw-Glicksberg is proved which splits the Banach space into a direct sum of the subspace spanned by the eigenvectors with unimodular eigenvalues and the space of the so-called flight vectors having 0 as a weak limit point.

Chapter 3 concerns positive contractions on \(L^ 1\), i.e. operators \(T\) on \(L^ 1\) such that \(\| T\| \leq 1\) and \(Tf\geq 0\) for \(f\geq 0\). For such operators an analogue of the Hopf decomposition is established. For an arbitrary positive contraction \(T\) on \(L^ 1\) the Cesáro means \(A_ nf=\frac{1}{n}\sum^{n-1}_{k=0}T^ kf\) can be divergent. But the remarkable Chacon-Ornstein theorem claims that for any positive \(g\) in \(L^ 1\) and for any \(f\) in \(L^ 1\) the sequence \(A_ nf/A_ ng\) converges a.e. on the set \(\{x:\sum_{n\geq 0}(T^ ng)(x)\geq 0\}\). This result yields the a.e. -convergence of \(A_ nf\) on the set \(\{x: \sum_{n\geq 0}(T^ ng)(x)>0\}\). The limit \(S_ nf/S_ ng\) was evaluated by Brunel. The corresponding result is presented in § 3.3. The rest of the chapter is devoted to the problem of the existence of invariant measures to the subadditive ergodic theorem for positive contractions on \(L^ 1\) and to the construction of Chacon’s example of divergence of Cesàro means for a positive contraction.

Chapter 4 deals with some more general objects than Chapter 3. First the case of arbitrary contractions on \(L^ 1\) is considered. It is proved that each such contraction \(T\) has a contractive modulus \(| T|\). For such operators \(T\) Chacon’s theorem is proved on the convergence of the ratios \(\sum^{n-1}_{k=0}T^ kf/\sum^{n-1}_{k=0}p_ k\) on \(\{\) x:\(\sum^{\infty}_{k=0}p_ k>0\}\), where \(\{p_ k\}\) is \(T\)-admissible, i.e. \(| f| \leq p_ k\) implies \(| Tf| \leq p_{k+1}\). Next, the vector-valued case and the case of power bounded operators are considered.

Chapter 5 concerns operators on \(C(K)\) and \(L^ p\), \(1<p<\infty\). The main results for operators on \(C(K)\) are Jamison’s theorem, which claims that for an irreducible Markov operator \(S\) on \(C(K)\) the pointwise convergence \((S^ nf)(\omega)\to 0\) implies \(\| S^ nf\| \to 0\), and Breiman’s strong law for Markov processes.

In the case of \(L^ p\) operators the main result of the chapter is the pointwise Akcoglu theorem for positive contractions on \(L^ p\) which claims that for any positive contraction \(T\) on \(L^ p\) the limit \(\lim_{n\to \infty}\frac{1}{n}\sum^{n-1}_{k=0}T^ kf\) exist a.e. An example due to Burkholder is given of a contraction on \(L^ 2\) for which the convergence fails. Another result given here is the dominated Stein estimate for the powers \(T^ kf\) where \(T\) is a self-adjoint positive contraction on \(L^ 2.\)

The other chapters concern some more general settings. Chapter 6 deals with multiparameter and amenable semigroups, Chapter 7 is devoted to local ergodic theorems. In Chapter 8 more general averages are investigated. Chapter 9 concerns ergodic theorems in von Neumann algebras, entropy and information, nonlinear nonexpansive mappings and other miscellaneous topics.

The book is completed by the supplement, written by A. Brunel, on Harris processes which is an important class of Markov processes. The book gives thorough and self-contained presentation of the theory. Different parts of the book can be read independently of each other.

Reviewer: V.V.Peller

### MSC:

37Axx | Ergodic theory |

28Dxx | Measure-theoretic ergodic theory |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

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

47A35 | Ergodic theory of linear operators |

22D40 | Ergodic theory on groups |

47B38 | Linear operators on function spaces (general) |

60G10 | Stationary stochastic processes |

47D03 | Groups and semigroups of linear operators |