##
**Dynamics of foliations, groups and pseudogroups.**
*(English)*
Zbl 1084.37022

The book mainly discusses dynamical and asymptotic properties of foliations on compact manifolds with the particular emphasis on results obtained by the author himself and his collaborators. Many of the results presented appear for the first time in a book. The book consists of 6 chapters.

The first chapter introduces foliations and pseudo-groups. Of particular importance is the holonomy pseudo-group defined by a foliation \({\mathcal F}\) on a compact manifold \(M\) and a “nice” cover of \(M\) by foliation charts (Section 1.3). Particularly useful pseudo-groups are so-called Markov pseudo-groups which are associated to subshifts of finite type. They can be used to construct explicitly closed foliated 3-manifold \((M,{\mathcal F})\) such that \({\mathcal F}\) admits an exceptional minimal set. Section 1.5 contains a summary of those properties of hyperbolic groups and spaces which are needed in the sequel.

In Chapter 2, the author presents a somewhat unified discussion of various notions of growth and growth types for groups, pseudo-groups and foliations. The usual growth of a finitely generated group and some of the well-known results about this growth are introduced. This mainly serves as a motivating background information for the notion of orbit growth (in Section 2.3) and expansion growth for pseudo-groups (in Section 2.4). Once again, of particular importance is the growth of the holonomy pseudo-group of a foliation \({\mathcal F}\) on \(M\). If \({\mathcal F}\) is of class \(C^1\) then this growth is at most exponential.

Section 3 is devoted to the discussion of entropy. In Section 3.1, the topological entropy of a continuous transformation of a compact Hausdorff topological space \(X\) and the entropy of a measure-preserving transformation are introduced as well and some examples are presented. The section concludes with a proof (due to Misiurewicz) of the variational pririciple which bounds the metric entropy of a continuous measure preserving transformation \(f\) of a compact metric space \(X\) by its topological entropy. In Section 3.2, a notion of (topological) entropy of a finitely generated pseudo-group \(G\) of a compact metric space \(X\) is defined. This entropy uses the concept of \((n,\varepsilon)\)-separation and depends on a good finite generating set \(G^1\) for \(G\). However, vanishing of this entropy is independent of the good finite generating set. In the case that \(X\) is the closure of a relatively compact open subset of \(\mathbb{R}^n\) then the entropy of a pseudo-group generated by a finite set \(G_1\) of Lipschitz homeomorphisms can be bounded by the product of the cardinality of \(G_1\) with a common Lipschitz constant for the elements of \(G_1\). Section 3.3 introduces a notion of entropy for a foliation of a compact Riemannian manifold \(M\) using leaf separation along transversals. This entropy depends in a controlled way on the choice of the metric and is related in Section 3.4 to the entropy of holonomy pseudo-groups defined by nice coverings of \(M\) for the foliation. If \({\mathcal F}\) is the one-dimensional foliation defined by a vector field \(X\) of unit length then the foliation entropy of \({\mathcal F}\) is just twice the topological entropy of the time – one map of the flow defined by \(X\). Thus the foliation entropy is a generalization of the classical notion of topological entropy. Section 3.5 is devoted to the discussion of the behavior of foliation entropy under gluing of foliated manifolds with boundaries and under tubulization, and Section 3.6 discusses \(C^2\)-foliations of codimension 1 where more precise informations can be obtained. For example, vanishing of this entropy implies vanishing of the Godbillon-Vey class.

Chapter 4 contains the measure-theoretic counter-part of Chapter 3. It begins with explaining the concept of a measure which is invariant under a pseudo-group. In the case of the holonomy-pseudo-group of a foliation, such a measure on a transversal \(T\) is called a transverse invariant measure. By a result of Plante, reproduced in Section 4.3, the existence of such a measure is guaranteed whenever the foliation possesses a leaf of nonexponential growth. Section 4.4 gives more precise informations in the codimension-one case. Section 4.5 shows how to use averaging to construct transverse invariant measures for foliations with vanishing entropy, and Section 4.6 is once again devoted to foliations of codimension one. In this case, positive entropy of a foliation has topological consequences (the existence of a resilient leaf), a result based on recent work of Hurder. For foliations \({\mathcal F}\) which do not admit transverse invariant measures, one can try to distinguish other classes of measures with particularly nice properties. Given any (leaf-wise) smooth Riemannian metric \(g\) on \(M\), a harmonic measure for \({\mathcal F}\) and \(g\) is a probability measure \(\mu\) such that \(\int\Delta(f)d\mu=0\) for every smooth function \(f\) where \(\Delta\) is the leaf-wise Laplacian along \({\mathcal F}\). Such measures are precisely the measures invariant under the leaf-wise heat diffusion. Moreover, a harmonic measure defines via disintegration a holonomy-invariant measure class on transversals. These facts (which are due to Garnett) are described in Section 4.7. Section 4.8 contains a brief introduction to Patterson-Sullivan measures for discrete subgroups of \(SO(n,1)\).

Chapter 5 passes to dimensions of spaces and pseudo-groups. It contains the definition of Hausdorff-dimension of a pseudo-group and of the transverse Hausdorff-dimension of a \(C^1\)-foliation \({\mathcal F}\) of a compact manifold as the Hausdorff dimension of the holonomy pseudo-group of \({\mathcal F}\) acting on a complete transversal. In Section 5.3, it is shown that if \({\mathcal F}\) is of class \(C^2\) and of codimension one and if the leaves of \({\mathcal F}\) are not dense, then this transverse Hausdorff dimension coincides with the usual Hausdorff dimension of the intersection of a complete transversal with the union of all compact leaves of \({\mathcal F}\).

The last chapter contains a collection of results related to the topics discussed above.

The book is written with care and clarity. It contains an extensive list of references which are carefully selected to guide the reader through the background material and to some proofs which are not contained in the volume.

The first chapter introduces foliations and pseudo-groups. Of particular importance is the holonomy pseudo-group defined by a foliation \({\mathcal F}\) on a compact manifold \(M\) and a “nice” cover of \(M\) by foliation charts (Section 1.3). Particularly useful pseudo-groups are so-called Markov pseudo-groups which are associated to subshifts of finite type. They can be used to construct explicitly closed foliated 3-manifold \((M,{\mathcal F})\) such that \({\mathcal F}\) admits an exceptional minimal set. Section 1.5 contains a summary of those properties of hyperbolic groups and spaces which are needed in the sequel.

In Chapter 2, the author presents a somewhat unified discussion of various notions of growth and growth types for groups, pseudo-groups and foliations. The usual growth of a finitely generated group and some of the well-known results about this growth are introduced. This mainly serves as a motivating background information for the notion of orbit growth (in Section 2.3) and expansion growth for pseudo-groups (in Section 2.4). Once again, of particular importance is the growth of the holonomy pseudo-group of a foliation \({\mathcal F}\) on \(M\). If \({\mathcal F}\) is of class \(C^1\) then this growth is at most exponential.

Section 3 is devoted to the discussion of entropy. In Section 3.1, the topological entropy of a continuous transformation of a compact Hausdorff topological space \(X\) and the entropy of a measure-preserving transformation are introduced as well and some examples are presented. The section concludes with a proof (due to Misiurewicz) of the variational pririciple which bounds the metric entropy of a continuous measure preserving transformation \(f\) of a compact metric space \(X\) by its topological entropy. In Section 3.2, a notion of (topological) entropy of a finitely generated pseudo-group \(G\) of a compact metric space \(X\) is defined. This entropy uses the concept of \((n,\varepsilon)\)-separation and depends on a good finite generating set \(G^1\) for \(G\). However, vanishing of this entropy is independent of the good finite generating set. In the case that \(X\) is the closure of a relatively compact open subset of \(\mathbb{R}^n\) then the entropy of a pseudo-group generated by a finite set \(G_1\) of Lipschitz homeomorphisms can be bounded by the product of the cardinality of \(G_1\) with a common Lipschitz constant for the elements of \(G_1\). Section 3.3 introduces a notion of entropy for a foliation of a compact Riemannian manifold \(M\) using leaf separation along transversals. This entropy depends in a controlled way on the choice of the metric and is related in Section 3.4 to the entropy of holonomy pseudo-groups defined by nice coverings of \(M\) for the foliation. If \({\mathcal F}\) is the one-dimensional foliation defined by a vector field \(X\) of unit length then the foliation entropy of \({\mathcal F}\) is just twice the topological entropy of the time – one map of the flow defined by \(X\). Thus the foliation entropy is a generalization of the classical notion of topological entropy. Section 3.5 is devoted to the discussion of the behavior of foliation entropy under gluing of foliated manifolds with boundaries and under tubulization, and Section 3.6 discusses \(C^2\)-foliations of codimension 1 where more precise informations can be obtained. For example, vanishing of this entropy implies vanishing of the Godbillon-Vey class.

Chapter 4 contains the measure-theoretic counter-part of Chapter 3. It begins with explaining the concept of a measure which is invariant under a pseudo-group. In the case of the holonomy-pseudo-group of a foliation, such a measure on a transversal \(T\) is called a transverse invariant measure. By a result of Plante, reproduced in Section 4.3, the existence of such a measure is guaranteed whenever the foliation possesses a leaf of nonexponential growth. Section 4.4 gives more precise informations in the codimension-one case. Section 4.5 shows how to use averaging to construct transverse invariant measures for foliations with vanishing entropy, and Section 4.6 is once again devoted to foliations of codimension one. In this case, positive entropy of a foliation has topological consequences (the existence of a resilient leaf), a result based on recent work of Hurder. For foliations \({\mathcal F}\) which do not admit transverse invariant measures, one can try to distinguish other classes of measures with particularly nice properties. Given any (leaf-wise) smooth Riemannian metric \(g\) on \(M\), a harmonic measure for \({\mathcal F}\) and \(g\) is a probability measure \(\mu\) such that \(\int\Delta(f)d\mu=0\) for every smooth function \(f\) where \(\Delta\) is the leaf-wise Laplacian along \({\mathcal F}\). Such measures are precisely the measures invariant under the leaf-wise heat diffusion. Moreover, a harmonic measure defines via disintegration a holonomy-invariant measure class on transversals. These facts (which are due to Garnett) are described in Section 4.7. Section 4.8 contains a brief introduction to Patterson-Sullivan measures for discrete subgroups of \(SO(n,1)\).

Chapter 5 passes to dimensions of spaces and pseudo-groups. It contains the definition of Hausdorff-dimension of a pseudo-group and of the transverse Hausdorff-dimension of a \(C^1\)-foliation \({\mathcal F}\) of a compact manifold as the Hausdorff dimension of the holonomy pseudo-group of \({\mathcal F}\) acting on a complete transversal. In Section 5.3, it is shown that if \({\mathcal F}\) is of class \(C^2\) and of codimension one and if the leaves of \({\mathcal F}\) are not dense, then this transverse Hausdorff dimension coincides with the usual Hausdorff dimension of the intersection of a complete transversal with the union of all compact leaves of \({\mathcal F}\).

The last chapter contains a collection of results related to the topics discussed above.

The book is written with care and clarity. It contains an extensive list of references which are carefully selected to guide the reader through the background material and to some proofs which are not contained in the volume.

Reviewer: Ursula Hamenstädt (Bonn)

### MSC:

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

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

57R30 | Foliations in differential topology; geometric theory |

37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |

37C45 | Dimension theory of smooth dynamical systems |

58H05 | Pseudogroups and differentiable groupoids |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

37B40 | Topological entropy |