Steven Hurder

Differentiability, rigidity and Godbillon-Vey classes for Anosov flows

The central geometric objects associated with an Anosov dynamical system on a compact manifold are the invariant stable and unstable foliations. While each stable and unstable manifold is as smooth as the system itself, the foliations that they form are believed to have only a moderate degree of regularity for most systems. We will analyze the exact degree of regularity of codimension-one stable and unstable foliations for low dimensional systems. Our main results relate the regularity of these foliations to cohomology classes associated to the system: the Anosov class, a new invariant of the flow which we introduce in this paper, and the Godbillon-Vey class of the weak-stable foliations, which we show is a well-defined invariant of the system

Homogeneous matchbox manifolds

We prove that a homogeneous matchbox manifold of any finite dimension is homeomorphic to a McCord solenoid, thereby proving a strong version of a conjecture of Fokkink and Oversteegen. The proof uses techniques from the theory of foliations that involve making important connections between homogeneity and equicontinuity. The results provide a framework for the study of equicontinuous minimal sets of foliations that have the structure of a matchbox manifold.

Cyclic cocycles, renormalization and eta-invariants

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The longitudinal cocycle and the index of Toeplitz operators

Abstract Let D be a self-adjoint leafwise elliptic operator on a foliated manifold. Compressing multiplication operators to the range of the positive spectral projection for D yields the class of leafwise Toeplitz operators. The extension generated by these operators is constructed. A topological formula for the index of a Toeplitz operator with invertible symbol is given. This index can also be obtained by pairing the K-theory class of the symbol with a certain cyclic cocycle. If one lifts an elliptic operator on a closed manifold to a leafwise elliptic operator on an associated flat foliated principal bundle, then this cocycle can be used to obtain refined invariants of the original operator.

Dual homotopy invariants of G-foliations

THIS PAPER develops a theory of dual homotopy invariants for G-foliations using the theory of minimal models. As an application of the theory which is constructed, we are able to extend the results of Heitsch on the independent variation of the secondary classes of foliations. A foliation with a non-zero rigid class is also shown to exist, based on an example of Schweitzer and Whitman. For Riemannian foliations, all of the indecomposable secondary classes are shown to be linearly independent in H*(FRI”). The work of Lazarov and Pasternack is used to show that all of the possibly variable indecomposable secondary classes are independently variable in H*(FR14). The third type of G-foliations considered are those with an integrable complex structure on their normal bundles. The results of Baum and Bott are used to establish that many of the secondary classes for these foliations are independently variable. For each of the three types of G-foliations considered, namely real, Riemannian and complex, it is shown that the homotopy groups of the corresponding classifying space BIG4 admit epimorphisms 7r~(BIo4)+RUn, where {u,} is a sequence depending on 4 and G, but which in general has a subsequence tending to infinity. If a manifold M is simply connected, then the invariants of a G-foliation 9 on M which we produce are functions on the homotopy groups of M. They can be viewed as generalizations of several other constructions of foliation invariants in the literature: There is a natural relation with the Chern-Simons invariants [9]. A means for producing such invariants was introduced by Haefliger in [18]. The various residue theorems for a G-foliation with singularities at a discrete set of points [2, 32 and 381 are special cases of this theory, where the residue is obtained by evaluating a dual homotopy class on the boundary of a disc about a singular point. For a Riemannian or complex foliation which is defined by a submersion[25], the secondary classes of the foliation are exactly cohomological representations of some of the dual homotopy invariants. The general theory of the invariants is developed in 02. We begin by showing that the algebra homotopy class of the truncated Chern-Weil homomorphism h(w) is a G-foliation invariant (Theorem 2.11); in fact, it is a universal invariant from which many other invariants of the foliation can be derived [23]. Applying the dual homotopy functor to h(o) yields a characteristic map h#: r*(I(G),)*r*(M) from the infinitedimensional vector space r*(l(G),) to the (pseudo) dual homotopy of the manifold (Theorem 2.12).

LS-category of compact Hausdorff foliations

The transverse (saturated) Lusternik-Schnirelmann category of foliations, introduced by the first author in [5, 9], is an invariant of foliated homotopy type with values in {1,2,...,1}. A foliation with all leaves compact and Hausdorff leaf space M/F is called compact Hausdorff. The transverse saturated category cat\| M of a compact Hausdorff foliation is always finite. In this paper we study the transverse category of compact Hausdorff foliations. Our main result provides upper and lower bounds on the transverse category cat\| (M) in terms of the geometry of F and the Epstein filtration of the exceptional set E. The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove that max{cat(M/F),cat\| (E)} � cat\| (M) � cat\| (E) + q We give examples to show that both the upper and lower bounds are realized, so the estimate is sharp. We also construct a family of examples for which the transverse category for a compact Hausdorff foliation can be arbitrarily large, though the category of the leaf spaces is constant.

Secondary classes and transverse measure theory of a foliation

1. The purpose of this note is to announce several theorems showing how the secondary classes of a foliation J of a compact manifold X depend upon the measure theoretic properties of the equivalence relation determined by the foliation. The relevant properties are: (i) amenability [14], which is equivalent to hyperfiniteness by ConnesFeldman-Weiss [3]; and (ii) the Murray-von Neumann type. A set B C X is saturated if it is the union of leaves of 7. The equivalence relation 7 has type I if there is a measurable subset of X which intersects almost every leaf exactly once; type II if it admits an invariant measure, finite or infinite in the given measure class but does not have an essential saturated set of type I; and type III if it does not have any essential saturated sets of types I or II. Every equivalence relation can be decomposed into parts of types I, II, and III. These types correspond to certain algebraic properties of the von Neumann algebra M(X, 7) associated with the equivalence relation [1, 13]. Let X be a compact manifold without boundary and J a C 2 , codimensionn foliation of X. The secondary classes are given by a map A* : H*(WOn) —• H*(X; R) with image spanned by the classes of the form A*(yicj). Here, yi is a basis element for the relative cohomology if*(gln, On), and cj is a Chern form of degree at most 2n. If degree cj = 2n, we say the class is residual. The Godbillon-Vey classes are those of the form A*(yicj) € # 2 n + 1 ( X ; R ) , with 2/1 G if(gln ,On) , the normalized basis element. The generalized GodbillonVey classes are those of the form A*(yiy/Cj), where yi = 1 is permitted. (For a convenient reference, see [11].) The residual secondary classes have the unusual property that they localize to the measurable saturated subsets of X: for each such B c X and residual yicj e # p (WO n ) , the restriction A*(2//Cj)|£ G H (X) is well defined [5]. The following theorems are stated for the secondary classes of 7 on X, but corresponding theorems also hold for the localized classes A*(yiCj)\B of the restriction 7\B.

Lectures on Foliation Dynamics

The study of foliation dynamics seeks to understand the asymptotic properties of leaves of foliated manifolds, their statistical properties such as orbit growth rates and geometric entropy, and to classify geometric and topological “structures” which are associated to the dynamics, such as the minimal sets of the foliation.

Transverse LS category for Riemannian foliations

We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation F on a closed manifold M. The essential transverse category cat e (M, F) is introduced in this paper, and we prove that cat e (M, F) is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category cat (M, F) is finite, and thus cat e (M, F) = cat(M, F) holds. A fundamental point of this paper is to use properties of Riemannian submersions and the Molino Structure Theory for Riemannian foliations to transform the calculation of cat e (M, F) into a standard problem about O(q)-equivariant LS category theory. A main result, Theorem 1.6, states that for an associated O(q)-manifold W, we have that cat e (M, F) = cat O(q) (Ŵ). Hence, the traditional techniques developed for the study of smooth compact Lie group actions can be effectively employed for the study of the LS category of Riemannian foliations. A generalization of the Lusternik-Schnirelmann theorem is derived: given a C 1 -function f: M → R which is constant along the leaves of a Riemannian foliation F, the essential transverse category cat e (M, F) is a lower bound for the number of critical leaf closures of f.

Exceptional minimal sets of C 1+α -group actions on the circle

We prove two extensions of Sacksteders Theorem for the action A : Γ × S 1 → S 1 of a finitely-generated group Γ on the circle by C 1+α -diffeomorphisms. If the action A has an exceptional minimal set K with a gap endpoint of exponential orbit growth rate, or if the action A on K has positive topological entropy, then the exceptional set K is hyperbolic. That is, A has a linearly contracting fixed-point in K . A key point of the paper is to prove a foliation closing lemma using the foliation geodesic flow technique.