HYPERBOLIC DYNAMICAL SYSTEMS AND THE NONCOMMUTATIVE INTEGRATION THEORY OF CONNES Jan Segert ABSTRACT We examine hyperbolic differentiable dynamical systems in the context of Connes noncommutative integration theory. We consider a C 2 diffeomorphism f of a compact manifold M with an invariant Borel probability measure µ. The global unstable manifolds partition M into equivalence classes, but this partition is not generally measurable, so the measure structure of the quotient space is pathological. The theory of Connes allows one to construct a generalized integration theory by associating groupoids to such pathological measure spaces. Using results of Ledrappier and Strelcyn on measurable partitions subordinate to unstable manifolds, we prove that the groupoid associated to the quotient by unstable manifolds, which we call the unstable groupoid, is a measure groupoid when restricted to a certain set of µ measure one. From a Riemannian metric on M, we construct in a natural way a transverse function on the unstable groupoid, and a homomorphism of the unstable groupoid into the multiplicative group of positive real numbers. We use these constructions to prove that SRB measures on M give rise to transverse measures on the unstable groupoid restricted to a certain set of measure one. We discuss the relevance of these results to the problem of classifying SRB measures for a given system. 1
1. Introduction A mathematical investigation always obeys the law of conservation of knowledge: we never get out more from it than we put in. The knowledge may be changed in form, it may be clearer and more exactly stated, but the total amount of the knowledge of nature given out by the investigation is the same as we started with. Henry Augustus Rowland, The Highest Aim of the Physicist (1899) This chapter is a brief introduction to the subject matter of this thesis. For the sake of brevity, here many terms are loosely defined, or undefined, and references are restricted mostly to works of an introductory nature. These deficiencies will be remedied in later sections. We shall be studying dynamical systems, in particular, differentiable maps of manifolds, or flows on manifolds. Good introductory references are Arnold and Avez [1], Eckmann and Ruelle [2]. Much of the motivation for the investigation of dynamical systems came from the study of classical mechanics, in particular, celestial mechanics. A leading question about the classical model of the solar system is whether it is stable, i.e., will some of the planets eventually escape, or collide, or crash into the sun? Many of the fundamental ideas of the present theory can be traced back to the work of Poincaré. Poincaré realized that even simple Hamiltonian systems can display a great complexity in the structure of orbits, some orbits being relatively regular, and some exhibiting apparent stochastic behavior. One class of examples which has been extensively studied is the integrable Hamiltonian systems [3,4], and Hamiltonian systems sufficiently close to integrable. The orbits of integrable Hamiltonian systems are rather simple, they lie on invariant tori. The work of Kolmogorov, Arnold, and Moser demonstrated that for Hamiltonian systems sufficiently close to integrable, some invariant tori persist, see e.g.[1,5] Interesting as this is mathematically, one cannot conclude anything about the stability of our solar system, because it is too far from being an integrable system, i.e., the planets are too heavy compared to the sun. At the other extreme is what are often called stochastic systems. The word stochastic is usually used in the loose sense that the time evolution has properties of random or statistical systems, e.g., flipping a coin. Poincaré studied 2
various examples of stochastic behavior in differentiable dynamical systems, a famous example is the orbit structure near a homoclinic point, see e.g.[5,6]. Before the advent of quantum statistical mechanics, there was great impetus to study problems arising from classical statistical mechanics. The question of replacing a time average an ensemble average was of central physical interest, see [7]. The mathematical study of related questions led to the development of ergodic theory, [2,7,8]. Ergodic theory concerns itself with the study of measure space automorphisms, and various stochastic properties thereof. Stochasticity in differentiable dynamical systems is generally related to hyperbolicity, which is, loosely speaking, sensitive dependence on initial conditions [2,9,10]. A hyperbolic system will have nearby points which separate at an exponential rate. In some sense, the initial conditions can never be prescribed precisely enough, and the system appears stochastic. There has been much progress in understanding various classes of hyperbolic systems. An important example is the geodesic flow on a manifold of negative curvature. The geometric and ergodic properties of this example have been widely studied [1,11]. This example was the prototype for Anosov s work. The so called Anosov systems [12] satisfy a very strong form of hyperbolicity, and have very strong stochastic properties. A very important contribution of Anosov was the demonstration that such systems are structurally stable. Further deep contributions on hyperbolic systems were made by Smale, whose Axiom A systems [13] were more general, but still satisfied a rather strong form of hyperbolicity. The next major development was the work of Pesin [14]. Pesin studied arbitrary differentiable dynamical systems, and was able to derive significant results from relatively weak hyperbolicity conditions. A central part of the study of hyperbolic systems has been the theory of stable manifolds. The basic idea is that for a hyperbolic dynamical system, under certain conditions, the set of points y such that as n tends to, f n y approaches agivenpointx, pursuant to some further restrictions, forms a submanifold, the so called stable manifold of x [6,14]. In various guises, this is a very old 3
notion. The central technical result is the stable manifold theorem for hyperbolic maps of a Banach space, which however was not fully understood until relatively recently [15,16]. Similarly, the unstable manifolds are the points which approach a point sufficiently fast for the inverse of the dynamical system. The stronger the hyperbolicity condition, the better the form of the stable manifold results one can obtain. Much of Anosov s work is based on consequences of the very good properties of the stable manifolds [12]. The stable manifolds for Axiom A system retain many of these properties [13,17]. Pesin s stable manifold theorem [18] holds for systems which preserve a smooth measure. Ruelle [19] obtained a stable manifold theorem without assuming an invariant smooth measure. Pesin s work is based to a large extent on the properties he was able to prove for the stable manifolds of systems with a weaker form of hyperbolicity [14,20]. In some sense, the stable (and unstable) manifolds have good local properties, but bad global properties. Globally, the stable manifold of a point is an immersed Euclidean space. We generally consider dynamical systems on a compact manifold X, so the stable manifold accumulates on itself, or can even be dense in X. Although the (global) stable manifolds partition X into equivalence classes, the quotient of X by this equivalence relation, i.e., the space of stable manifolds, generally has terrible properties. As a topological space, the quotient may have no non-trivial open sets. Even as a measure space, the quotient is often not countably separated. The usual analytic tools are of no use. For this reason, the quotient space has not been of use, although naively it appears to be a very useful construct. For example, one might hope to decompose an invariant measure into pieces supported on the invariant manifolds. This fails, because the disintegration of a measure is valid only for a measurable decomposition [21], and the decomposition into global stable (or unstable) manifolds is in general not measurable. We quote from Eckmann and Ruelle [2] p 639:... the unstable manifolds partition the space into equivalence classes. it might seem natural to define SRB measures by using this partition for a 4
decomposition of ρ into pieces ρ α, carried by different unstable manifolds: ρ = ρ α m(dα), where α parametrizes the V u s, and m is a measure on the space of equivalence classes. In reality, this space of equivalence classes does not exist in general (as a measurable space) because of the folding and accumulation of the global unstable manifolds... The global stable and unstable manifolds, in all their geometrical complexity, reflect important properties of the dynamical system. This is clearly seen in the study of attractors [2,22]. Attractors are, loosely speaking, subsets onto which points converge. Attractors, in a large variety of circumstances, are a union of global unstable manifolds. The geometrical complexity of systems with such attractors reflects the complexity of the global unstable manifolds. Another indication is given by the metric entropy. The metric entropy [23] is an invariant of a measurable dynamical system with invariant measure defined by Kolmogorov in the 1950 s. The metric entropy, or entropy for short, measures the amount of information generated as the dynamical system evolves. If a system has sensitive dependence on initial conditions, we expect information regarding the initial condition to be generated as the system evolves. For differentiable dynamical systems, the entropy is intimately related to the properties of the unstable manifolds [2]. Although measurable decomposition of X into global unstable manifolds is not possible, one can cover X by small subsets, such that the decomposition of each subset into pieces of unstable manifolds is measurable. One can then locally disintegrate the invariant measure into pieces supported on pieces of unstable manifolds. The entropy is related to the smoothness of the disintegrated measures. The original work along these lines was done by Sinai, on systems with a transverse foliation [24]. Pesin proved a result of this type for a class of systems satisfying weak hyperbolicity conditions, and preserving a smooth measure. The metric entropy is in these cases equal to the integral of a certain 5
geometric quantity [20]. This is sometimes called Pesin s formula. Recently, a very general result has been proved by Ledrappier and Young [25,26], which applies for arbitrary Borel measures, and assumes no hyperbolicity conditions. The positivity of the entropy is related to the presence of hyperbolicity. In both the study of attractors and metric entropy, a special class of measures has certain important properties. These are the SRB measures, named after Sinai, Ruelle, and Bowen, each of whom made important contributions to their study. The SRB measures can be characterized by a number of properties, perhaps the the simplest characterization is that their conditional measures along the unstable manifolds are smooth. These measures are important also for the study of physical systems [2]. The existence of SRB measures for a given dynamical system is an important open question. In this thesis, we introduce a new tool to the study of hyperbolic dynamical systems. We study the space of global stable and unstable manifolds using Connes theory of noncommutative measure theory [27,28]. Connes theory generalizes measure theory to singular measure spaces, e.g., the quotient space of an ergodic equivalence relation. This theory has its origins in the virtual group program of Mackey [29,30]. The main technical notion is that of a measure groupoid. There are connections between virtual groups and von Neumann algebras, as had been noted by various authors, e.g., [31,32]. Connes introduced concepts related to traces and weights on von Neumann algebras to the setting of measure groupoids. The resulting noncommutative integration theory generalizes ordinary integration theory. In particular, one may treat spaces which are pathological as ordinary measure spaces, such as the quotient by global stable or unstable manifolds. The central part of the noncommutative integration theory concerns the generalizations of functions and measures which can satisfactorily be defined and applied on the badly behaved quotient space. The generalization of a function is a measurable functor from the measure groupoid to the category of standard measure spaces. The generalization of a measure is called a transverse measure, and can be related to an ordinary measure by a theorem 6
which generalizes the classical notion of disintegration of a measure relative to a measurable partition. We apply this theory to the quotient of dynamical system by the unstable (or stable) manifolds. The properties of the quotient space, considered as a singular space in the sense of Connes, of course depend on the properties of the dynamical system. Likewise, the disintegrated measure depends on the original measure invariant under the dynamical system. Clearly, the properties of the stable (or unstable) manifolds are important. The stronger the hyperbolicity, the better behaved are the stable manifolds and the quotient space. The Anosov case has the strongest hyperbolicity properties, and correspondingly, the quotient space has the best behavior. In the general case, we have Pesin theory stable manifolds, whose properties are not as nice. We study the various cases, and derive corresponding regularity properties of the quotient, considered as a measurable groupoid with transverse measure. A chapter by chapter outline follows. In chapter 2, we shall discuss hyperbolic dynamical systems. We define the terms used loosely in the preceding. We shall emphasize the various forms of the stable manifold theorems, and related results. We discuss the physical importance of measures which are smooth along the unstable manifolds, and present the theorems which connect this property to the metric entropy. Although we can of course make no claim of completeness, this chapter is reasonably self contained regarding the material covered. This chapter is expository, and contains no new results. In chapter 3, we review aspects of measure theory. The chapter covers some basic results in measure theory, including results on standard spaces which will be used later. We review the theory of measurable partitions, which is important in the sequel. We study with concrete examples the nonmeasurability of the partition into global stable manifolds. This chapter is expository, and contains no new results. 7
Chapter 4 is an introduction to Connes noncommutative measure theory. We construct concretely a simple version of this theory, which should indicate the major ideas without getting bogged down in the technical detail necessary for the general case. The basic idea is to replace the notion of a function. A function associates to each point a number. This is replaced by associating to each point a measure space with a mass. We show the equivalence of these approaches in the ordinary (commutative) setting. This chapter is based on the ideas of Connes. We formulate and prove a number of results implicit in [27], but not explicitly stated. Chapter 5 presents the general form of Connes noncommutative measure theory. The notion of a measure space is replaced by a (noncommutative) measure groupoid. The functions on a measure space are replaced by the notion of transverse functions on the groupoid, which can be integrated using a transverse measure. Much of this chapter is an exposition of the work of Connes [27] and Kastler [28]. We also prove some new results which are necessary for the following chapter. Chapter 6 is the heart of this work. It contains wholly original material. We describe an application of the noncommutative integration theory to hyperbolic dynamical systems. We study SRB measures in this context. We construct the space of global unstable manifolds as a principal groupoid, which we call the unstable groupoid. We prove that this is a measure space in the sense of the noncommutative theory, i.e., the unstable groupoid is a measure groupoid. We next construct in a canonical way a transverse function on the unstable groupoid associated to a Riemann metric. At this stage, we must excise a set of measure zero in order to prove that this transverse function is proper. We show how an SRB measure naturally defines a transverse measure on the unstable groupoid. A theorem of Ledrappier and Strelcyn [33] on the existence of a certain type of measurable partition plays an important role in the proofs. We then discuss the implications of these constructions. We exhibit a technical obstacle which prevents us from concluding that there is an injection from the set of SRB 8
measures to the set of transverse measures on the unstable groupoid. If this injection could be proved, this would result in the possibility of proving results on the existence and uniqueness of SRB measures from corresponding results on transverse measures on the unstable groupoid. However, in general, very little can be said about the classification of transverse measures. We analyze the problem of proving the injection, and present some conditions from which the result would follow. We conclude this introduction by indulging in some speculation about the significance of the noncommutative integration theory in the study of measures on hyperbolic dynamical systems. We consider an analogy with differential geometry. A differentiable manifold is composed of local coordinate patches, and every notion of differential geometry, the exterior derivative, the Levi-Civita connection, is concretely described in terms of some local coordinates. Certain special types of coordinates, e.g., Riemann normal coordinates, greatly simplify some concrete calculations. But the power of differential geometry lies in the coordinate independence, in the fact that the specific choices of local coordinates make no difference. Now consider the present situation in the study of measures on hyperbolic dynamical systems. Consider for example the Pesin entropy formula. This is proved by using a certain kind of measurable partition, whose elements are subsets of unstable manifolds. The proof becomes tractable using these partitions. Consider such partitions as the analogues of, say, Riemann normal coordinates in differential geometry. They have properties amenable to to performing a certain type of calculation. But further insight into differential geometry comes from considering coordinate independent notions. One might think of the noncommutative theory of the quotient by unstable manifolds as the analogue in this setting of coordinate independent differential geometry. This idea should be taken with a grain of salt, but might be a useful reference point for the reader. 9