Nonsingular Flows Research Articles

ON QUOTIENTS OF NONSINGULAR ACTIONS WHOSE SELF-JOININGS ARE GRAPHS

We study the notions of minimal self-joinings (MSJ) and graph self-joinings (GSJ) (analogous to simplicity in the finite measure preserving case) for nonsingular actions of locally compact groups. We show that for a nonsingular action of a group G with MSJ, every quotient comes from a closed subgroup of the center of G whose action is totally non-ergodic. Thus, totally ergodic nonsingular flows with MSJ are prime. We then show an analogous result to Veech's theorem, namely that for a nonsingular action of a group G with GSJ, every quotient comes from a locally compact subgroup of the centralizer whose action is totally non-ergodic.

An ergodic maximal equality for nonsingular flows and applications

This equality implies that a nonnegative f in LI(#) satisfies f f l o g f d # l} if and only if f f'd# < o0. (For a related topic, see [1], [9], [111 and [14].) About ten years ago, Marcus and Petersen [8] proved equality (2), under the assumption that all the Tt are measure preserving. Thus our result may be regarded as a generalization of their theorem to noninvariant measures. Further we shall apply the proof of (2) to investigate an integrability problem for the maximal function Mf defined by

A note on the classification of nonsingular flows with transverse similarity structures

A note on the classification of nonsingular flows with transverse similarity structures

Zeros of integrals along trajectories of ergodic nonsingular flows

Zeros of integrals along trajectories of ergodic nonsingular flows

Volumes of Flows

If $F$ is an oriented nonsingular flow on a Riemannian manifold $M$, the volume of $F$ is defined as the $n$-dimensional measure of the unit vector field tangent to $F$, as a section of ${T_ * }\left ( M \right )$ with the induced metric. It is shown that, for any metric of the two-dimensional torus, and for any homotopy class of flows on the torus, there is a unique smooth flow of minimal volume within the homotopy class. It has been shown that the Hopf foliation on the round threesphere absolutely minimizes the volume of flows on ${S^3}$. In higher dimensions this is not the case; the Hopf fibrations are not even local minima of the volume functional for flows on the round five-sphere. It is not known whether a volume-minimizing flow on ${S^5}$ exists.

Volumes of flows

If F F is an oriented nonsingular flow on a Riemannian manifold M M , the volume of F F is defined as the n n -dimensional measure of the unit vector field tangent to F F , as a section of T ∗ ( M ) {T_ * }\left ( M \right ) with the induced metric. It is shown that, for any metric of the two-dimensional torus, and for any homotopy class of flows on the torus, there is a unique smooth flow of minimal volume within the homotopy class. It has been shown that the Hopf foliation on the round threesphere absolutely minimizes the volume of flows on S 3 {S^3} . In higher dimensions this is not the case; the Hopf fibrations are not even local minima of the volume functional for flows on the round five-sphere. It is not known whether a volume-minimizing flow on S 5 {S^5} exists.

Vector fields with transverse foliations, II

AbstractWhen does a non-singular flow on a 3-manifold have a 2-dimensional foliation everywhere transverse to it? A complete answer is given for a large class of flows, those with 1-dimensional hyperbolic chain recurrent set. We find a simple necessary and sufficient condition on the linking of periodic orbits of the flow.

Nonsingular holomorphic flows

Soit M une variete de Kahler compacte et v un champ vectoriel holomorphe. Alors soit v a un zero soit sa partie reelle v+v a une section droite

Any flow is an orbit factor of any flow

AbstractWe prove that given any two ergodic non-singular flows , St, the first can be time changed to t so that St is a factor of t. A corresponding result for transformations is that if , T are any two ergodic non-singular transformations then there is a tower over that has T as a factor.

Flows associated with product type odometers

AbstractAn AC-flow is the associated flow of a product type odometer (PTO). We give examples of AC-flows and compute their L∞-point-spectra. We also introduce an invariant for isomorphism of aperiodic conservative ergodic nonsingular flows which is a closed subset of the unit interval and contains 0 and 1. We give a necessary condition for the associated flow of an approximately finite ergodic group to be finite measure preserving.

Homological identities for closed orbits

The classical Lefschetz formula expresses a relationship between the fixed points of certain transformations and homological invariants. When applied to continuous systems, one obtains the Hopf formula relating the singularities of a flow to the Euler characteristic of the underlying manifold. The main goal of this paper is to prove an analogous formula that relates the closed orbits of a stable flow to a homological invariant of the manifold. There are serious obstruction to finding such a formula for flows. The first is that there are flows on any closed manifold of dimension > 4 of zero Euler characteristic with no periodic orbits whatsoever [27]. These flows, however, are unstable by Pugh's C a closing lemma [19]. So it is natural to restrict attention to flows that satisfy Axiom A and the No Cycles property, since Smale has shown that these flows are O-stable [23]. Secondly there are gradient flows on any closed manifold that are Axiom A-No Cycles and have no closed orbits [24]. So it is natural to study only nonsingular flows. The possibility that a Lefschetz type formula for such flows existed was suggested to us by R. Bowen. Even nonsingular Morse-Smale flows, however, are difficult to deal with since Asimov's geometric index is not an invariant of the nonsingular homotopy class [2]. We must further restrict the class of flows to assume that the closed orbits relate in some manner to the homology of the underlying manifold. We define in Sect. 2 the class of homology proper flows. Such a flow has the property that closed orbits of large period lie in large homology classes. In Sect. 7 we show that a homology proper Axiom A-No Cycles flow does satisfy a Lefschetz-type formula that relates invariants of the closed orbits to a manifold invariant. This formula is closely related to Franks' study of Smale flows on S 3 in which the linking numbers of closed orbits are used to compute the Alexander polynomial of a certain link. Indeed the manifold invariant used in Sect. 7 is a close analogue of an Alexander polynomial that we call the Alexander

Non-singular Periodic Flows on T3 and Periodic Homeomorphisms of T2

A homeomorphism f of a space M is pointwise periodic if for each m ∈ M there exists an integer k such that fk(m) =m, where fk is the kth iterate of f. Montgomery proves [5] that if M is a connected topological manifold, then f is periodic; i.e., there exists an integer n such that fn = id. Noting this, Weaver [7] proves that if M is an orientable 2-manifold of class C1, U⊆M open and C ⊆ U a compact connected set and if g : U → M of class C1 is such that (i) g(C) = C and (ii) whenever the derivative of g at points x∈C has rank 2 it is orientation preserving, then f = g|c : C → C periodic implies that all but a finite number of points of C have as least period the period of f.

The spectrum of the adjoint representation and the hyperbolicity of dynamical systems

Let (M,g) denote a smooth compact Riemannian manifold. Anosov [I] defined the global hypcrbolicity of a C2 diffeomorphism (resp. a nonsingular flow f’) on M :f (resp. fr) is h yperbolic or, as we shall say, Anosov, if the tangent bundle TM splits as a sum of invariant subbundles TM = E’ ~3 E(resp. TM = B @ E@ [A’, where [X7 is th I e ine bundle spanned by S = d/dt(ft)) such that Tf exponentially contracts E+ in positive time while exponentially contracting Ein negative time. In this paper we exhibit several criteria which are equivalent to Anosov’s hyperbolicity condition. Our work continues the studies of a number of authors, which we can describe after fixing some terminology. Let C(E) denote the real Banach space of continuous sections of TM or, in the flow case, sections of TM/[X] with norm ),I 17 1) = sup(g,(~, 7) / .Y E M}. Define the adjoint representation f* off (resp. fi off’) as the bounded operator on C(E):

The periodic structure of non-singular Morse-Smale flows

The periodic structure of non-singular Morse-Smale flows

Plugging Flows

A plug construction is a local modification of a nonsingular flow which severs certain kinds of recurrence properties. In this paper we investigate the effect of plug constructions on minimal sets, the nonwandering set, and the chain recurrent set and the explosions of these sets when a plug construction is perturbed.

Non-singular flows on $S^3$ with hyperbolic chain-recurrent set

Non-singular flows on $S^3$ with hyperbolic chain-recurrent set

Point Spectra of Non-Singular Flows

An example is given of a family of non-singular transformations whose point spectra are uncountable.

Foliations with all leaves compact

(Received 20 October 1975) §1. THE PLr~OSE of this paper is to present some information about the following Question: I[ M is a compact manifold [oliated by compact submani[olds (everything smooth), is there an upper bound on the volume of the leaves?t In particular, if M is a compact manifold supporting a nonsingular flow in which each orbit is periodic, is there an upper bound on the lengths of the orbits? In his thesis, Reeb ([12], for an analytic example see [3, p. 68]) describes a smooth flow on a non-compact manifold such that all orbits are periodic and such that the lengths of the orbits are not locally bounded. After our research was completed, the third author found a smooth flow on a closed 5-manifold[15, 16], which showed that the answer, in.general, was no. The former example shows that the question is global and cannot be answered by simply considering the structure of the foliation in a neighborhood of individual compact leaves. The latter example shows that some additional hypothesis on M is required. This example is worth keeping in mind while reading this paper due to its close connection with our main result. The existence of an upper bound on the volume of the leaves has rather important consequences which provide a description of the local, as well as global, structure of the foliation. The boundedness of volume near any given leaf is equivalent to the finiteness of the holonomy group of that leaf, and also to the hausdroff separation property for the topology of the leaf space near the leaf ([4], [8], see also §4). Hence, in the presence of a bound on the volume, a structure theorem due to Ehresmann [4, Theorem 4.3] provides a nice picture of the local behavior of the foliation. In the absence of such a bound the geometrical possibilities are somewhat formidable. Concerning the known cases of the Question, if the leaves have condimension 1 then the bound exists by a relatively elementary argument [12]. For periodic flows on compact 3-manifolds Epstein [3] has demonstrated the existence of a bound by a surprisingly delicate argument. Our two principal results were obtained in trying to understand Epstein's argument. Our first result is that in the presence of a certain homological assumption, the answer to the Question is yes. THEOREM 1. Suppose M is a compact smooth manifold which is smoothly foliated by compact leaves of dimension d. Suppose that the leaves are oriented in a continuous manner, and that the images of the fundamental classes of the leaves all lie in some open hal[space of the d-dimensional real homology of M. Then there is an upper bound on the volumes of the leaves of M. Consequently, all the holonomy groups of the foliation are finite. Another way of stating the homologicai condition is to say there exists a closed d-form to on M, such that to has a positive integral along each leaf. Here M may have boundary, in which case we assume that the boundary is a union of leaves. The with-boundary version of the Theorem follows from the without-boundary version by doubling M along its boundary. Two particular situations to which the theorem applies are when (i) each leaf of the foliation

Plugging flows

A plug construction is a local modification of a nonsingular flow which severs certain kinds of recurrence properties. In this paper we investigate the effect of plug constructions on minimal sets, the nonwandering set, and the chain recurrent set and the explosions of these sets when a plug construction is perturbed.

Flows Associated with Ergodic Non-Singular Transformation Groups

In this paper we shall give a nice invariant for the weak equivalence of ergodic non-singular transformation groups. It is a one-parameter ergodic non-singular flow associated with an ergodic non-singular transformation group. Since in 1960 an example of an ergodic non-singular transformation without <r-finite invariant measures was given in Ergodic theory [16], the structure and the classification of ergodic non-singular transformations have been studied by many authors ([2], [4], [5], [6], [7], [8] ~[H] and [13]). Among these works, Krieger's weak equivalence theory is fundamental in the classification problem of ergodic nonsingular transformation groups without er-finite invariant measures. This classification is closely connected with the classification of type III factors in the theory of von Neumann algebras ([15]). The Tomita-Takesaki theory of generalized Hilbert algebras ([18]) plays important roles in the analysis of type III factors. Using this theory, A. Connes [3] introduced algebraic invariants — the S-set S(M) and the T-set T(M) — for a factor M and obtained a classification of type III factors. M. Takesaki [19] introduced the dual action of the modular automorphism group and obtained the structure theorem of type III factors. In the classification problem of ergodic non-singular transformation groups G, W. Krieger [10] and the present authors [6] introduced invariants r(G) and T(G) respectively, both of which are