# Lyapunov 1-forms for flows

arXiv:math/0210473v2 [math.DS] 10 Feb 2003

Lyapunov 1-forms for ?ows

M. Farber? , T. Kappeler? , J. Latschev? and E. Zehnder February 10, 2003

Abstract In this paper we ?nd conditions which guarantee that a given ?ow Φ on a compact metric space X admits a Lyapunov one-form ω lying in a ˇ ˇ 1 (X ; R). These conditions are prescribed Cech cohomology class ξ ∈ H formulated in terms of the restriction of ξ to the chain recurrent set of Φ. The result of the paper may be viewed as a generalization of a well-known theorem of C. Conley about the existence of Lyapunov functions.

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Introduction

C. Conley proved in [1, 2] that any ?ow Φ : X × R → X on a compact metric space X decomposes into a chain recurrent ?ow and a gradient-like ?ow. More precisely, he proved the existence of a Lyapunov function for the ?ow, i.e. a continuous function L : X → R, which decreases along any orbit of the ?ow lying in the complement X ? R of the chain recurrent set R ? X of Φ and is constant on the connected components1 of R. Theorem 1 (C. Conley, [1, 2]) Let Φ : X × R → X , Φ(x, t) = x · t, be a continuous ?ow on a compact metric space X . Then there exists a continuous function L : X → R, which is constant on the connected components of the chain recurrent set R = R(Φ) of the ?ow Φ, and satis?es L(x · t) < L(x) for any x ∈ X ? R and t > 0. This important result led Conley to his program of understanding very general ?ows as collections of isolated invariant sets linked by heteroclinic orbits. Our aim in this paper is to go one step further and to analyze the ?ow within the chain-recurrent set R, where it is typically complicated. As a new tool, we study the notion of a Lyapunov one-form for Φ, which is a natural generalization of the notion of a Lyapunov function and has been introduced in

? Partially supported by a grant from the Israel Academy of Sciences and Humanities and by the Herman Minkowski Center for Geometry; part of this work was done while M. Farber visited FIM ETH in Zurich ? Partially supported by the European Commission under grant HPRN-CT-1999-00118 1 C. Conley (see [2], Theorem 3.6D) proved that the chain transitive components of R coincide with the connected components of R.

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a di?erent context in Farber’s papers [5, 6]. We prove that under some natural ˇ ˇ 1 (X ; R) can be represented assumptions, a given Cech cohomology class ξ ∈ H by a continuous closed Lyapunov one-form for the ?ow Φ. The notion of a continuous closed one-form on a topological space generalizes the notion of a continuous function. For convenience of the reader we recall the relevant de?nitions and the main properties of continuous closed one-forms in §2, referring for more details to the papers [5, 6], where they were originally introduced. For the purpose of this introduction, let us say that continuous closed one-forms are analogues of the familiar smooth closed one-forms on differentiable manifolds. Any continuous closed one-form ω on a topological space ˇ ˇ 1 (X ; R), which plays X canonically determines a Cech cohomology class [ω ] ∈ H a role analogous to the de Rham cohomology class of a smooth closed one-form. For any continuous curve γ : [0, 1] → X , the line integral γ ω ∈ R is de?ned and has the usual properties; in particular, it depends only on the homotopy class of the curve relative to the end-points. De?nition 1 Consider a continuous ?ow Φ : X × R → X on a topological space X . Let Y ? X be a closed subset invariant under Φ. A continuous closed one–form ω on X is called a Lyapunov one–form for the pair (Φ, Y ) if it has the following two properties: (L1) For every x ∈ X ? Y and every t > 0,

x·t

ω < 0,

x

where the integral is calculated along the trajectory of the ?ow. (L2) There exists a continuous function f : U → R de?ned on an open neighborhood U of Y such that ω |U = df and f is constant on any connected component of Y . Any continuous function L : X → R determines the closed one-form ω = dL (see §2) and in this special case, condition (L1) reduces to the requirement L(x · t) < L(x) for any t > 0 and x ∈ X ? Y , while condition (L2) means that L is constant on any connected component of Y . Hence for ω = dL, the above de?nition reduces to the classical notion of a Lyapunov function, see [13]. The following remark illustrates De?nition 1. Given a ?ow Φ on X and a ˇ Lyapunov one–form ω for (Φ, Y ), representing a nonzero Cech cohomology class ˇ 1 (X ; R), the homology class z ∈ H1 (X ; Z) of any periodic orbit of [ω ] = ξ ∈ H Φ satis?es ξ, z ≤ 0, with equality if and only if the periodic orbit is contained in Y . Using this ˇ 1 (X ; R) fact one constructs ?ows such that no nonzero cohomology class ξ ∈ H contains a Lyapunov 1-form. ˇ 1 (X ; R) a In this paper we will associate with any cohomology class ξ ∈ H subset Rξ ? R of the chain recurrent set R = R(Φ) of the ?ow Φ, see section 2

§3 for details. The set Rξ is closed and invariant under the ?ow and can be characterized as the projection of the chain recurrent set of the natural lift of the ?ow to the Abelian cover of X associated with the class ξ . A (δ, T )-cycle is a pair (x, t) ∈ X × R satisfying t ≥ T and d(x, x · t) < δ . Here d denotes the distance function on X . See Figure 1. If X is locally path-connected, any (δ, T )-cycle with small enough δ determines a closed loop, which ?rst follows the ?ow line from x to x · t and then returns from x · t to x by a path contained in a suitably small ball. This leads to the notion of a homology class z ∈ H1 (X ; Z) associated to a (δ, T )-cycle, see De?nition 4. The class z is uniquely de?ned if X is homologically locally 1-connected; without this assumption the homology class z associated with a (δ, T )-cycle might not ˇ 1 (X ; R) × H1 (X ; Z) → R can be unique. The natural bilinear pairing , : H

Figure 1: (δ, T )-cycle be understood as ξ, z =

γ

ω, where ω is a representative closed one–form for

ˇ 1 (X ; R) and γ : [0, 1] → X is a loop representing the class z ∈ H1 (X ; Z). ξ∈H Despite the fact that the homology class z associated to a (δ, T )-cycle might depend (for wild X ) on the choice of the connecting path between x · t and x, the construction is such that the value ξ, z ∈ R only depends (for small enough δ > 0) on the (δ, T )-cycle itself; see §3 for details. The following Theorem is our main result: Theorem 2 Let Φ be a continuous ?ow on a compact, locally path connected, ˇ 1 (X ; R). Denote by Cξ the subset metric space X and ξ a cohomology class in H Cξ = R ? Rξ (1.1)

of the chain recurrent set R of the ?ow Φ. Assume that the following two conditions are satis?ed (A) ξ |Rξ = 0;

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(B) there exist constants δ > 0, T > 1, such that every homology class z ∈ H1 (X ; Z) associated to an arbitrary (δ, T )-cycle (x, t) with x ∈ Cξ satis?es ξ, z ≤ ?1. Then there exists a Lyapunov one-form ω for (Φ, Rξ ) representing the cohomology class ξ . Moreover, the subset Cξ is closed. Conversely, if for the given cohomology class ξ there exists a Lyapunov oneform for the pair (Φ, Rξ ) in the class ξ and if the set Cξ is closed then (A) and (B) hold true. Corollary 1 Let Φ : X ×R → X be a continuous ?ow on a compact, locally path ˇ ˇ 1 (X ; R) satisfying connected metric space. Any Cech cohomology class ξ ∈ H ξ |R = 0 (where R = R(Φ) denotes the chain recurrent set of the ?ow), contains a Lyapunov one-form ω for (Φ, R). Corollary 1 follows directly from Theorem 2 since under the assumption ξ |R = 0 the set Rξ coincides with R (compare De?nition 5) and so the set Cξ is empty. Corollary 1 also admits a simple, independent proof based on Conley’s Theorem 1. Corollary 2 Suppose Φ : X × R → X is a ?ow on a compact locally path connected metric space, whose chain recurrent set consists of ?nitely many rest points and periodic orbits. Then a Lyapunov one–form for (Φ, Y ), with suitable ˇ 1 (X ; R) if and only if the Y ? X , exists in a nontrivial cohomology class ξ ∈ H homology classes of the periodic orbits are contained in the half space Hξ := {z ∈ H1 (X ; Z) | ξ, z ≤ 0}. If the above condition holds, then the set Y coincides with the union of the rest points and of those periodic orbits, for which the corresponding homology classes z ∈ H1 (X ; Z) satisfy ξ, z = 0. This Corollary is a direct consequence of our main Theorem 2. The class of ?ows meeting its assumptions includes the Morse-Smale ?ows on closed manifolds. Another interesting special case arises when Rξ = ?. From Theorem 2 we also deduce the following result: Corollary 3 Let Φ : X ×R → X be a continuous ?ow on a compact, locally path ˇ 1 (X ; R) be a nonzero Cech ˇ connected, metric space X and let ξ ∈ H cohomology class. The following two conditions are equivalent: (i) Rξ = ? and the ?ow satis?es condition (B ) of Theorem 2; (ii) there exists a Lyapunov one–form for (Φ, ?) representing the class ξ .

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ˇ 1 (X ; Z), then either of the above conditions is If the class ξ is integral, i.e. ξ ∈ H equivalent to the existence of a continuous locally trivial ?bration p : X → S 1 ? C with the following properties. The function t → arg(p(x · t)) is di?erentiable, the derivative d arg(p(x · t)) < 0 dt ˇ 1 (X ; Z) is negative for all x ∈ X , t ∈ R, and the cohomology class p? (?) ∈ H 1 1 ˇ (S ; Z) is the fundamental class of the circle coincides with ξ , where ? ∈ H S 1 oriented counterclockwise. In particular, for any angle θ ∈ S 1 the set K = p?1 (θ) is a cross section of the ?ow Φ. Recall that a closed subset K ? X is a cross section of the ?ow Φ if the ?ow map K × R → X is a surjective local homeomorphism (see S. Schwartzman [11]). A cross section K is transversal to the ?ow and the orbit of every point in X intersects K in forward and backward time. We give a proof of Corollary 3 in section 5. Under slightly stronger assumptions, S. Schwartzman [12] proved, among other results, the equivalence of the property of having a cross section and the existence of a ?bration p : X → S 1 as stated in the second part of Corollary 3. In subsequent work we will compare in detail our results with Schwartzman’s beautiful paper [11]. The second part of Corollary 3 may be viewed as a generalization of Fried’s results on the existence of cross sections to ?ows on manifolds [9, Theorem D]. The assumption of D. Fried to ensure the existence of cross sections is formulated in terms of the notion of homological directions of a ?ow, which we now recall. A sequence (xn , tn ) ∈ X × (1, ∞) is a closing sequence based at x ∈ X if the sequences xn and xn · tn tend to x. We will assume that X is a compact polyhedron. Then any closing sequence determines uniquely a sequence of homology classes zn ∈ H1 (X ; Z). Here zn denotes the homology class of a loop, which starts at xn , follows the ?ow until xn · tn , and then returns to xn along a ”short” path. Let DX be the factorspace DX = H1 (X ; R)/R+ , where this space is topologized as the disjoint union of the unit sphere with the origin. Any closing sequence as above determines a sequence of ”homology directions” z ?n ∈ DX , the equivalence classes of zn in DX . The set of homology directions DΦ ? DX of the ?ow Φ is de?ned as the set of all accumulation points of all sequences z ?n corresponding to closing sequences in X . As noted by D. Fried [9] it is enough to consider closing sequences (xn , tn ) with tn → ∞. Proposition 1 Let X be a ?nite polyhedron and let ξ ∈ H 1 (X ; Z) be an integral cohomology class. Let Φ : X × R → X be a continuous ?ow such that the chain recurrent set Rξ is isolated in R. Then condition (B) of Theorem 2 is equivalent to the Fried’s condition that any homology direction z ? = lim z ?n ∈ DX of any closing sequence (xn , tn ) ∈ X × (1, ∞) with xn ∈ Cξ satis?es ξ, z ? < 0. See §6 for a proof. A comparison of the results of D. Fried [9] with the results of this paper shows that our setting is more general in two respects: we allow spaces X of a 5

ˇ more general nature and arbitrary real Cech cohomology classes ξ . In [9] X is required to be a compact manifold, possibly with boundary, and the class ξ has to be integral. The equivalence of our condition (B) with the condition of D. Fried (Proposition 1) holds only under these additional assumptions. In the papers [5], [6] two di?erent generalizations cat(X, ξ ) and Cat(X, ξ ) of the classical notion of Lusternik - Schnirelman category cat(X ) were introduced; here X is a ?nite polyhedron and ξ ∈ H 1 (X ; R) a cohomology class. Using these new concepts, an extension of the Lusternik - Schnirelman theory for ?ows was constructed, see [5], [6]. The main results of [5], [6] allow to estimate the number of ?xed points of a ?ow under the assumption that (1) the ?xed points are isolated in the chain recurrent set, and (2) the ?ow admits a Lyapunov closed 1-form lying in the class ξ . The results of the present paper ?t nicely in this program and explain the nature of the assumption (2). Note that (1) is similar in spirit (although formally not equivalent) to the property that the set Rξ is isolated in the chain recurrent set R of the ?ow.

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Closed one-forms on topological spaces

In this section we recall the notion of a continuous closed one-form on a topological space, which has been introduced in [5, 6]. De?nition 2 Let X be a topological space. A continuous closed one-form on X is de?ned by a an open cover U = {U } of X and by a collection {?U }U ∈U of continuous functions ?U : U → R with the following property: for any two subsets U, V ∈ U the di?erence ?U |U ∩V ? ?V |U ∩V : U ∩ V → R (2.1)

is a locally constant function (i.e. constant on each connected component of U ∩ V ). Two such collections {?U }U ∈U and {ψV }V ∈V are called equivalent if their union {?U , ψV }U ∈U ,V ∈V satis?es condition (2.1). The equivalence classes are called continuous closed one–forms on X . Any continuous function f : X → R (viewed as a family consisting of a single element) determines a continuous closed one–form, which we denote by df (the di?erential of f ). A continuous closed one–form ω vanishes, ω = 0, if it is represented by a collection of locally constant functions. The sum of two continuous closed one–forms determined by collections {?U }U ∈U and {ψV }V ∈V is the continuous closed one–form corresponding to the collection {?U |U ∩V + ψV |U ∩V }U ∈U ,V ∈V . Similarly, one can multiply continuous closed one–forms by real numbers λ ∈ R by multiplying the corresponding representatives with λ. With these operations, the set of continuous closed one–forms on X is a real vector space. Continuous closed one–forms behave naturally with respect to continuous maps: If h : Y → X is continuous and {?U }U ∈U determines a continuous 6

closed one–form ω on X , then the collection of continuous functions φU ? h : h?1 (U ) → R determines a continuous closed one–form on Y which will be denoted by h? ω . As a special case of this construction, we will often use the operation of restriction of a closed one-form ω to a given subset A ? X ; in this case h is the inclusion map A → X and the form h? ω is simply denoted as ω |A . A continuous closed one–form ω on X can be integrated along continuous paths in X . Namely, let ω be given by a collection {?U }U ∈U , and γ : [0, 1] → X be a continuous path. We may ?nd a ?nite subdivision 0 = t0 < t1 < · · · < tN = 1 of the interval [0, 1] such that for each 1 ≤ i ≤ N the image γ ([ti?1 , ti ]) is contained in a single open set Ui ∈ U . Then we de?ne the line integral

N

ω :=

γ i=1

?Ui (γ (ti )) ? ?Ui (γ (ti?1 )).

(2.2)

The standard arguments show that the integral (2.2) is independent of all choices and in fact depends only on the homotopy class of γ relative to its endpoints. Consider the following exact sequence of sheaves over X 0 → RX → CX → BX → 0, (2.3)

where RX is the sheaf of locally constant functions, CX is the sheaf of real-valued continuous functions, and BX is the sheaf of germs of continuous functions modulo locally constant ones. More precisely, BX is the sheaf corresponding to the presheaf U → CX (U )/RX (U ). By the de?nitions above, the global sections of the sheaf BX are in one-to-one correspondence with continuous closed 1-forms on X . Hence the space of all closed 1-forms on X is H 0 (X ; BX ). The exact sequence of sheaves (2.3) generates the cohomological exact sequence 0 → H 0 (X ; RX ) → H 0 (X ; CX ) → H 0 (X ; BX ) → H 1 (X ; RX ) → 0

d []

(2.4)

In this exact sequence, H 0 (X ; CX ) = C (X ) is the space of all continuous functions f : X → R and the map d assigns to any continuous function f its ˇ di?erential df ∈ H 0 (X ; BX ). The group H 1 (X ; RX ) is the Cech cohomology 1 ˇ H (X ; R) (see [14], chapter 6); the map [ ] assigns to any closed one-form ω its ˇ ˇ 1 (X ; R). This proves: Cech cohomology class [ω ] ∈ H Lemma 1 A continuous closed 1-form ω ∈ H 0 (X ; BX ) equals df for some ˇ continuous function f : X → R if and only if its Cech cohomology class [ω ] ∈ 1 ˇ ˇ ˇ 1 (X ; R) can be H (X ; R) vanishes, [ω ] = 0. Any Cech cohomology class ξ ∈ H realized by a continuous closed 1-form on X . ˇ 1 (X ; R) → H 1 (X ; R) Note also that there is a natural homomorphism H ˇ from Cech cohomology to singular cohomology. Using Lemma 1 and the wellknown identi?cation H 1 (X ; R) ? Hom(H1 (X ); R), it can be described as a pairing ˇ 1 (X ; R) × H1 (X ; Z) → R, , :H 7 where [ω ], [γ ] =

γ

ω.

(2.5)

In other words, choosing a representative closed 1-form ω for a cohomology ˇ 1 (X ; R) and a closed loop γ in X representing a homology class class ξ ∈ H z ∈ H1 (X ; Z), the number ξ, z ∈ R equals the line integral γ ω , which is independent of the choices. Here is a generalization of the well-known Tietze extension theorem: Proposition 2 Let X be a metric space and A ? X a closed subset. Let ω ˇ 1 (A; R) denote the Cech ˇ be a continuous closed one–form on A, and let ξ ∈ H ′ 1 ˇ cohomology class of ω . Then for any cohomology class ξ ∈ H (X ; R), satisfying ξ ′ |A = ξ , there exists a continuous closed one–form ω ′ on X representing the cohomology class ξ ′ , such that ω ′ |A = ω . Proof. Choose an arbitrary continuous closed one–form ?′ representing the class ξ ′ . Then ?′ |A is cohomologous to ω , i.e. ?′ |A ? ω = df , where f : A → R is a continuous function. By the Tietze Extension Theorem for functions, we ?nd a continuous function f ′ : X → R extending f . Then ω ′ = ?′ ? df ′ is a closed one–form in the class ξ ′ satisfying ω ′ |A = ω . The statement of Proposition 2 can be expressed as follows: a continuous closed 1-form ω on a closed subset A ? X can be extended to a continuous ˇ 1 (A; R) can be closed 1-form on X if and only if the cohomology class [ω ] ∈ H 1 ˇ extended to a cohomology class lying in H (X ; R).

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The chain recurrent set Rξ

The goal of this section is to introduce a new chain recurrent set Rξ = Rξ (Φ) ? ˇ X , which is associated with a ?ow Φ on X together with a Cech cohomology 1 ˇ class ξ ∈ H (X ; R). The set Rξ appears in the statement of our main Theorem 2. Throughout this section we will assume that X is a locally path connected compact metric space. Recall that a space X is locally path connected if for every open set U ? X and for every point x ∈ U there exists an open set V ? U with x ∈ V such that any two points in V can be connected by a path in U . Equivalently, X is locally path connected i? the connected components of open subsets are open (see [14], page 65).

3.1

De?nition of Rξ

Recall the de?nition of the chain-recurrent set R = R(Φ) of the ?ow Φ. Given any δ > 0, T > 0, a (δ, T )-chain from x ∈ X to y ∈ X is a ?nite sequence x0 = x, x1 , . . . , xN = y of points in X and numbers t1 , . . . , tN ∈ R satisfying ti ≥ T and d(xi?1 · ti , xi ) < δ for all 1 ≤ i ≤ N . Note that a (δ, T )-cycle (see §1) is a (δ, T )-chain of a special kind (with n = 1).

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Figure 2: (δ, T )-chain The chain recurrent set R = R(Φ) of the ?ow Φ is de?ned as the set of all points x ∈ X such that for any δ > 0 and T ≥ 1 there exists a (δ, T )chain starting and ending at x. It is immediate from this de?nition that the chain recurrent set is closed and invariant under the ?ow; R contains all ?xed points and periodic orbits. The chain recurrent set R contains the set of all nonwandering points and, in particular, the positive and negative limit sets of any orbit [1, §II.6]. The set R = R(Φ) is a disjoint union of its chain transitive components2 . Lemma 2 Given a locally path connected compact metric space X and a number ε > 0, there exists δ = δ (ε) > 0 such that any two points x, y ∈ X with d(x, y ) < δ can be connected by a continuous path γ : [0, 1] → X contained in some open ε-ball. Proof. By the de?nition of local path-connectedness, each point x ∈ X has a neighborhood Vx contained in the ε-ball Bx around x such that any two points in Vx can be connected by a path in Bx . Choose a ?nite subcover of the covering {Vx }x∈X of X and choose δ (ε) as the Lebesgue number of this ?nite cover. De?nition 3 A pair (ε, δ ) of real numbers ε = ε(ξ ) > 0 and δ = δ (ξ ) > 0 is ˇ 1 (X ; R) if (1) ξ |B = 0 for called a scale of a nonzero cohomology class ξ ∈ H any ball B ? X of radius 2ε and (2) any two points x, y ∈ X with d(x, y ) < δ can be connected by a path in X contained in a ball of radius ε. Such a scale always exists. In fact, since we may realize the class ξ by a continuous closed 1-form ω = {φU }U ∈U with a ?nite open cover U and then take for ε half of the Lesbegue number of U . Using Lemma 2 we may then ?nd δ = δ (ξ ) > 0 satisfying condition (2) in De?nition 3. ˇ We want to evaluate Cech cohomology classes on broken chains of trajectories of the ?ow which start and end at the same point. This can be done as follows. ˇ 1 (X ; R) (see De?nition 3). Suppose Let ε = ε(ξ ), δ = δ (ξ ) be a scale for ξ ∈ H we are given a closed (δ, T )-chain, i.e. a (δ, T )-chain from a point x to itself. We have a sequence of points x0 = x, x1 , . . . , xN ?1 , xN = x of X and a sequence of

2 Recall that x, y ∈ R belong to the same chain transitive component if for any δ > 0 and T > 1 there exist (δ, T )-chains from x to y and from y to x.

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numbers t1 , . . . , tN ∈ R with ti ≥ T , such that d(xi?1 · ti , xi ) < δ for any 1 ≤ i ≤ N . We want to associate with such a chain a homology class z ∈ H1 (X ; Z). Choose continuous paths σi : [0, 1] → X , where 1 ≤ i ≤ N , connecting xi?1 · ti with xi and lying in a ball Bi of radius ε. We obtain a singular cycle which is a combination of the parts of the trajectories from xi?1 to xi?1 · ti and the paths σi . De?nition 4 The homology class z ∈ H1 (X ; Z) of this singular cycle is said to be associated with the given closed (δ, T )-chain. Note that the obtained class z may depend on the choice of paths σi (if the space X is wild, i.e. not locally contractible). However the value

N xi?1 ·ti N

ξ, z ∈ R,

where

ξ, z =

i=1 xi?1

ω

+

i=1 σi

ω,

(3.1)

is independent of the paths σi . Indeed, if we use two di?erent sets of curves ′ σi and σi then the di?erence of the corresponding expressions in (3.1) will be N ′ ′ the integral over the sum of singular cycles i=1 (σi ? σi ), each cycle σi ? σi being contained in a ball of radius 2ε. Since we know that the restriction of the cohomology class ξ on any such ball vanishes, we see that the right side of (3.1) is independent of the choice of the curves σ1 , . . . , σN . The homology class z ∈ H1 (X ; Z) associated with a closed (δ, T )-chain is uniquely de?ned if X is homologically locally connected in dimension 1 (see [14], chapter 6, page 340 for the de?nition) and δ > 0 is su?ciently small. Now we are ready to de?ne the subset Rξ of the chain reccurent set R: De?nition 5 Let ε = ε(ξ ) and δ = δ (ξ ) be a scale of a cohomology class ξ ∈ ˇ 1 (X ; R). Then Rξ = Rξ (Φ) denotes the set of all points x ∈ X with the H following property: for any 0 < δ ′ < δ and T > 1 there exists a (δ ′ , T )-chain from x to x such that ξ, z = 0 for any homology class z ∈ H1 (X ; Z) associated with this chain. Roughly, the set Rξ can be characterized as the part of the chain recurrent set of the ?ow in which the cohomology class ξ does not detect the motion. It is easy to see that Rξ is closed and invariant with respect to the ?ow. Note also that Rξ = Rξ′ whenever ξ ′ = λξ , with λ ∈ R, λ = 0. Thus, Rξ ˇ 1 (X ; R). depends only on the line through ξ in the real vector space H Any ?xed point of the ?ow belongs to Rξ . The points of a periodic orbit belong to Rξ if the homology class z ∈ H1 (X ; Z) of this orbit satis?es ξ, z = 0. Example. It may happen that the points of a periodic orbit belong to Rξ although ξ, z = 0 for the homology class z of the orbit. This possibility is illustrated by the following example. Consider the ?ow on the planar ring Y ? C shown on Figure 3. In the polar coordinates (r, φ) the ring Y is given by the inequalities 1 ≤ r ≤ 3 and the ?ow is given by the di?erential equations r ˙ = (r ? 1)2 (r ? 3)2 (r ? 5)2 , 10 ˙ = sin r · π . φ 2

Figure 3: The ?ow on the planar ring Y . Let Ck , where k = 1, 2, 3, denote the circle r = 2k ? 1. The circles C1 , C2 , C3 are invariant under the ?ow. The motion along the circles C1 and C3 has constant angular velocity 1. Identifying any point (r, φ) ∈ C1 with (5r, φ) ∈ C3 we obtain a torus X = Y / ? and a ?ow Φ : X × R → X . The images of the circles ′ ′ ′ C1 , C2 , C3 ? Y represent two circles C1 = C3 and C2 on the torus X . 1 Let ξ ∈ H (X ; R) be a nonzero cohomology class which is the pullback of a cohomology class of Y . One veri?es that in this example the set Rξ (Φ) coincides with the whole torus X . In particular, Rξ (Φ) contains the periodic ′ ′ ′ ′ orbits C1 = C3 and C2 although clearly ξ, [Ck ] = 0.

3.2

Rξ and dynamics in the free Abelian cover

Now we will give a di?erent characterization of Rξ using the dynamics in the covering space associated with the class ξ . We will use chapter 2 in [14] as a reference for notions related to the theory of covering spaces. Recall our standing assumption that X is a locally path connected compact metric space. For simplicity of exposition we will additionally assume that X is connected. ˇ ˇ 1 (X ; R) determines a homomorphism Any Cech cohomology class ξ ∈ H hξ : π1 (X, x0 ) → R, hξ ([α]) = ξ, [α] =

α

ω ∈ R,

(3.2)

where α : [0, 1] → X is a continuous loop α(0) = α(1) = x0 , [α] ∈ π1 (X, x0 ) denotes its homotopy class, and ω is a continuous closed one-form in the class ξ . The map hξ is called homomorphism of periods. The kernel of hξ is a normal subgroup H = Ker(hξ ) ? π1 (X, x0 ). We ? ξ → X , corresponding to want to construct a covering projection map pξ : X ? H , i.e. (pξ )# π1 (Xξ , x ?0 ) = H . The uniqueness of such a covering projection map follows from [14], chapter 2, Corollary 3. To show the existence we may use [14], chapter 2, Theorem 13; according to this Theorem (see also Lemma 11

11 in [14], chapter 2) we have to show that for some open cover U of X the subgroup π1 (U , x0 ) ? π1 (X, x0 ) is contained in H . Here π1 (U , x0 ) ? π1 (X, x0 ) denotes the subgroup generated by homotopy classes of the loops of the form α = (γ ? γ ′ ) ? γ ?1 where γ ′ is a closed loop lying in some element of U and γ is a path from x0 to γ ′ (0). To show that this condition really holds in our situation, let us realize ξ by a closed one-form ω = {fU }U ∈U , U being an open cover of X . We claim that π1 (U , x0 ) ? H for this cover U . Indeed, for any loop of the form α = (γ ? γ ′ ) ? γ ?1 where γ ′ lies in some U ∈ U , ξ, [α] =

α

ω=

γ′

ω = 0,

ˇ ˇ 1 (X ; R) since γ ′ lies in U and ξ |U = 0. Thus: Any Cech cohomology class ξ ∈ H ? ξ → X with connected total uniquely determines a covering projection map p : X ? ξ , such that (pξ )# π1 (X ?ξ , x space X ?0 ) = Ker(hξ ). Lemma 3 Let X be a connected and locally path connected compact metric ˇ 1 (X ; R) be a Cech ˇ space and let ξ ∈ H cohomology class. The group of covering ? ξ → X is a ?nitely generated free Abelian transformations of the covering map X group. ? ξ → X can be identi?ed Proof. The group of covering transformations of X with π1 (X, x0 )/H , which is isomorphic to the image of the homomorphism of periods hξ (π1 (X, x0 )) ? R. It is a subgroup of R and hence it is Abelian and has no torsion. Therefore it is enough to show that the image of the homomorphism of periods hξ (π1 (X, x0 )) ? R is ?nitely generated. Let ω = {fU }U ∈U be a continuous closed 1-form with respect to an open cover U representing ξ . Find an open cover V of X and a function κ : V → U ? ? U . We may such that for any V ∈ V , the set U = κ(V ) ∈ U satis?es V realize ω with respect to the open cover V as ω = {gV }V ∈V , where gV = fU |V for U = κ(V ). The path connected components of open subsets of X are open (since X is locally path connected) and hence the family of path connected components of the sets V ∈ V form an open cover of X . Using compactness, we may pass to a ?nite subcover; therefore, without loss of generality we may assume that V is ?nite and the sets V ∈ V are path connected. For any V1 , V2 ∈ V the function gV1 ? gV2 : V1 ∩ V2 → R is locally constant. We claim that the set SV1 V2 ? R of real numbers gV1 (x) ? gV2 (x) ∈ R, where x varies in V1 ∩ V2 , is ?nite. Assume the contrary, i.e. there exists an in?nite sequence xn ∈ V1 ∩ V2 , where n = 1, 2, . . . , such that gV1 (xn ) ? gV2 (xn ) = gV1 (xm ) ? gV2 (xm ) for n = m. (3.3)

By compactness we may assume that xn converges to a point x∞ ∈ X . Denote U1 = κ(V1 ), U2 = κ(V2 ), where U1 , U2 ∈ U . Then x∞ belongs to U1 ∩ U2 and thus xn ∈ U1 ∩ U2 for all large n. Let W ? U1 ∩ U2 denote the path connected component of x∞ in U1 ∩ U2 . Since W is open and contains x∞ , it follows that xn belongs to W for all large enough n. The function fU1 (x) ? fU2 (x), where 12

x ∈ U1 ∩ U2 , is continuous and locally constant hence it is constant for x ∈ W . We obtain that fU1 (xn ) ? fU2 (xn ) = fU1 (xm ) ? fU2 (xm ) for all large enough n and m. But this contradicts (3.3) since fUi |Vi = gVi and therefore fUi (xn ) = gVi (xn ) for i = 1, 2 and all n. The union S= SV1 V2

V1 ,V2 ∈V

of all subsets SV1 V2 is a ?nite subset of the real line. We will show now that the subgroup of R generated by S contains the group of periods hξ (π1 (X, x0 )). Let γ : [0, 1] → X be an arbitrary loop, γ (0) = γ (1) = x0 . We may ?nd division points t0 = 0 < t1 < t2 < · · · < tN = 1 and open sets V1 , . . . VN ∈ V such that γ ([ti?1 , ti ]) ? Vi for i = 1, 2, . . . , N . Then γ (ti ) ∈ Vi ∩ Vi+1 for i = 1, 2, . . . , N , where we understand that VN +1 = V1 . According to the de?nition of the line integral (see (2.2)) we have hξ ([γ ]) = ξ, [γ ] =

γ N

ω=

=

i=1

[gVi (γ (ti )) ? gVi (γ (ti?1 ))] =

N

=

i=1

gVi (γ (ti )) ? gVi+1 (γ (ti )) ,

which shows that any period hξ ([γ ]) ∈ R lies in the subgroup generated by the ?nite set S ? R. This implies that the group of periods is ?nitely generated and completes the proof of the Lemma. 2

Lemma 4 Assume that X is connected and locally path connected. Let ω be ?ξ → X a continuous closed 1-form on X . Consider the covering map pξ : X ˇ ˇ 1 (X ; R) of ω . Then determined by the Cech cohomology class ξ = [ω ] ∈ H ? p? ξ (ω ) = dF, where F : Xξ → R is a continuous function. ?ξ , ω ? = 0 vanishes for any closed loop γ in X ? where ω ? denotes p? ξ (ω ), by the construction of the covering Xξ . De?ne Proof. Note that the integral

γ x ?

F (? x) =

x ?0

ω ?,

?ξ , x ?∈X

? ξ is a base point, and the integration is taken along an arbitrary where x ?0 ∈ X ? path in Xξ connecting x ?0 with x ?. It is easy to see that F (? x) is independent of the choice of the path, F is continuous, and dF = p? ξ (ω ). 13

2 ? ξ is now used to characterize the chain recurrent set Rξ as The covering X follows. Proposition 3 Let X be a connected and locally path connected compact metric space. Given a continuous ?ow Φ : X × R → X and a cohomology class ˇ 1 (X ; R), consider the free Abelian covering pξ : X ? ξ → X associated with ξ∈H ? ? ? ? ξ . Fix ξ (see above) and the canonical lift Φ : Xξ × R → Xξ of the ?ow Φ to X ? ? ξ , which is invariant under the group of covering translations a metric d on X and such that the projection pξ is a local isometry. Then the chain recurrent set ? , the image of the chain recurrent set Rξ = Rξ (Φ) ? X coincides with pξ (R(Φ)) ? ? R(Φ) ? Xξ of the lifted ?ow under the projection. ??X ? ξ of radius ε0 (with respect Proof. Let ε0 > 0 be such that for any ball B ? to the metric d) the following holds: ?∩B ? = ? for any element g = 1 of the group of covering translations (1) g B ? of Xξ , and ? is an isometry. (2) the projection pξ restricted to B We may satisfy (1) since the group of covering transformations of the cov? ξ acts properly discontinuously (see [14], page 87). Note that ξ |B = 0, ering X ? ). where B = pξ (B Let δ0 = δ (ε0 ) > 0 be the number given by Lemma 2. Then the pair (ε0 , δ0 ) is a scale of ξ in the sense of De?nition 3. ? ξ belongs to the chain recurrent set R(Φ). ? Suppose that a point x ? ∈ X Then for any δ > 0 and T > 0 there exists a (δ, T )-chain of the form ?(? x ?0 = x ?, x ?1 , . . . , x ? N ?1 , x ?N = x ?, t1 , . . . , tN ∈ R, such that d xi?1 · ti , x ?i ) < δ and ti ≥ T for all i = 1, 2, . . . , N . We will assume below that δ < δ0 . Projecting downstairs, we ?nd a sequence x0 = x = pξ (? x), x1 , . . . , xN = pξ (? x) = x ∈ X, with xi = pξ (? xi ) satisfying d(xi?1 · ti , xi ) < δ for i = 1, 2, . . . , N . This sequence forms a (δ, T )-chain in X which starts and ends at x. For any homology class z ∈ H1 (X ; Z) associated with this chain one has ξ, z = 0 since we can ?nd a ? ξ . This shows that loop representing this class admitting a lift to the covering X ? is contained in Rξ . pξ (R(Φ)) ? is such that the point To prove the inverse inclusion, assume that x ? ∈ X x = pξ (? x) ∈ X belongs to Rξ . Hence for any δ > 0 and T > 1 we can ?nd a (δ, T )-chain x0 = x, x1 , . . . , xN = x, ti ∈ R, such that d(xi?1 · ti , xi ) < δ , ti ≥ T , and for any associated homology class z ∈ H1 (X ; Z) one has ξ, z = 0. We will assume that δ < δ0 , where δ0 is given as above. Choose continuous curves σi : [0, 1] → X such that σi (0) = xi?1 · ti and σi (1) = xi for i = 1, 2, . . . , N and the image σi ([0, 1]) is contained in a ball of radius ?0 . The concatenation of the parts of trajectories from xi?1 to xi?1 · ti and the paths σi , where i = 1, 2, . . . , N , 14

forms a closed loop γ , which starts and ends at x. This loop lifts to a closed loop ? ξ which starts and ends at x in the cover X ? since the homology class z = [γ ] of the loop satis?es ξ, z = 0. The lift γ ? of γ is a concatenation of parts of trajectories ? and the lifts σ of the lifted ?ow Φ ?i of the paths σi , where i = 1, . . . , N . We obtain ? ξ , where i = 0, 1, . . . , N , such that pξ (? points x ?i ∈ X xi ) = xi and x ?0 = x ?=x ?N . Besides, we have σ ?i (0) = x ?i?1 · ti , σ ?i (1) = x ?i for i = 1, 2, . . . , N . Since each σi lies in a ball of radius ?0 in X it follows from our assumption (1) above that each ??X ? ξ ; from assumption (2) we ?nd that d ?(? path σ ?i lies in a ball B xi?1 · ti , x ?i ) < δ ? ξ starting and ending for all i = 1, . . . , N . Thus we have found a (δ, T )-chain in X ? which is equivalent to Rξ ? pξ (R(Φ)). ? at x ?. This proves that pξ ?1 (Rξ ) ? R(Φ), 2

4

Proof of Theorem 2

In this section we will prove our main Theorem 2. The proof consists of two parts: the necessary conditions (easy) and the su?cient conditions (more di?cult).

4.1

Necessary conditions

If ω is a Lyapunov one-form for (X, Rξ ) then by De?nition 1, ω |U = df where f is a continuous function de?ned on an open neighborhood U ? Rξ . Hence ˇ 1 (X ; R) denotes the the restriction of ξ on Rξ vanishes, ξ |Rξ = 0, where ξ ∈ H cohomology class of ω . Thus condition (A) in Theorem 2 is necessary. The following Proposition implies that condition (B) of Theorem 2 is satis?ed for any ?ow admitting a Lyapunov one-form for (Φ, Rξ ) and having the property that the set Cξ = Cξ (Φ) is closed. Proposition 4 Let Φ : X × R → X be a continuous ?ow on a compact, locally path connected, metric space X . Let ω be a Lyapunov one–form for (Φ, Y ), see De?nition 1, where Y ? X is a closed ?ow-invariant subset. Let C ? X be a closed, ?ow-invariant subset such that Y ∩ C = ?. Then there exist numbers δ > 0 and T > 1, such that any homology class z ∈ H1 (X ; Z) associated with any (δ, T )-cycle (x, t) with x ∈ C , satis?es ξ, z ≤ ?1. ˇ 1 (X ; R) denotes the cohomology class of ω . Here ξ = [ω ] ∈ H Proof. As Y ∩ C = ? it follows that on C the function x → x ω < 0 is continuous and negative. Since C is compact, there exists a positive constant c > 0, such that

x·1 x·1

ω < ?c

x

for all x ∈ C. 15

(4.1)

ˇ 1 (X ; R), see De?nition 3. Let (ε, δ ) be a scale of the cohomology class ξ ∈ H Let η > 0 be such that for any continuous curve σ : [0, 1] → X , lying in a ball B ? X of radius ε one has | σ ω | < η . We de?ne T = [(1 + η )/c] + 2, (4.2)

where [a] denotes the integer part of a number a. Unlike δ , the number T not only depends on the class ξ but on the chosen representative ω . We will show that δ > 0 and T > 1 satisfy our requirements. Indeed, let (x, t) be a (δ, T )-cycle with x ∈ C , and let γ be a closed loop in X , obtained by ?rst following the trajectory x · τ , where τ ∈ [0, t], and then returning from the end point x · t to x along a short path σ lying in a ball of radius ε. Then we have

x·t

ξ, [γ ] =

γ

ω=

x

ω+

σ

ω

For the second integral we have σ ω < η by construction. Since t ≥ T and T is an integer, we can estimate the ?rst integral as

x·t T x·i x·t

ω=

x i=1 x·(i?1)

ω+

x·T

ω < ?T c,

x·t

where we have used T times the inequality (4.1) and the estimate x·T ω < 0. By our choice of T , we have T c > 1 + η (cf. (4.2)) and so we see that ξ, z < ?1 for any homology class z ∈ H1 (X ; Z) associated with any (δ, T )-cycle in C . 2

4.2

Constructing a Lyapunov 1-form: the ?rst step

In the remainder of this section we prove the existence claim of Theorem 2. The proof is split into several Lemmas. In a ?rst step we construct a Lyapunov oneform for the ?ow restricted to Cξ ; later on we will extend the obtained closed one-form to a Lyapunov one-form de?ned on the whole space X . We start with the following combinatorial Lemma: Lemma 5 Any nonempty word w of arbitrary (?nite) length in an alphabet consisting of L letters can be written as a product (concatenation) w = w1 w2 . . . wl of l ≤ L nonempty words, such that in each word wi , the ?rst and last letters coincide. Proof. We will use induction on L. For L = 1 our claim is trivial. We are left to prove the claim of the lemma assuming that it is true for words in any 16

alphabet consisting of less than L letters. Consider a word w in an alphabet with L letters. Let a be the ?rst letter of w. Finding the last appearance of a in w, we may write w = w1 w′ , where w1 starts and ends with a and w′ does not include a. We may now apply the induction hypothesis to w′ , which allows to write w′ = w2 w3 . . . wl , where l ≤ L and in each wi the ?rst and the last letters coincide. This clearly implies the lemma. 2 Let ω be an arbitrary, continuous closed one–form on X representing a coˇ 1 (X ; R). Our ?nal goal will be to modify ω so that at the homology class ξ ∈ H end we obtain a Lyapunov one–form for (Φ, Rξ ). Lemma 6 Under assumption (B) of Theorem 2 there exist ? > 0 and ν > 0 such that for any x ∈ Cξ and t ≥ 0 we have

x·t

ω ≤ ??t + ν,

x

(4.3)

where the integral is calculated along the trajectory of the ?ow. In particular,

x·t t→+∞

lim

ω = ?∞

x

and the convergence is uniform with respect to x ∈ Cξ . Proof. Let ε > 0 be such that for any ball B ? X of radius ε one has ξ |B = 0 (see De?nition 3) and for any continuous curve σ : [0, 1] → B holds | σ ω | < 1/2. Let δ > 0 be such that condition (B) of Theorem 2 holds for some T > 1 and, additionally, for any points x, y ∈ X with d(x, y ) < δ there is a continuous path σ connecting x and y and lying in a ball of radius ε. By (B),

x·t

ω < ?1/2,

x

(4.4)

whenever x ∈ Cξ , t ≥ T and d(x, x · t) < δ . Using the compactness of X , we ?nd a constant M > 0 such that for any point x ∈ X and any time 0 ≤ t ≤ T ,

x·t

ω < M.

x

(4.5)

Next we choose points y1 , y2 , . . . , yk in X , such that the open balls of radius δ/2 with centers at these points cover X . Hence, for any x ∈ X there exists an index i ∈ {1, . . . , k }, such that d(x, yi ) < δ/2. Given x ∈ Cξ and t ≥ 0, we consider the sequence of points xj = x · (jT ) ∈ Cξ , j = 0, 1, . . . , N, 17 where N = [t/T ].

As explained above, for any j = 0, . . . , N there exists an index 1 ≤ ij ≤ k , such that d(xj , yij ) < δ/2. Thus, any point x ∈ Cξ determines a sequence of indices i 0 , i1 , . . . , iN ∈ { 1 , 2 , . . . , k } , (4.7) (4.6)

which, to a certain extent, encode the trajectory starting at x. If it happens that in the sequence (4.7) one has for some r < s, ir = is , then the part of the trajectory between xr = x · (rT ) and xs = x · (sT ) is a (δ, T )-cycle (in view of (4.6)) and by (4.4)

xs

ω < ?1/2.

xr

(4.8)

Let 1 ≤ m ≤ k be an index which appears in the sequence (4.7) most often. Clearly, it must appear at least [(N + 1)/k ] times. Let α be the smallest number with iα = m and β the largest number with iβ = m, so that 0 ≤ α < β ≤ N . Then, using (4.8), we have

xβ

ω ≤ ?

xα

N +1?k . 2k

(4.9)

x

To complete the argument, we need to estimate the remaining integrals x α ω x·t (corresponding to the beginning of the trajectory) and xβ ω (corresponding to the end of the trajectory). View the sequence i0 , i1 , . . . , iα as a word w in the alphabet {1, 2, . . . , k } and apply Lemma 5. As a result we may split the sequence w = i0 , i1 , . . . , iα into l ≤ k subsequences w1 , w2 , . . . , wl , each beginning and ending with the same symbol. If wj = ir , ir+1 , . . . , is is one of the subsequences, where r ≤ s, x then ir = is and using (4.8) we ?nd xrs ω ≤ 0. In other words, the integral corresponding to each subsequence wj is nonpositive (in fact it is less than ?1/2 if the subsequence wj has more than one symbol). Now we want to estimate the contribution of the integrals corresponding to the word breaks in w = w1 w2 . . . wl . If wj ends with the symbol is and the x following subsequence wl+1 starts with is+1 then we have xss+1 ω ≤ M (see (4.5)). Thus, any word break contributes at most M to the integral. x The integral x α ω is the sum of the contributions corresponding to the words wj (which are all nonpositive) and contributions of the word breaks (each is at most M ). Since there are l ? 1 ≤ k ? 1 word breaks, we obtain

xα

ω ≤ (l ? 1)M ≤ (k ? 1)M.

x

(4.10)

x·t ω xN

Similarly, xβN ω ≤ (k ? 1)M. For the remaining integral we have which again follows from (4.5), since t ? N T < T . 18

x

< M,

Summing up, we ?nally obtain the estimate

x·t

ω < (k ? 1)M ?

x

N +1?k + (k ? 1)M + M. 2k

(4.11)

Hence (4.3) holds true with the constants ?= 1 2kT 1 and ν = (2k ? 1)M + . 2 2

Lemma 7 Conditions (A) and (B) of Theorem 2 imply that the set Cξ = R ? Rξ is closed. Proof. Since ξ |Rξ = 0, we conclude that for any continuous closed one-form ω in the class ξ the restriction ω |Rξ is the di?erential of a function and hence there exists a constant C > 0 such that for any x ∈ Rξ and any t > 0,

x·t

ω < C.

x

(4.12)

Assume that the set Cξ is not closed, i.e. there exists a sequence of points

xn ·t

xn ∈ Cξ converging to a point x0 ∈ Rξ . By Lemma 6,

xn

ω < ??t + ν . Taking

t = t0 = (ν + 2C )/?, we obtain

xn ·t0

ω < ?2C

xn

(4.13)

x0 ·t0

for any n = 1, 2, . . . . Passing to the limit with respect to n we ?nd

x0

ω ≤

?2C , contradicting the estimate (4.12). 2

Lemma 8 Let ω be a continuous closed one-form on X realizing a class ξ ∈ ˇ 1 (X ; R). Assume that conditions (A) and (B) of Theorem 2 hold. Let f : H Cξ → R be the function de?ned by

x·t

f (x) := sup

t≥0 x

ω.

(4.14)

Then: (i) f is well de?ned and continuous; 19

(ii) ω1 = ω |Cξ + df is a continuous closed one–form on Cξ representing the cohomology class ξ |Cξ ; (iii) for any x ∈ Cξ and for any t > 0,

x·t

ω1 ≤ 0 ,

x

(4.15)

i.e. ω1 is a Lyapunov one-form for the restricted ?ow Φ|Cξ in a weak sense; (iv) there exists a number T > 1, such that for any x ∈ Cξ and any t ≥ T ,

x·t

ω1 ≤ ?1 .

x

(4.16)

Proof. Let t0 > 0 be the time such that ??t0 + ν = 0 with ? > 0 and ν > 0 as in Lemma 6. Then the supremum in (4.14) is achieved for t ∈ [0, t0 ], and hence we may write

x·t

f (x) = max

0≤t≤t0

ω.

x

The continuity of f now follows from the uniform continuity of the integral with respect to (x, t) ∈ Cξ × [0, t0 ]. Claim (ii) follows from (i). To prove (iii), we ?nd

x·t x·t x·t

ω1 =

x x

(ω + df ) =

x x·t

ω + [f (x · t) ? f (x)]

x·(t+τ ) x·τ

=

x

ω + sup

τ ≥0 x·τ x·t

ω ? sup

τ ≥0 x

? sup

τ ≥0 x·τ x

ω

= sup

τ ≥t x

ω

ω ≤ 0.

We next prove (iv). Let M denote the maximal value of the continuous function f : Cξ → R and m its minimal value. We apply Lemma 6 and obtain

x·t x·t

ω1 =

x x

ω + [f (x · t) ? f (x)] ≤ ??t + ν + (M ? m).

Hence, choosing T such that ??T + ν = ?2 ? (M ? m), claim (iv) follows. 2

20

4.3

Second step: smoothing

In this subsection we describe a procedure for smoothing a continuous closed one-form along the ?ow, which will be used in the proof of Theorem 2 below. It is a modi?cation of a well-known method for continuous functions, see for example [11]. We use this construction to smooth the closed one-form ω1 which is constructed in the proof of Lemma 8. A function f : X → R is said to be di?erentiable along a continuous ?ow d f (x · t)|t=0 exists for any x ∈ X . More Φ : X × R → X if the derivative dt generally, a continuous closed one–form ω on X is said to be di?erentiable with respect to the ?ow Φ if the derivative ω ˙ (x) := d dt

x·t

ω |t=0

x

(4.17)

exists for any x ∈ X . In this case, ω ˙ : X → R is a function on X , which we call the derivative of ω with respect to the ?ow Φ. If ω is represented as ω = {?U }U ∈U with respect to an open cover U of X then for x ∈ U and t su?ciently small we x·t have x ω = ?U (x · t) ? ?U (x) and we see that a continuous closed one–form is di?erentiable with respect to the ?ow Φ if and only if the local de?ning functions ?U are. Lemma 9 Assume that conditions (A), (B) of Theorem 2 hold. Then there ˇ 1 (X ; R) with the exists a continuous closed one-form ω2 on X in class ξ ∈ H following properties: (i) ω2 is di?erentiable with respect to the ?ow Φ; (ii) the derivative ω ˙ 2 : X → R is a continuous function; (iii) for some σ > 0 one has ω ˙ 2 (x) ≤ ?σ for all x ∈ Cξ ; (iv) ω2 |U = 0 for some open neighborhood U ? X of Rξ . In particular, ω2 |Cξ is a Lyapunov one-form for the ?ow Φ|Cξ . Proof. Using assumptions (A), (B) and Lemma 7, we ?nd a closed neighborhood V ? X of Rξ , such that ξ |V = 0 and V ∩ Cξ = ?. Here we use ˇ the continuity property of the Cech cohomology theory, see [3], Chapter 10, Theorem 3.1. Let ω1 be the closed one-form on Cξ given by Lemma 8. Using the Tietze Extension Theorem (see Proposition 2), we may ?nd a closed one-form ?1 on ˇ 1 (X ; R). X , such that ?1 |Cξ = ω1 , ?1 |V = 0, and [?1 ] = ξ ∈ H ? ξ → X corresponding to the Cech ˇ Consider the covering map pξ : X coho? mology class ξ , see subsection 3.2. By Lemma 4 we have pξ (?1 ) = dF1 , where ? ξ → R is a continuous function. Let C ?ξ denote the preimage p?1 (Cξ ). F1 : X ξ ?ξ and t ≥ 0, F1 (x · t) ≤ F1 (x), by Lemma 8, (iii). Then for any point x ∈ C Moreover, statement (iv) of Lemma 8 implies that there exists T > 0, such that F1 (x · t) ? F1 (x) ≤ ?1, for 21 ?ξ , t ≥ T. x∈C (4.18)

Let ? : R → [0, ∞) be a C ∞ -smooth function with the following properties: (a) the support of ? is contained in the interval [?T ? 1, T + 1]; (b) ?|[?T,T ] = const = σ > 0; (c) ?(?t) = ?(t); (d) ?′ (t) ≥ 0 for t ≤ 0; (e)

R ?(t)dt

= 1.

? ξ → R by F2 (x) = F (x · t)?(t)dt. It is clear that Using ? we de?ne F2 : X R 1 F2 is continuous. Since F2 (x · s) = R F1 (x · t)?(t ? s)dt, we see that F2 is ? ξ . If x ∈ C ?ξ , we ?nd, using (4.18) di?erentiable with respect to the ?ow on X and the properties of ?, dF2 (x · s) |s=0 ds

T +1

= ?

?T ?1 ?T

F1 (x · t)?′ (t)dt (4.19)

= ≤ ?

[F1 (x · (?t)) ? F1 (x · t)] · ?′ (t)dt ?′ (t)dt = ?σ.

?T ?1 ?T ?T ?1

? ξ . Using Let G denote the group of covering transformations of the covering X the homomorphism of periods (3.2), one sees that the class ξ determines a ? ξ and any g ∈ G we have monomorphism α : G → R, such that for any x ∈ X F1 (gx) ? F1 (x) = α(g ). (4.20) Since (gx) · t = g (x · t), we ?nd F1 ((gx) · t) = F1 (g (x · t)) = F1 (x · t) + α(g ) and, multiplying by ?(t) and integrating gives F2 (gx) ? F2 (x) = α(g ) (4.21) ? ξ and g ∈ G. Formula (4.21) says that the action of the covering for any x ∈ X translations changes F2 by adding a constant, and therefore, F2 determines a continuous closed one-form on X . More precisely, dF2 = p? ξ (ω2 ) for some continuous closed one-form ω2 on X . Since F2 is di?erentiable with respect to ? ξ and the derivative d F2 (x · s) is continuous (see (4.19)), the the ?ow on X ds derivative ω ˙ 2 : X → R is a well-de?ned, continuous function. Clearly, as ?1 vanishes on V , the form ω2 vanishes on the open set U ? X of points x ∈ X with x · [?T ? 1, T + 1] ? V . Since Rξ ? U , this proves (iv). ?ξ → R Comparing (4.20) and (4.21) we ?nd that the function F1 ? F2 : X is invariant under the covering translations. Hence F1 ? F2 = f ? pξ , where f : X → R is a continuous function. Therefore ?1 ? ω2 = df , i.e. ω2 lies in the cohomology class ξ . We know that ω2 is di?erentiable with respect to the ?ow and ω ˙ 2 ≤ ?σ < 0 on Cξ . 2

22

4.4

Third step: extension

Now we complete the proof of the existence claim of Theorem 2. Let L : X → R be a Lyapunov function for (Φ, R). Such a function exists according to Theorem 1 of C. Conley. We apply the smoothing procedure from the previous subsection to L. Namely, let ρ : R → [0, ∞) be a C ∞ -smooth function, such that supp(ρ) = [?1, 1], R ρ(t)dt = 1, ρ(?t) = ρ(t) and ρ′ (t) > 0 for all t ∈ (?1, 0) and set L1 (x) =

R

L(x · t)ρ(t)dt.

˙ 1 (x) = We ?nd (precisely as in the previous subsection) that the derivative L d L1 (x · s)|s=0 exists and is given by ds ˙ 1 (x) = L

0 ?1

[L(x · (?t)) ? L(x · t)] ρ′ (t)dt.

(4.22)

˙ 1 : X → R is a continuous function and This identity implies that L ˙ 1 (x) < 0 L for any x ∈ X ? R.

Let ω2 be the closed one-form on X given by Lemma 9. We will set ω3 = ω2 + λ(dL1 ), (4.23)

where λ > 0 and dL1 is the di?erential of the function L1 (see section 2). In view of the construction of ω2 and L1 , for any λ, the form ω3 is a continuous closed 1-form on X representing the cohomology class ξ and satis?es condition (L2) of De?nition 1. We now show that for λ large enough ω3 satis?es condition (L1) and hence it is a Lyapunov one-form for (Φ, Rξ ). Indeed, ω3 is di?erentiable along the ?ow ˙ 1 . By Lemma 9, ω and has the derivative ω ˙3 = ω ˙ 2 + λL ˙ 2 < 0 on Cξ . Hence we may ?nd an open neighborhood W of Cξ , so that ω ˙ 2 < 0 on W . By claim (iv) of Lemma 9, ω ˙ 2 = 0 vanishes on some open neighborhood U of Rξ , whereas ˙ 1 < 0 on U ? Rξ . Hence we see that for any λ > 0 the inequality ω L ˙ 3 < 0 holds on W and on U ? Rξ . Finally we shall show that ω ˙ 3 < 0 on X ? Rξ for λ > 0 su?ciently large. The function x→? ω ˙ 2 (x) , ˙ 1 (x) L x ∈ X ? (U ∪ W ) (4.24)

˙ 1 < 0 on X ? R). Since X ? (U ∪ W ) is well-de?ned and continuous (recall that L is compact, the function (4.24) is bounded. Choose λ > 0 to be larger than the maximum of (4.24). Then ω ˙ 3 (x) < 0 holds for all x ∈ X ? Rξ , as desired. This completes the proof of Theorem 2. 23

2 The arguments above prove the following, slightly stronger statement: Corollary 4 Under the assumptions (A) and (B) of Theorem 2, there exists a continuous closed one-form ω on X lying in the cohomology class [ω ] = ξ ∈ ˇ 1 (X ; R) which satis?es condition (L2) and the following stronger version of H condition (L1): ω is di?erentiable with respect to the ?ow Φ (in the sense explained in section §4.3), the derivative ω ˙ : X → R is continuous and ω ˙ < 0 on X ? Rξ . 2

5

Proof of Corollary 3

By Theorem 2, (i) implies (ii). Conversely, assume that there exists a Lyapunov one–form ω for (Φ, ?) representing the class ξ . Proposition 4 shows that condition (B ) of Theorem 2 is satis?ed. We are left to prove that Rξ = ?. Consider ? ξ → X corresponding to the class ξ (see §3.2). It is enough to the covering pξ : X ? of the lifted ?ow Φ ? in X ? ξ is empty (see show that the chain recurrent set R(Φ) ? ? Proposition 3). By Lemma 4, pξ (ω ) = dF , where F : Xξ → R is a continuous ? ξ and t > 0. In parfunction. By assumption (ii), F (x · t) < F (x) for all x ∈ X ? ξ , is negative and ticular, the function φ(x) = F (x · 1) ? F (x), de?ned on x ∈ X invariant under the group of covering translations (see (4.20)); hence it equals ψ ? pξ , where ψ : X → R is a continuous function on X . By the compactness ? ξ . Choose a metric d of X there exists σ > 0 such that φ(x) < ?σ for all x ∈ X ? ξ , which is invariant under the group of covering translations. There exists on X ? ξ with d(x, y ) < δ one has |F (x) ? F (y )| < σ/2. δ > 0 such that for any x, y ∈ X ? ξ and the numbers Now, assume that the points x0 = x, x1 , . . . , xN = x ∈ X ? in X ?ξ , t1 , . . . , tN ∈ R represent a (δ, T )-chain with T > 1 of the lifted ?ow Φ i.e. ti ≥ T and d(xi?1 · ti , xi ) < δ for i = 1, . . . , N . Then for any i = 1, 2, . . . , N we have F (xi?1 · ti ) ? F (xi?1 ) < ?σ and F (xi ) ? F (xi?1 · ti ) < σ/2,

which imply that F (xi ) ? F (xi?1 ) < ?σ/2 and hence F (xN ) ? F (x0 ) < 0. The last inequality contradicts x0 = x = xN . This proves that there are no closed ?ξ . (δ, T )-chains in the covering X Thus we have shown that (i) and (ii) are equivalent. ˇ 1 (X ; Z). As Now assume that (ii) holds and the class ξ is integral, i.e. ξ ∈ H X is locally path connected and compact, it has ?nitely many path connected components. Thus without loss of generality we may assume that X is path connected. Let ω be a Lyapunov one-form for (Φ, ?) satisfying the properties (iii) and (iv) of Corollary 4. De?ne a map p : X → S 1 ? C by choosing a point x0 ∈ X and setting

x

p(x) = exp 2πi

x0

ω ,

24

where the line integral is taken along any path connecting x0 with x. Since ξ is integral, the value p(x) ∈ S 1 does not depend on the choice of the path. The function

x·t

t → arg(p(x · t)) = 2π

x0

ω

is di?erentiable and the derivative d arg(p(x · t)) < 0 dt is negative. The equality ξ = p? (?) is immediate from the de?nition of p. It remains to prove that p de?nes a locally trivial ?bration. Pick η ∈ R and let Kη = p?1 (exp(2πiη )) ? X . For any x ∈ Kη , let fx : R → R be a continuous function such that fx (0) = η and p(x · t) = exp(2πifx (t)) for all t ∈ R. The function fx is uniquely determined. It is di?erentiable and there exists ε > 0 such that for all x ∈ Kη and t ∈ R, d fx (t) < ?ε. dt

?1 Let gx = fx be the inverse function. De?ne G : Kη × R → X by G(x, t) = x · gx (t). Then G is continuous and the diagram

Kη × R e ?

?→

G

X ? p (5.1)

S1 commutes, where e(x, t) = exp(2πit). This proves that p is a locally trivial ?bration and that Kη is a cross section of the ?ow Φ. 2

6

Proof of Proposition 1

Suppose that the set Cξ ? R = R(Φ) is closed but condition (B) of Theorem 2 is violated. Then there exists a sequence of points xn ∈ Cξ and numbers tn > 0 such that the distances d(xn , xn · tn ) tend to 0 as tn → ∞, and ξ, zn > ?1, (6.1)

where zn ∈ H1 (X ; Z) denotes the homology class obtained by “closing” the trajectory xn · t for t ∈ [0, tn ]. Using the compactness of Cξ we may additionally assume that xn converges to a point x ∈ Cξ . Since we assume that the class ξ is integral, we may rewrite (6.1) in the form ξ, zn ≥ 0. Thus we obtain a closing sequence (xn , tn ) such that for any homology direction z ? ∈ DX associated with it, ξ, z ? ≥ 0, i.e. the condition of Fried [9] also fails to hold. 25

Conversely, we now show that the condition of Fried [9] holds assuming that condition (B) of Theorem 2 is satis?ed. Fix a norm || || on the vector space H1 (X ; R). As X is a polyhedron, there exists δ > 0 so that for any δ/2-ball B in X the inclusion B → X is null-homotopic. Further, there exists a constant C > 0 such that for any homology class z ∈ H1 (X ; Z) associated with a (δ, T )cycle (x, t) in X one has ||z || ≤ Ct. (6.2)

Let ω be a continuous closed 1-form in the class ξ . By Lemma 6 there exist ? > 0 and ν > 0 such that

x·t

ω ≤ ??t + ν

x

(6.3)

for all x ∈ Cξ and t > 0. Let η > 0 be such that γ ω < η for any curve lying in a ball of radius δ/2. The estimate (6.3) implies that ξ, z ≤ ??t + ν + η (6.4)

for any (δ, T ) cycle (x, t) with x ∈ Cξ , where z ∈ H1 (X ; Z) denotes the associated homology class. Since ξ, z ≥ ?c||z ||, where c > 0, we obtain that the homology class z of any (δ, T )-cycle (x, t) with x ∈ Cξ satis?es ||z || ≥ ν+η ? ·t? . c c (6.5)

Now, let (xn , tn ) be a closing sequence (as de?ned in §1), where xn ∈ Cξ , such that xn converges to a point x ∈ Cξ and tn → ∞. Let zn ∈ H1 (X ; Z) denote the homology class determined by closing (xn , tn ). Then (6.5) implies that ||z || ||zn || → ∞. By (6.2), ?t ≤ ? , which when substituted into (6.4) leads to C ? ν +η zn ≤ ? + . ξ, ||zn || C ||zn || Therefore, we obtain for the homology direction estimate ξ, zn ∈ DX of the class zn the ||zn ||

zn ? ≤ ? < 0, ||zn || 2C 2

if n is large. This shows that Fried’s condition [9] is satis?ed.

7

Examples

Example 1. Here we describe a class of examples of ?ows Φ : X × R → X , for which there exists a cohomology class ξ satisfying the conditions (A) and (B) of Theorem 2. 26

Let M be a closed smooth manifold with a smooth vector ?eld v . Let Ψ : M × R → M be the ?ow of v . Assume that the chain recurrent set R(Ψ) is a union of two disjoint closed sets R(Ψ) = R1 ∪ R2 , where R1 ∩ R2 = ?. Out of this data we will construct a ?ow Φ on X = M × S1 such that Rξ (Φ) = R1 × S 0 , Cξ = R2 × S 1 . Here ξ ∈ H 1 (X ; Z) denotes the cohomology class induced by the projection onto the circle X → S 1 and S 0 ? S 1 is a two-point set. Let θ ∈ [0, 2π ] denote the angle coordinate on the circle S 1 . We will need ? ? two vector ?elds w1 and w2 on S 1 , w1 = cos(θ) · ?θ and w2 = ?θ . The ?eld 0 1 w1 has two zeros {p1 , p2 } = S ? S corresponding to the angles θ = π/2 and θ = 3π/2. Let fi : M → [0, 1], where i = 1, 2, be two smooth functions having disjoint supports and satisfying f1 |R1 = 1, f2 |R2 = 1. Consider the ?ow Φ : X × R → X determined by the vector ?eld V = v + f1 w1 + f2 w2 . Any trajectory of V has the form (γ (t), θ(t)), where γ ˙ (t) = v (γ (t)), i.e. γ (t) is a trajectory of v . It follows that the chain recurrent set of V is contained in R(Ψ) × S 1 . Over R1 we have the vertical vector ?eld w1 along the circle which has two points S 0 ? S 1 as its chain recurrent set. Over R2 we have the vertical vector ?eld w2 which has all of S 1 as the chain recurrent set. We see that R1 × S 0 = Rξ (Φ), R2 × S 1 = Cξ . Hence ξ |Rξ = 0 (and Cξ is closed). Clearly condition (B) of Theorem 2 is satis?ed as well. Example 2. Let X = T 2 , thought of as R2 /Z2 with coordinates x and y on 2 R . Any cohomology class ξ ∈ H 1 (T 2 ; R) can be written as ξ = ?[dx] + ν [dy ], where dx and dy are the standard coordinate 1-forms. We consider the ?ow of the following vector ?eld V = f (x, y ) · a ? ? +b ?x ?y ,

where b = 0, a/b ∈ Q and f : T 2 → [0, 1] is a smooth function vanishing at a single point p ∈ T 2 . The chain recurrent set R is the whole torus, R = T 2 , while ? T 2, if ?a + νb = 0, ? Rξ = ? ?1 f (0) = {p}, otherwise.

Assuming in addition that ?a + νb = 0, the set Cξ = T 2 ? {p} is not closed. Nevertheless, a Lyapunov one-form in the class ξ = 0 exists if and only if ?a + νb < 0. In this case ω = ?dx + νdy is such a Lyapunov 1-form. This example shows that the existence of a Lyapunov one-form for (Φ, Rξ ) does not imply Cξ to be closed.

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Example 3. Consider the standard irrational ?ow on the torus X = T 2 , i.e. ? ? the ?ow of the vector ?eld V = a ?x + b ?y , where b = 0 and a/b ∈ / Q. Choose 1 a cohomology class ξ = ?[dx] + ν [dy ] ∈ H (X ; R) such that ?a + νb = 0. Then R = X and Rξ = ?, but condition (B ) of Theorem 2 is not satis?ed and so there is no Lyapunov one-form for (Φ, Rξ ) in the class ξ . This example shows that condition (B ) is not a consequence of the fact that Cξ is closed.

References

[1] C. Conley, Isolated invariant sets and the Morse index, CBMS regional conference series in mathematics, no. 38, AMS, 1976. [2] C. Conley, The gradient structure of a ?ow: I, Ergod. Th. & Dynam. Sys., 8(1988), 11–26. [3] S. Eilenberg, N. Steenrod, Foundations of Algebraic Topology, Princeton, New Jersey, 1952. [4] H. Fan and J. Jost, Novikov – Morse theory for Dynamical Systems, Calculus of Variations, 2002. [5] M. Farber, Zeroes of closed 1-forms, homoclinic orbits and Lusternik– Schnirelman theory, ”Topological Methods in Nonlinear Analysis”, 19(2002), 123 - 152. [6] M. Farber, Lusternik–Schnirelman Theory and Dynamics, in: “Lusternik - Schnirelmann Category and Related Topics”, O. Cornea et al. editors, Contemporary Mathematics, vol. 316(2002). [7] J. Franks, A Variation on the Poincar? e-Birkho? Theorem, Contemporary Mathematics, 81(1988), 111–117. [8] J. Franks, Homology and dynamical systems, CBMS regional conference series in mathematics, no. 49, AMS, 1982.. [9] D. Fried, The geometry of cross sections to ?ows, Topology, vol. 21 (1982), 353–371. [10] F. B. Fuller, On the surface of section and periodic trajectories, Amer. J. Math., 87 (1965), 473–480. [11] S. Schwartzman, Asymptotic cycles, Ann. of Math., vol. 66, 1957, 270–284. [12] S. Schwartzman, Global cross-sections of compact dynamical systems, Proc. Nat. Acad. Sci. USA 48(1962), 786 - 791. [13] M. Shub, Global Stability of Dynamical Systems, Springer Verlag, 1986.

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[14] E. Spanier, Algebraic Topology, Springer Verlag, 1966.

Michael Farber, Department of Mathematics, Tel Aviv University, Tel Aviv, 69978, Israel, mfarber@tau.ac.il

Thomas Kappeler, Institute of Mathematics, University of Z¨ urich, 8057 Z¨ urich, Switzerland, tk@math.unizh.ch

Janko Latschev, Institute of Mathematics, University of Z¨ urich, 8057 Z¨ urich, Switzerland, janko@math.unizh.ch

Eduard Zehnder, Department of Mathematics, ETH Z¨ urich, 8092 Z¨ urich, Switzerland, eduard.zehnder@math.ethz.ch

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