METRIC CYCLES, CURVES AND SOLENOIDS

We prove that every one-dimensional real Ambrosio-Kirchheim current with zero boundary (i.e. a cycle) in a lot of reasonable spaces (including all finite-dimensional normed spaces) can be represented by a Lipschitz curve parameterized over the real line through a suitable limit of Cesàro means of this curve over a subsequence of symmetric bounded intervals (viewed as currents). It is further shown that in such spaces, if a cycle is indecomposable, i.e. does not contain “nontrivial” subcycles, then it can be represented again by a Lipschitz curve parameterized over the real line through a limit of Cesàro means of this curve over every sequence of symmetric bounded intervals, that is, in other words, such a cycle is a solenoid.


Introduction
The paper is aimed at studying the notion of so-called solenoids or solenoidal vector charges, i.e. one-dimensional currents (in the classical sense of Whitney in a Euclidean spaces or in the sense of metric currents of Ambrosio-Kirchheim in a generic metric space) with zero boundary given by a limit of Cesàro means of some Lipschitz curve parameterized over the real line (viewed as currents as well). This notion is well related to the subject of optimal transportation, though in a quite indirect way. Namely, one of the topics close to the optimal transportation is weak KAM theory and the existence of Mather's minimal measures. The strong analogy between Mather's theory of minimal measures in Lagrangian dynamics and the optimal mass transportation theory has been shown by L.C. Evans and further explored by many researchers. In the study of Mather's minimal measures recently the result on decomposition of one-dimensional cycles (currents without boundary) into "elementary" cycles represented by Lipschitz curves has proved to be very helpful [2,4]. The "elementary" cycles used in such a decomposition are exactly solenoids. For normal currents in a Euclidean space such a result has been first proven in [14]. In [11] an analogous decomposition result has been proven for Ambrosio-Kirchheim metric currents in arbitrary metric spaces. Namely, it has been shown that if T is a metric current in a complete metric space E with, say, a compact support (we just mention this case for simplicity), and has ∂T = 0 (i.e. is a cycle), then there is a finite positive measure η over the space Lip 1 (R; E) of 1-Lipschitz curves in E parameterized over the real line such that for η-a.e. θ there is a limit M(S θ ) dη(θ) = η(Lip 1 (R; E)), M(T ) standing for the mass of T , so that in particular, S θ is a metric current of unit mass for η-a.e. θ ∈ Lip 1 (R; E), and, further, we may assume θ(R) ⊂ supp S θ ⊂ supp T for η-a.e. θ. The currents S θ associated to curves θ are usually called, following [14], elementary solenoids. A closely related object is that of an asymptotic cycle introduced by S. Schwartzman in [12] and further studied in [13]. The latter appears quite natural as well in the study of the problem of representation of homology classes of manifolds (see [7,9,8,6]). Last but not least one has to mention the applications of a decomposition of every cycle in solenoids in the problems of approximation of a given vector field by some vector field with better properties (i.e. harmonic approximation), see [5] and references therein.
What has been mentioned above well justifies the interest in a more profound study of solenoids, i.e. cycles T satisfying in the weak sense of currents for some curve θ ∈ Lip 1 (R; E). Even a weaker notion seems to be interesting, namely, we say that a current T is represented by some curve θ ∈ Lip 1 (R; E), if the above limit exists just for some increasing subsequence of t (it is worth making the following comparison: if the notion of a solenoid is directly related to that of a Schwartzman asymptotic cycle, then the set of cycles represented by the same curve corresponds in a similar way to the notion a Schwartzman balanced cluster [7]). In this paper we show that in a lot of reasonable spaces including all finite-dimensional normed ones every cycle (i.e. a real one-dimensional metric current without boundary), up to normalization by its mass, is represented by some curve. We further show that if a cycle T represented by a curve is indecomposable, i.e. intuitively, does not contain subcycles different from λT , λ ∈ [0, 1], then it is a solenoid. The results we provide seem in fact to be interesting already in classical settings (E being a Euclidean space or a smooth connected Riemannian manifold, the cycles being intended in the sense of classical Whitney currents, i.e. elements of some space dual to a space of smooth differential forms), but the use of the theory of Ambrosio-Kirchheim currents gives us the possibility to obtain some generalization to more general, and not even finite-dimensional, metric spaces. However, we are not interested in providing the most general results here and limit ourselves mainly to illustrating the described phenomena. Thus when possible we provide the results for the settings where they are easier to obtain (say, in finite-dimensional normed spaces rather than in general metric spaces) and provide only those generalizations to metric space settings which are almost "free of charge" by the use of the theory of Ambrosio-Kirchheim currents. Such generalizations, when they seem to be too technical, are located in the appendix.

Notation and preliminaries
The metric spaces are always in the sequel assumed to be complete. A metric space will be called connected by rectifiable arcs, if every couple of points in this space can be connected by a Lipschitz curve of finite length. The parametric length of a Lipschitz curve θ : [a, b] → E will be denoted by ℓ(θ). If d is the distance of the metric space E, we let For a compact set K ⊂ E, we let diam * K to stand for the diameter of K with respect to the distance d * . It is worth recalling that For a set D ⊂ E of a metric space E with distance d we employ the usual notation The notation Lip(X, Y ) (resp. Lip k (X, Y ) and Lip b (X, Y )) for metric spaces X and Y stands for the set of all Lipschitz maps (resp. all Lipschitz maps with Lipschitz constant k, the set of bounded Lipschitz maps) f : X → Y . We omit the reference to Y in case Y = R and write just Lip(X), Lip k (X), Lip b (X) respectively.
For metric currents we use the same notation as in [10] (i.e. almost identical to that of [1], with the only exception of the notation for the mass measure). In particular, D k (E) = Lip b (E) × (Lip(E)) k stands for the space of metric k-forms, its elements (i.e. k-forms) being denoted by f dπ, where f ∈ Lip b (E), π ∈ (Lip(E)) k , M k (E) stands for the space of k-dimensional metric currents, M(T ) stands for the mass of a current T , and µ T stands for the mass measure associated to this current. The one-dimensional current associated to a Lipschitz curve θ : [a, b] → E will be denoted by [[θ]], namely, is called a cycle, if ∂T = 0, and if T and S are cycles and T ≤ S, we say that T is a subcycle of S.
The current T ∈ M k (T ) will be called tight if so is the measure µ T , and, similarly, the sequence {T j } ⊂ M k (T ) will be called tight if so is the sequence of measures {µ Tj }. All the measures considered in the sequel, unless explicitly stated otherwise, are assumed to be finite positive Borel measures over a metric space E where they are defined. Recall that it is consistent with Zermelo-Fraenkel set theory to assume that all such measures are tight (and hence, in particular, all the metric currents are tight); and it is always true when E is a Polish (i.e. complete separable metric) space. The narrow topology on measures is defined by duality with the space C b (E) of continuous bounded functions.

Cycles and curves
Before introducing the concept of a solenoid we study the following weaker notion.
in the weak sense of currents.
Note that this definition makes sense since We also remark that the same curve θ ∈ Lip 1 (R; E) may represent different cycles (corresponding to different diverging sequences {s k }). Finally, it is clear that for a cycle T ∈ M 1 (E) represented by some curve one has M(T ) ≤ 1. We will show that in a lot of reasonable metric spaces (including, of course, all finite-dimensional normed spaces) every cycle with bounded support, normalized by its mass, is represented by some curve. In fact, the main idea, as well as the main difficulty, is in the proof of this result for finite-dimensional normed spaces which we provide in this section. Since this section is mainly propaedeutic, we are not interested in obtaining the most general representation result, and will be satisfied with the finite-dimensional one. Some easy generalizations to infinite-dimensional spaces are given in section A. Proof. Combine Corollary 3.8, Lemma 3.9 and Lemma 3.6 below.
The rest of the section contains the technical assertions used in the proof of Theorem 3.2. The complete metric space E here and in the sequel is supposed to be generic (i.e. not necessarily finite-dimensional normed) unless otherwise explicitly stated. We will first make the following useful observations.
we have that in the narrow sense of measures, so that in particular the sequence of measures ], one has for each subsequence of j such that {µ Tj (B)} is convergent, the relationship In fact, which means and minding that lim j M(T j ) = 1 we get (3.1).
in the weak sense of currents for some θ ∈ Lip 1 (R; E) and some sequence s k ↗ +∞, then again in the weak sense of currents, whenever t k /s k → 1. In fact, assuming without loss of generality that t k > s k , we get as k → +∞, which shows the claim.
The following statement holds true. Proof. Since for some θ j : R → E with Lip θ j ≤ 1, we define the curve θ : R → E inductively as follows.
Let K ⊂ E be such a compact set that µ Tj (K c ) ≤ 1/4. Note that for every ε > 0 and for each j ∈ N the set θ −1 j ((K) ε ) ⊂ R is unbounded both from above and from below. In fact, suppose the contrary, i.e. that the latter set is bounded, say, from above, i.e. θ j (t) ̸ ∈ (K) ε for all t >t j ε . Then which contradicts the choice of K (the latter provides µ Tj (K) > 3/4). Let K j ν ⊂ E be such a compact set that 1 Let d w stand for the distance over {T j } provided by lemma A.1 from [11], and denote by ∥ · ∥ 0 the Kantorovich-Rubinstein norm metrizing the narrow topology on positive finite Borel measures overX (see [3][theorem 8.3.2]). Set ε := 1. Let r 1 := s 1 1 and θ coincide with and, analogously, θ [r j , r j + α + j ] be an arbitrary curve with Lipschitz constant bounded by 1 connecting θ j (r j ) with θ j+1 (r j + α + j ). Clearly, with this construction ] }, and therefore, the latter is uniformly tight by the Prokhorov theorem for nonnegative measures (theorem 8.6.4 from [3]). Thus, by lemma A.1 from [11], the convergence in (3.2) is also in the weak topology of currents. But as j → +∞, and hence in the weak sense of currents, which shows the thesis.
The following auxiliary statements have been used in the proof of the above Theorem 3.2.
. . , m, and each θ i is a simple closed curve, so that, in particular, ∂T = 0. Then T is represented by some curve.
Choose an x 0 ∈ E and let σ i stand for the Lipschitz curves parameterized by arclength and connecting x 0 to θ i (0), andσ i stand for the same curves covered in the opposite direction (i.e.σ i (t) : , and, finally, set Define now inductively Obviously Let θ k i stand for a Lipschitz curve defined as the composition of the curves σ i with θ i covered n k i times and then withσ i , where n k . We then define θ over each interval [t k , t k+1 ] as the composition of the curves θ k i , i = 1, . . . , m, and parameterized with velocity not greater than one, which is possible because Extending the definition of θ to the whole R defining θ(t) for t < 0 in a symmetric way, we will have which shows the statement. By lemma C.1 from [10] there are polyhedral currents T ′ ν ∈ M 1 (E) with uniformly bounded supports and such that One has that T ν are polyhedral and Proof. We refer to theorem A.1 from [10] for the existence of an A ′ ∈ M 1 (E) which provides the minimum of the mass functional T → M(T ) among all T ∈ M 1 (E) satisfying ∂T = ∂A, so that, in other words, M(A ′ ) be the classical Monge-Kantorovich cost of optimal transportation the measure (∂A) + to (∂A) − (where φ ± stand for the positive and negative parts of a finite signed Borel measure φ respectively). The same theorem applied in the case when ∂A is a finite sum of signed Dirac masses, shows that one can choose A ′ so that µ A ′ be concentrated over geodesics (i.e. segments) connecting these masses, that is, in this case one may choose A ′ polyhedral.

Indecomposable cycles
In this section we will be interested in the following notion. We will study now the representation of indecomposable cycles by curves. The following observation makes sometimes calculations (or just writing them) a bit easier.

Remark 4.2. If T ∈ M 1 (E) is an indecomposable cycle represented by a curve and M(T ) = 1, then
In fact, denoting The following auxiliary result will be used in our construction.  We have then On the other hand, that is, αT ′ ≤ T . This implies αT ′ = λT for some λ ∈ [0, 1], since T is indecomposable.
and hence keeping in mind as j → ∞. We have then In fact, recalling (4.1) and supposing that for some sequence of j (not relabeled) one has we get passing to the limit in (4.3) while taking into account (4.1) and (4.2), that and passing to a limit as j → +∞, we get M(T ′ ) − 1 ≥ 0, i.e. M(T ′ ) = 1.
It suffices now to notice that and therefore T ′ = T as claimed.
The following lemma has been used in the above proof.
whenever the first limit exists (by Remark 3.4 with [0, s j+1 ] instead of [−s j , s j ]). We have then hence for some k ∈ N one has for all j ≥ k and for some C > 0 independent of j and ε. LettingK to be such a compact set that µ Rj (K c ) ≤ Cε, j = 1, . . . , k − 1, then for K :=K ∪K one has µ Rj (K c ) ≤ Cε so that the sequence {R j } is compact in the weak topology of currents by compactness theorem 5.2 from [1].

Solenoids
Let us introduce now another definition. Clearly, being a solenoid is stronger than just being representable by a curve. The following result holds true. Proof. Let the limit being, as usual, in the weak sense of currents, for some sequence s ′ j ↗ +∞. We will assume lim j s ′ j /s ′ j+1 = 0 since otherwise there is nothing to prove. In fact, if lim j s ′ j /s ′ j+1 = α > 0 for some subsequence of {s ′ j }, then for every increasing sequence {r j }, up to passing to a further subsequence of {s ′ j } (not relabeled) we may consider any subsequence of {r j } (again not relabeled) such that r j ∈ [s ′ j , s ′ j+1 ], and hence lim inf j r j /s ′ j+1 ≥ α > 0, which by Proposition 4.3 means that for every subsequence of {r j }, and hence, for the whole sequence {r j }. The proof will be achieved in two steps.
Consider now an arbitrary increasing sequence {r i }, lim i r i = +∞. For every i ∈ N we find a j = j(i) such that r i ∈ [s j(i+1)−1 , s j(i+1) ] and calculate Clearly j(i) → +∞ when i → +∞. For the sake of brevity of notation we write j := j(i + 1) − 1. We consider several cases.
) , so that as j → ∞, while lim j s j n j /s j+1 = 1. Otherwise, setting as i → ∞, and keeping in mind that lim i s j ρ i /r i = 1, we get that the above limit is T . Case 2. lim i r i /s j =: α > 1. Consider first for simplicity the case r i = ns j for some n ∈ N. The case of a generic α > 1 follows by applying Proposition 4.3 (with σ instead of θ and ns j instead of s j ).
Theorem 5.2 implies that if E is the space where every tight cycle with bounded support is represented by a curve (a fairly general collection of such spaces including of course, all finite-dimensional vector spaces, are provided by Theorem 3.2, Proposition A.1 and Corollary A.2), then every indecomposable tight cycle in M 1 (E) with bounded support is, up to normalization by its mass, a solenoid.
Note that Theorem 5.2 gives only sufficient conditions for a cycle to be a solenoid. In fact, solenoids are not necessarily indecomposable as the following example shows. Example 5.3. Let {z 1 , z 2 , z 3 } ⊂ E := R 2 be vertices of an equilateral triangle with side 3, and let T i ∈ M 1 (R 2 ) be a cycle represented by the oriented unit circumference S i with center z i , having density 1/6π, i = 1, . . . , 3 (for definiteness, assume that the orientation of all the three cycles are, say, clockwise). In this way, defined T := T 1 + T 2 + T 3 we get M(T ) = 1. We show now that T is a solenoid (although, clearly it is not indecomposable: in fact, T i ≤ T and T i ̸ = λT for all λ ∈ [0, 1]). We further abuse slightly the notation and let S i stand both for the cycle (as a metric current), i.e. the oriented circumference (so that T i = S i /6π), and its support, i.e. the circumference itself, the exact meaning being always clear from the context.
We have to define a curve θ ∈ Lip 1 (R; E) such that the limit where the limits of currents are intended in the weak sense. which concludes the proof of (5.1).
Appendix A. Some easy generalizations to infinite-dimensional spaces We provide here a couple of easy, though quite technically looking results generalizing Theorem 3.2 to not necessarily finite-dimensional spaces. We are not really interested in obtaining the most general results, since the main idea is well illustrated by Theorem 3.2, and thus we limit ourselves here to provide some fairly general results that are almost "free of charge" by using the theory of metric currents. Proof. When E is a finite-dimensional normed space, the result is given by Theorem 3.2. Suppose that E is a Banach space with metric approximation property. Let {T n } be a sequence of currents over finite-dimensional subspaces E n of E be provided by lemma A.6 from [11] (minding the remark A.7 from the same paper), namely, T n ⇀ T in the weak sense of currents, M(T n ) → M(T ) as n → ∞, and ∂T n = 0. Thus each T n (as a cycle over a finite-dimensional normed space) is represented by some curve in Lip 1 (R; E n ), hence in Lip 1 (R; E), and again applying Lemma 3.6 we get the result.
Of course the above Proposition A.1 provides a rather wide class of spaces where every cycle is (up to a normalization by its mass) represented by some curve, which includes all finite-dimensional normed spaces, Hilbert spaces and a lot of reasonable Banach spaces (for instance, Lebesgue spaces). However, if we are interested in generic metric spaces without any linear structure, the following corollary might be helpful. To formulate it, we recall that a closed subset X ⊂ Y of a metric space Y is called a 1-Lipschitz neighborhood retract, if there is an open U ⊂ Y satisfying X ⊂ U and a map p ∈ Lip 1 (Ū , X) such that p(x) = x for all x ∈ X.
Corollary A.2. Let X be a quasiconvex metric space isometrically embedded in Banach space Y with metric approximation property as a 1-Lipschitz neighborhood retract of the latter. Then for every tight cycle T ∈ M 1 (X) with bounded support one has that T /M(T ) is represented by some curve.
Proof. Assume without loss of generality M(T ) = 1. Let j : X → Y be an isometric embedding. Then j # T is represented by some curve in Y by Proposition A.1, and thus j # T is represented by a curve in j(X) by Lemma A.3 (applied withV := j(X)). In other words, for some θ ′ ∈ Lip 1 (R; j(X)) and some sequence s k ↗ +∞. Therefore, where θ(t) := j −1 (θ(t)), which concludes the proof.
We estimate now