AVERAGE LINKING NUMBERS

GONZALO CONTRERAS, RENATO ITURRIAGA

To the memory of Ricardo Mañé.

1. Introduction.

Let ^{1} γ_{1}, γ_{2} be two disjoint oriented knots in the three dimensional sphere S^{3} the linking number ℓ(γ_{1}, γ_{2}) is defined, as the algebraic intersection number of γ_{1} with an orientable surface N transversal to γ_{1}, with oriented boundary γ_{2}. For better understanding of our results we need the two following interpretations of the linking number (See [R]):

1) For any regular planar projection it is the algebraic crossing number of γ_{1} over γ_{2}.

2) It is the degree of the map

T : T^{2} → S^{2}

(t_{1}, t_{2}) → \frac{γ_{1}(t_{1}) − γ_{2}(t_{2})}{|γ_{1}(t_{1}) − γ_{2}(t_{2})|}

where T^{2} is the two dimensional torus and S^{2} is the two dimensional sphere. Hence

ℓ(γ_{1}, γ_{2}) = \frac{1}{vol(S^{2})} ∫_{S^{1}×S^{1}} det(DT)dsdt

= \frac{1}{4π} ∫_{0}^{t_{1}} ∫_{0}^{t_{2}} \frac{(γ′_{1} × γ′_{2})(γ_{1} − γ_{2})}{|γ_{1} − γ_{2}|^{3}}

where γ_{i} : [0, t_{i}] → S^{3} is a parametrization of γ_{i}; i = 1, 2. This formula is known as the Gauss Formula.

Given a flow φ_{t} generated by a vector field F, and two closed orbits γ_{1}, γ_{2} we define the average linking number as \frac{1}{T_{1}T_{2}} ℓ(γ_{1}, γ_{2}) where T_{i} is the period of γ_{i}. In order to define the average linking number for general orbits we need a convinient set of short curves.

Define a good set of short curves as a system of piecewise differentiable paths joining points x, y ∈ S^{3}, depending in a measurable way on x and y such that the Gauss integrals of every pair of nonintersecting pairs of the system, and the Gauss integrals of every pair of nonintersecting { paths of the system, segments of orbits φ_{t}(p), 0 ≤ t ≤ 1 }, are bounded independently of the paths by a constant k (see [A]). In section 3 we will prove the existence of such a set.

1Research partially supported by CONACYT, Mexico and CNPq, Brazil.

For any x ∈ S^{3} and T ∈ R let x̂_{T} denote the knot formed by the orbit fromx to φ_{T} (x) and then close it by a path α_{φ_{T} (x),x} joining x to T (x) of a good set of short curves. For any two points in different orbits p, q and times T_{1}, T_{2} define, when possible, the average linking number as

ℓ(x, y) := lim_{T_{1}, T_{2}→∞} \frac{1}{T_{1}T_{2}} ℓ(x̂_{T} , ŷ_{T})

The main pourpose of this paper is to decide when does the limit when T_{1}, T_{2} tend to infinity exists and does not depend on the set of short curves. Our main Theorem is

Theorem 1. Let μ, ν be two invariant measures on S^{3}. Then the function J : S^{3} × S^{3} → R defined by

J(p, q) = \frac{F (p) × F (q) · (p − q)}{|p − q|^{3}}

is in L^{1}(μ × ν)

From Birkhoff’s Theorem we obtain the following

Theorem 2. a) For μ × ν almost every pair of points the limit

ℓ̂(x, y) = lim_{T_{1},T_{2}→∞} \frac{1}{T_{1}T_{2}} ∫_{0}^{T_{1}} ∫_{0}^{T_{2}} \frac{(γ′_{1} × γ′_{2}, γ_{1} − γ_{2})}{|γ_{1} − γ_{2}|^{3}}

exist.

b) If there is no periodic orbit with positive μ and ν measure this limit is for μ × ν-almost every pair the same as ℓ(x, y), in particular the asymptotic linking number does not depend on the choice of the set of short curves. If the measures μ and ν are ergodic, the asymptotic linking ℓ(x, y) is given by the integral ∫ ∫ J(p, q) dμ(p) dν(q).

E. Ghys and J.M. Gambaudo have anounced a proof that the average linking is a topological invariant for volume preserving suspension flows. The same assertion of this theorem was proven by Arnold in [A] for smooth measures and in [C] for Gibbs measures. The proof in [A] relays on the fact that the singularity of the formula has order 2 which is less than the codimension of the diagonal, the singular set of J. The proof in [C] relays on the observation that the singularity of the Gauss formula has actually order 1 and that the Hausdorff dimension of a Gibbs measure is greater than 1.

The idea of the proof is that the singularity of the Gauss formula disappears when we integrate along the orbits in a small flow box. This is because the integral of the Gauss formula is the integral of the Jacobian of T therefore in small time intervals the area of the image of T should be contained in one half of the sphere. However since we actually want bound the integral of the absolute value of the Jacobian of T it was easier to carry on an analitical proof, rather than to worry about of regions of injectivity of T and the sign of its Jacobian.

Finally, as an application, in [VV] Verjovsky and Vila following the same philosophy extended other topological invariants, such as Witten’s invariant of links, to an average value for an invariant smooth measure of a flow. In their paper, the only obstruction to define the average Witten’s invariant for singular invariant measures is the existence of the average linking number for such a measure, which is proven here on theorem 1. We also quote the work of Freedman and He [FH] where they relate the asymptotic crossing number (the integral of |J(x, y)|) with the energy of a vector field with a smooth invariant measure.

The second author is grateful with R. Vila for showing his work and both of us are grateful with E. Ghys for valuable conversations in the congress in honor to 25 anniversary of the seminar of Dynamical Systems of Impa at Rio de Janeiro. The first author is grateful to CIMAT, Mexico, for its Hospitality.

2. Proof of the Theorems.

Lemma 2.1. For any compact subset K ⊂ R^{3}, there exist ε > 0, A > 0, such that if p ∈ K, |x − p| < ε, |y − p| < ε, then

J(x, y) = \frac{H^{T} B H}{|H|^{3}} + ϕ(x, y)

where |ϕ(x, y)| < A is bounded, H = y − x and B is the matrix defined by B · H = F (p) × (DF (p) · H) for all H ∈ R^{3}.

Proof.

We have that

J(x, y) = −\frac{F (x) × F (y)}{|H|} · \frac{H}{|H|^{3}} ,

where y = x + H. Since F (y) = F (x) + DF (x) · H + ψ(x, H), with |ψ(x, H)| ≤ a |H|^{2}, uniformly on all p ∈ K, |x − p| < ε, then

\begin{aligned}J(x, y) |H|^{3} & = H · [F (x) × (F (x) + DF (x) · H + ψ(x, H))] \\ & = H · F × DF (x) · H + H · F (x) × ψ(x, H) \end{aligned}

but if A := a sup_{d(x,K)≤ε} |F (x)|,

\frac{H · F (x) × ψ(x, H)}{|H|^{3}} ≤ a |F (x)| ≤ A

Given p ∈ R^{3} and ε > 0, let E(p) and N(p) be defined by

\begin{matrix}E(p) = E(p, ε) = \{ q ∈ R^{3} | (q − p) · F (p) = 0 , |q − p| < ε \} , \\ N(p) = N(p, ε) = ⋃_{t∈[−ε,ε]} φ_{t}(E(p)) . \end{matrix}

For s, t ∈ R and q ∈ E(p), write p_{s} = φ_{s}(p), q_{t} = φ_{t}(q). Given q ∈ E(p) let τ : [−ε, ε] → R be such that q_{τ(s)} ∈ E(p_{s}, 1), i.e. (q_{τ(s)} − p_{s}) · F (p_{s}) = 0.

Lemma 2.2. For any compact subset K ⊂ R^{3}, there exist ε > 0, M > 0, such that if p ∈ K, q ∈ E(p, ε), |s| , |t| < ε, then

J(p_{s}, q_{t}) = \frac{Q_{s} B_{s} Q_{s} + (t − τ(s)) Q_{s} (F_{s} × A_{s} F_{s})}{|H_{st}|^{3}} + ψ(p, q, s, t)

where |ψ(p, q, s, t)| ≤ M, F_{s} = F (p_{s}), A_{s} = DF (p_{s}), H_{st} = q_{t} − p_{s}, Q_{s} = q_{τ(s)} − p_{s} and B_{s} is the matrix defined by B_{s} H = F_{s} × (A_{s} H) for all H ∈ R^{3}.

Proof.

We have that q_{τ(s)} = p_{s} + Q_{s}, and

\begin{aligned}F (q_{τ(s)}) & = F (p_{s}) + DF (p_{s}) Q_{s} + G_{1}(p, q, s) |Q_{s}|^{2} , |G_{1}(p, q, s)| < a_{1} , for some a_{1} > 0, \\ q_{t} − p_{s} & = Q_{s} + (q_{t} − q_{τ(s)}) = Q_{s} + (t − τ(s)) F (q_{τ(s)}) + G_{2} |t − τ(s)|^{2} , |G_{2}| < a_{2} \\ H_{st} & = Q_{s} + (t − τ(s)) F (p_{s}) + G_{3} (|t − τ(s)|^{2} + |t − τ(s)| |Q_{s}|) , |G_{3}| < a_{3} . \end{aligned}

Observe that, since Q_{s} ⊥ F (p_{s}), writing Δt := t − τ(s),

|H_{st}|^{2} ≥ |Q_{st}|^{2} + |Δt|^{2} |F_{s}|^{2} − a_{3}^{2} (|Δt|^{2} + |Δt| |Q_{s}|)^{2}
≥ |Q_{s}|^{2} + \frac{1}{2} |Δt|^{2} |F_{s}|^{2}
[⋆]  |H_{st}|^{2} ≥ \frac{1}{2} [|Q_{s}|^{2} + |Δt|^{2} |F_{s}|^{2}] ,

if ε > 0 is small enough. Therefore |Q_{s}| ≤ |H_{st}|, and |Δt| ≤ α |H_{st}|, for some α = α(K, ε) > 0. Now

H_{st} B_{s} H_{st} = (Q_{s} + Δt F_{s} + a_{3} (|Δt|^{2} + |Δt| |Q_{s}|)) · (F_{s} × A_{s} H_{st})
= Q_{s} F_{s} × A_{s} H_{st} + a_{3} O (|H_{st}|^{2}) O (|H_{st}|)

= Q_{s} F_{s} × A_{s} (Q_{s} + (Δt) F_{s} + O (|H_{st}|^{2}) ) + O (|H_{st}|^{3})

= Q_{s} B_{s} Q_{s} + Q_{s} (F_{s} × A_{s} F_{s}) Δt + O (|H_{st}|^{3}) .

By lemma 2.1, we have that

J(p_{s}, q_{t}) = \frac{H_{st} B_{s} H_{st}}{|H_{st}|^{3}} + ϕ(p_{s}, q_{t})

with |ϕ(p_{s}, s, t)| < A. This completes the proof of the lemma.

Lemma 2.3. For any compact subset K ⊂ R^{3} there exist ϵ > 0, a, b > 0, such that if q ∈ E(p, ϵ), |s|, |t| < ϵ then

a|Q_{0}| ≤ |Q_{s}| ≤ b|Q_{0}|

where Q_{s} is defined as in Lemma 2.

Proof.

Define τ : ∪_{p∈K}({p}×E(p, ϵ)×[−ϵ, ϵ]) → R by q_{τ(p,q,s)} ∈ E(p_{s}, 1).We first show that τ is differentiable on q. By definition of τ, we have that

G(p, q, τ(p, q, s)) := F (p_{s}) · (φ(q, τ(p, q, s)) − p_{s}) = 0,

since

\frac{∂G}{∂τ} = F (p_{s}) · \frac{∂G}{∂τ}|_{(q,τ)} = F (p_{s}) · F (q_{τ}) > 0,

by the implicit function theorem we have that τ is differentiable respect to q and

\frac{∂τ}{∂q} = \frac{−F (p_{s}) · \frac{∂φ}{∂q}|_{(q,τ)}}{F (p_{s}) · F (q_{τ})} .

Moreover, there exists C = C(K, ϵ) > 0 such that |\frac{∂τ}{∂q} | < C. We have that

Q_{s} = q_{τ(s)} − p_{s} = (q_{τ(s)} − p_{s}) + (q_{s} − p_{s}).

Since q_{τ(s)} − q_{s} = (τ(s) − s) F (q_{σ}) for some σ between s and τ(s), we have that

| q_{τ(s)} − q_{s}| ≤ C |q − p| max_{q} |F (q)| ≤ B|Q_{0}|.

For B := C max_{q} |F (q)|. By Gronwall’s inequality

|q_{s} − p_{s}| ≤ D |q − p| = D |Q_{0}|,

for some uniform D = D(K, ϵ). Therefore if b = C + D, we have that

|Q_{s}| ≤ (C + D) |Q_{0}| = b |Q_{0}|.

The other inequality is obtained from this one by changing the roles of p and p_{s} and reversing the time; (observe that φ(q_{s}, τ(p_{s}, q_{s}, −s) = q). Therefore

Q_{0} ≤ b Q_{s}.

Now take a = \frac{1}{b} .

Lemma 2.4. For any compact subset K ⊂ R there exist ε > 0, M > 0 such that if p is in K and q ∈ E(p, ε) then

∫_{−ε}^{ε} ∫_{−ε}^{ε} J(p_{s}, q_{t}) ds dt < Mε

Proof.

We first prove the case when p ≠ q. By lemma 2, we have that

J(p_{s}, q_{t}) = \frac{Q_{s} B_{s} Q_{s}}{|H_{st}|^{3}} + \frac{Δt Q_{s} (F_{s} × A_{s} F_{s})}{|H_{st}|^{3}} + ψ(p, q, s, t) ,

with ψ(p, q, s, t) ≤ M_{1}. We bound the integral of each term. From lemma 3 and [⋆], we have that

a |Q_{0}| ≤ |Q_{s}| ≤ b |Q_{0}|

|H_{st}|^{2} ≥ \frac{1}{2} [|Q_{s}|^{2} + |Δt|^{2}|F_{s}|^{2}] ≥ \frac{1}{2} [|Q_{s}|^{2} + |Δt|^{2}D]

where D = min_{|q−p|<ε} |F (q)| > 0. Writing α = \frac{1}{4}a and β = \frac{1}{2} \sqrt{D}, we have

|H_{st}|^{2} ≥ \frac{1}{2}(a^{2}|Q_{0}|^{2} + |Δt|^{2}D),

|H_{st}| ≥ (α^{2}|Q_{0}|^{2} + β^{2}|Δt|^{2})^{\frac{1}{2}} .

Since B_{s}H = F (p_{s}) × (DF (p_{s}) · H), let B = max_{p_{s}} ∥B_{s}∥, then

\frac{|Q_{s}B_{s}Q_{s}|}{|H_{st}|^{3}} ≤ \frac{B |Q_{s}|^{2}}{|H_{st}|^{3}} ≤ \frac{B b^{2}|Q_{0}|^{2}}{(α^{2}|Q_{0}|^{2} + β^{2}|Δt|^{2})^{\frac{3}{2}}} = \frac{B b^{2}|Q_{0}|^{2}}{β( \frac{α^{2}}{β^{2}} |Q_{0}|^{2} + |Δt|^{2})^{\frac{3}{2}}}

Observe that, since Δt = t − τ(s),

∫_{−ε}^{ε} ∫_{−ε}^{ε} |J| dt ds = ∫_{−ε}^{ε} ∫_{−ε−τ(s)}^{ε−τ(s)} |J| d(Δt) ds

Since ∫ \frac{dx}{r^{3}} = \frac{1}{a^{2}} \frac{x}{r}, r = \sqrt{x^{2} + a^{2}} we have that

∫_{−ε}^{ε} ∫_{−ε}^{ε} \frac{Q_{s} B_{s} Q_{s}}{|H_{st}|^{3}} dt ds ≤ ∫_{−ε}^{ε} 2 ∫_{0}^{2ε} \frac{B b^{2} |Q_{0}|^{2}}{β \sqrt{\frac{α^{2}}{β^{2}} |Q_{0}|^{2} + |Δt|^{2}}} d (Δt) ds

≤ \frac{2 ε 2 B b^{2} |Q_{0}|^{2}}{\frac{α^{2}}{β} |Q_{0}|^{2}} \frac{Δt}{\sqrt{\frac{α^{2}}{β^{2}} |Q_{0}|^{2} + |Δt|^{2}}} |_{0}^{2ε}

≤ \frac{4 ε B b^{2} β}{α^{2}} (1 + 1) = \frac{8 ε B b^{2} β}{α^{2}} = εM_{2}

Now we bound the integral of the second term. Let A := max_{p_{s}} |F (p_{s}) × (DF (p_{s}) · F (p_{s}))|. Then, since ∫ \frac{x}{r^{3}} dx = −\frac{1}{r}, r = \sqrt{x^{2} + a^{2}}, we have that

∫_{−ε}^{ε} ∫_{−ε}^{ε} \frac{(t − τ(s)) Q_{s} (F_{s} × A_{s}F_{s})}{|H_{st}|^{3}} dt ds ≤ ∫_{−ε}^{ε} ∫_{−ε}^{ε} \frac{|Δt| b |Q_{0}| A}{(\sqrt{α^{2} |Q_{0}|^{2} + β^{2} |Δt|^{2}})^{3}} dt ds

≤ ∫_{−ε}^{ε} 2 ∫_{0}^{2 ε} \frac{b |Q_{0}| A |Δt|}{(β\sqrt{\frac{α^{2}}{β^{2}} |Q_{0}|^{2} + |Δt|^{2}})^{3}} d(Δt) ds

≤ \frac{4 ε b A}{β} |Q_{0}| ∫_{0}^{2 ε} \frac{|Δt| d(Δt)}{(\sqrt{\frac{α^{2}}{β^{2}} |Q_{0}|^{2} + |Δt|^{2}})^{3}}

≤ \frac{4 ε b A}{β^{3}}|Q_{0}| \frac{1}{\sqrt{\frac{α^{2}}{β^{2}} |Q_{0}|^{2} + |Δt|^{2}}} |_{2ε}^{0}

≤ \frac{4 ε b A}{β^{3}} \frac{|Q_{0}|}{\frac{α}{β} |Q_{0}|} ≤ \frac{4 b A}{α β^{2}} ε = ε M_{3} .

From lemma 2, |ψ(p, q, s, t)| ≤ M_{1}, therefrore, if ε < 1,

∫_{−ε}^{ε} ∫_{−ε}^{ε} |J(p_{s}, q_{t})| ds dt ≤ ε M_{2} + ε M_{3} + 4 ε^{2} M_{1} ≤ ε M_{4} , where M_{4} := M_{1} + M_{2} + M_{3} .

Now suppose that p = q, we have that

\begin{aligned}F (q_{s}) & = F (q) + s A F (q) + s^{2} K(q, s) \\ q_{s} & = q + s F (q) + \frac{s^{2}}{2} A F (q) + s^{3} G(q, s) \end{aligned}

where A = DF (q), K(q, s), and G(q, s) are bounded. Then

|q_{s} − q|^{3} J(q, q_{s}) = F (q) × (F (q) + s A F (q) + s^{2} K) · (s F (q) + \frac{s^{2}}{2} AF (q) + s^{3} G)
= (s F × A F + s^{2} F × K) · (s F + \frac{s^{2}}{2} AF + s^{3} G)
= s^{4} F × A F · G + \frac{s^{4}}{2} F × K · A F + s^{5} F × K · G
|q_{s} − q|^{3} |J(q, q_{s})| ≤ s^{4} D ,

for some D = D(K, ε) > 0, for all q ∈ K, |s| < ε. Furthermore,

|q_{s} − q| = |s F (q) + \frac{s^{2}}{2} AF (q) + s^{3} G| ≥ a s ,

for a = \frac{1}{2} min_{q∈K} |F (q)| if ε = ε(K) > 0 is small enough. Therefore

|J(q, q_{s})| ≤ \frac{s^{4} D}{a^{3} s^{3}} ≤ \frac{ε D}{a^{3}} ,

and

∫_{−ε}^{ε} ∫_{−ε}^{ε} J(p_{s}, p_{t}) ds dt ≤ \frac{ε^{3} D}{a^{3}} ≤ ε M_{5} for all p ∈ K,

for M_{5} := \frac{D}{a^{3}} .

Corollary 2.5. For any compact subset K ⊂ R there exist δ, ε > 0, M > 0 such that if p, q are in K.

∫_{−δ}^{δ} ∫_{−δ}^{δ} J(p_{s}, q_{t})dsdt < Mε

Proof.

Take ϵ as in the previous lemma. As the vector field never vanishes N(p, δ) is a neighbourhood of , p hence it contains a ball of radius r. Moreover this r may be taken independent of the point p. For any two points p, q if δ is small enough then either:

There exist s_{0}, t_{0} with |s_{0}|, |t_{0}| < 2δ such that q_{s_{0}} is in E(p_{t_{0}}) and in this case

∫_{−δ}^{δ} ∫_{−δ}^{δ} J(p_{s}, q_{t})dsdt < ∫_{−ε}^{ε} ∫_{−ε}^{ε} J(p_{s}, q_{t})dsdt < Mε

Or the distance between q_{s} and p_{t} with |s_{0}|, |t_{0}| < δ is bounded by rand hence the integral is bounded.

Proof Theorem 1

Since J is bounded outside of a neighbourhood of the diagonal Δ := {(p, p) | p ∈ S^{3}}, it is enough to prove that J is in L^{1}(μ × ν) on a neighbourhood of Δ. Since the compact set Δ may be covered by a finite number of open sets

V(p, δ) := {(q_{s}^{1}, q_{t}^{2}) : (q^{1}, q^{2}) ∈ E(p, δ) × E(p, δ)|s|, |t| < δ}

with p ∈ S^{3} and δ > 0 from corollary 5. It is enough then to prove that the restriction J|_{V(p,δ)} is in L^{1}(μ × ν). Let μ̂,ν̂ be the transversal measures on E(p, ε) defined by μ̂(A) = μ(∪_{|t|<ε}ϕ_{t}(A)), ν̂(A) = ν(∪_{|t|<ε}ϕ_{t}(A)).Then by lemma 4 we have that

∫_{V (q,δ)} |J(x, y)| dμ(x) dν(y) = ∫_{E(q,δ)} ∫_{E(q,δ)} (∫_{−δ}^{δ} ∫_{−δ}^{δ} |J(p_{s}, y_{t})| dμ̂(p) dν̂(y))

≤ ∫_{E(q,δ)} ∫_{E(q,δ)} M δ dμ′(p) dν(y) ≤ M δ .

Proof of Theorem 2: Part (a) is a direct consequence of the Birkhoff theorem and Theorem 1. Part (b) is a consequence of the existence of a good set of short curves because

\begin{aligned}|∫_{0}^{T_{1}+1} ∫_{0}^{T_{2}+1} − ∫_{0}^{T_{1}} ∫_{0}^{T_{2}} | & ≤ |∫_{0}^{T_{1}} ∫_{T_{2}}^{T_{2}+1}| + |∫_{T_{1}}^{T_{1}+1} ∫_{0}^{T_{2}} | + |∫_{0}^{1} ∫_{0}^{1}| \\ & ≤ k T_{1} + k T_{2} + k \end{aligned}

and hence

lim_{T_{1}, T_{2}→∞} \frac{1}{T_{1} T_{2}} ∫_{0}^{T_{1}+1} ∫_{0}^{T_{2}+1} = lim_{T_{1}, T_{2}→∞} \frac{1}{T_{1} T_{2}} ∫_{0}^{T_{1}} ∫_{0}^{T_{2}}

3. Existence of a good set of short curves.

The following lemmas prove the existence of a good set of short curves if they are poligonal curves made of small straight line segments which are always transversal to the direction of the flow.

Lemma 3.1. Let L(t) = a + t b be a small line segment in R^{3} and let γ(s) be a small differentiable curve such that

(i) For any s, t ∈ [−ε, ε], the vectors L′(t) = b, γ′(s) and γ(s) − L(t) are not coplanar.

(ii) The vectors \frac{γ(s)−L(t)}{∥γ(s)−L(t)∥} are always in a hemisphere of S^{2}.

Then the Gauss (absolute) integral

∫_{−ε}^{ε} ∫_{−ε}^{ε} |I(s, t)| ds dt < \frac{1}{2} vol (S^{2}) ,

where

I(s, t) := \frac{(γ′(s) × L′(t)) · (γ(s) − L(t))}{|γ(s) − L(t)|} .

Proof.

The integral I(s, t) is the Jacobian of the function T (s, t) = \frac{γ(s)−L(t)}{|γ(s)−L(t)|} . The condition (i) implies that I(s, t) has always the same sign. We will see that T : [−ε, ε] × [−ε, ε] → S^{2} is injective and hence

∫_{−ε}^{ε} ∫_{−ε}^{ε} |I(s, t)| ds dt = area (Γ ([−ε, ε] × [−ε, ε])) < \frac{1}{2} vol (S^{2}) .

We show that T is injective. By the transversality condition (i), if s_{1} ≠ s_{2}, then the planes generated by (γ(s_{1}) and L) and (γ(s_{2}) and L) are different. Therefore if γ(s_{1}) − L(t_{1}) is parallel to γ(s_{2}) − L(t_{2}) then s_{1} = s_{2}. If γ(ŝ) − L(t_{1}) = λ (γ(ŝ) − L(t_{2})) then L(t_{1}) = λ L(t_{2}) and t_{1} = t_{2}. □

Lemma 3.2. Let F be a Lipschitz nonsingular vectorfield on R^{3} and let A_{1}, A_{2}, A_{3} > 0 be such that

|F (x) − F (y)| ≤ A_{1} |x − y| , A_{2} < |F (x)| < A_{3} for all x, y ∈ R^{3} .

Then there exists ε = ε(A_{1}, A_{2}, A_{3}) > 0 such that if L(t) = a + t b, ∥b∥ = 1, |t| < 1 is a small line segment and γ(s), |s| < ε is a small orbit segment of F such that

(i) The angles ⟨(γ′(s), b) > \frac{π}{3}, ⟨(γ′(s), −b) > \frac{π}{3} for all |s| < ε.

(ii) The vectors Ł′(0) = b, γ′(0) and γ(0) − L(0) are not coplanar.

Then the Gauss (absolute) integral

∫_{−1}^{1} ∫_{−ε}^{ε} |I(s, t)| ds dt < 2 vol(S^{2}) .

Proof.

It is enough to show that ∫_{−1}^{1} ∫_{0}^{ε} |I| < vol(S^{2}). Let g(s) := (γ′(s) × L′(t))· (γ(s) − L(t)) = (γ′(s) × b) · (γ(s) − a), and let

s_{0} := min{ s > 0  | g(s) = 0 }
t_{0} := max{ s > s_{0}  | g(s) = 0 for all s_{0} ≤ s ≤ t_{0} } .
s_{n+1} := min{ s > t_{n}  | g(s) = 0 }
t_{n+1} := max{ s > s_{n+1} | g(s) = 0 for all s_{n+1} ≤ s ≤ t_{n+1} } .

By lemma 3.1,

∫_{−1}^{1} ∫_{0}^{s_{0}} |I(s, t)| ds dt < \frac{1}{2} vol (s^{2}) .

Let N := \frac{b×γ′(0)}{|b×γ′(0)|}, c := N × b and consider the orthonormal basis

\{ b, N, c \} of R^{3}. Given u_{1}, u_{2} ∈ T := \{ w ∈ S^{3} | ⟨(w, b) > \frac{π}{3} , ⟨(w, −b) > \frac{π}{3} \}, consider their cylindrical coordinates

u_{i} = λ_{i} b + r_{i} cos θ_{i} c + r_{i} sin θ_{i} N , some λ_{i} , r_{i} , θ_{i} .

Let A_{4} > 0 be such that if u_{i} ∈ T, |u_{i}| > A_{2}, i = 1, 2, then

|u_{1} − u_{2}| > A_{4} |θ_{1} − θ_{2}| .

Let θ_{n}, φ_{n} be defined by

γ′(s_{n}) = λ_{n} b + r_{n} cos θ_{n} c + r_{n} sin θ_{n} N ,
γ′(t_{n}) = ℓ_{n} b + k_{n} cos φ_{n} c + k_{n} sin φ_{n} N

Observe that since γ(s_{n}) − L(t) is coplanar with b and γ′(s_{n}) then γ(s_{n}) − L(t) = μ_{n}(t) b + ρ_{n}(t) cos θ_{n} c + ρ_{n}(t)μ sin θ_{n} N with the same θ_{n}. Similarly γ(t_{n}) − L(t) has the same angle φ_{n} as γ′(t_{n}).

For |t| < 1, t_{n} < s < s_{n+1}, the point G(s, t) := \frac{γ(s)−L(t)}{|γ(s)−L(t)|} remains on the sector

H_{n} := { v ∈ S^{2} | v = λ b + r cos θ c + r sin θ N , r > 0, λ ∈ R, θ ∈ [φ_{n}, θ_{n+1}] }

Moreover, by the argument of Lemma 3.1 since I(s, t) ≠ 0, for t_{n} < s < s_{n+1}, the map G|_{[t_{n},s_{n+1}]×[−1,1]} is inective. Hence

∫_{−1}^{1} ∫_{t_{n}}^{s_{n+1}} |I| < Area of sector (H_{n}) = \frac{vol (S^{2})}{2 π} |θ_{n+1} − φ_{n}| .

Also ∫_{−1}^{1} ∫_{s_{n}}^{t_{n}} |I| = 0, because I(s, t) = 0 on s_{n} ≤ s ≤ t_{n}. By (*) we have that

A_{4} |θ_{n+1} − φ_{n}| < |γ′(s_{n+1} − γ′(t_{n})| = |F (γ(s_{n+1})) − F (γ(t_{n}))|
< A_{1} |γ(s_{n+1}) − γ(t_{n})| < A_{1} A_{3} |s_{n+1} − t_{n}|
|θ_{n+1} − φ_{n}| < A_{5} |s_{n+1} − t_{n}| ,

where A_{5} := \frac{A_{1} A_{3}}{A_{4}} > 0. Therefore

∫_{−1}^{1} ∫_{0}^{ε} |I(s, t)| ds dt = ∫_{−1}^{1} ∫_{0}^{s_{0}} |I| + ∑_{n=0}^{∞} ∫_{−1}^{1} ∫_{t_{n}}^{s_{n+1}} |I|

≤ \frac{1}{2} vol (S^{2}) + ∑_{n=0}^{∞} \frac{vol (S^{2})}{2 π} A_{5}|, |s_{n+1} − t_{n}|

≤ \frac{1}{2} vol (S^{2}) + \frac{vol (s^{2})}{2 π} A_{5} ε

≤ vol (S^{2}) ,

if we take ε < \frac{π}{A_{5}} . □

References.

[A] V. I. Arnold, The asymptotic Hopf invariant and its applications, Sel. Math. Sov. 5 (1986), 327-345 (an English translation)

[C] G. Contreras. Average linking numbers of closed orbits of hyperbolic flows, on Workshop on dynamical systems, ICTP, 1988. Pitman Research notes in Math., 221, 106-108, Longman, Essex, New York, (1990).

[C] G. Contreras. Average linking of closed orbits of hyperbolic flows. Ph.D. Thesis, Impa. To appear. Journal of the L.M.S.

[VV] A. Verjovski & R. Vila. The Jones-Witten invariant for flows on a 3dimensinal manifold. To appear. Comm. Math. Phys.

[FH] M.H. Freedman & Z.X. He. Divergence-free fields: Energy and asymptotic crossing number. Ann. of Math., 134 (1991), 189-229.

[R] Rolfsen. Knots and Links. Publish or Perish, Wilmington, Delaware, 1977.

Gonzalo Contreras.  Renato Iturriaga.
Matemática. PUC-Rio.  CIMAT.
R. Marquês de São Vicente, 225 Ap. Postal 402
Rio de Janeiro 22453-900  36.000 Gto. Guanajuato.
Brasil.  México.
gonzalo@mat.puc-rio.br  ituriaga@buzon.main.conacyt.mx