Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 14, 2004, 39–53

© P. N. Ivanshin

ABSTRACT. We consider a manifold M with a foliation F given by a locally free action of a commutative Lie group H. Also we assume that there exists an integrable Ehresmann connection on (M,F) invariant with respect to the action of the group H. We get the structure of the restriction of the algebra C0(M) to the leaves in three partial cases. Also we consider a classification of the quasiinvariant measures and means on the leaves of F.

1. Introduction.

In the present paper we consider a manifold M with a foliation F given by a locally free action of a commutative Lie group H. Let us denote this action by R : M × H → M. Let dim H = n, and π : ℝn → H be the universal covering. Then h↦Rπ(h) is the locally free action of ℝn on M. Thus, without loss of generality, we can set H = ℝn.

Also we assume that there exists an integrable Ehresmann connection [1] on (M,F) which is invariant with respect to the action of H.

Let us denote by C0(M) the algebra consisting of continuous functions on M vanishing at infinity. The main object of our investigation is the algebra C0 (M)∣L obtained by restriction of C0(M) to a leaf of the foliation.

Recently the author proved that, under the assumptions given above, there exists an almost everywhere continuous bijection φ : M → P × H∕HP , where P is a connected total transversal tangent to the Ehresmann connection and HP = {h ∈ H∣hP = P}. The main problem is to describe C0(M)∣L for L ∈ F under different assumptions. We consider three distinct cases:

1) P∕HP is a Hausdorff space;

2) any leaf L ∈ F is dense on M, there exists an HP -invariant metric on the transversal P, and the isotropy groups of all the leaves coincide;

3) a base of HP is a contraction on P.

2. The first case.

Let us say that Bɛ → C0(M) as ɛ → 0 if for any f ∈ C0(M) and a sequence (ɛn)n∈ℕ ( ɛn → 0 as n →∞) there exists a sequence (fn), fn ∈ Bɛn such that fn → f in the usual ∥⋅∥0 or sup norm.

□ The first assumption implies that Bɛ,x → C0(M) as ɛ → 0 for all x ∈ P, otherwise P∕HP is not a T 1 -space and moreover not a Hausdorff one. From Corollary 27 [4] it follows that there exists a metric on P invariant with respect to the action of HP . (Our assertions imply that Q = P∕HP is a manifold with a boundary given by ⋃ iPi′. So we take a slice [4] σ : Q → P, such that σ is continuous on Q ∖ π(⋃ iPi′). The last condition of Corollary 27 [4] is met since the set P ∖⋃ iPi′ (P,Q ∖ π(⋃ iPi′),π) is a cover of Q with the covering group HP . Let us extend σ to ⋃ iPi′ by continuity. (In fact we get a multivalued map). Let us define a metric d on P as follows: On the factor Q d(x,y) can be an arbitrary one. Then we must put into consideration the set of continuous sections Σ = ⋃ h∈HPhσ. Let us point out that hσ ⋂ h′σ ⊂⋃ iPi′. Then we define metric on P by gluing metrics on the images hσ, i.e. for any h ∈ HP we put d(hx,hy) = d(x,y). Since the set I of disjunct continuous sections is at most countable one and ⋃ i∈Iσi(Q) = P this metric is a correctly defined one on the whole manifold P . This definition is correct since by construction metrics coincide on the borders of the sets. Thus we apply the construction from [4].)

Let us prove now that Bɛ → C0(M) ( ɛ → 0). The proof is by induction on the dimension of the subset of P . Let us first define the step of the induction as follows: fn (tγ) = f(γ) γ ∈ P′, t ∈ [0, 1∕n] from the normal foliation on U2∕n(⋃ iPi′), fn (tγ) = f(tγ), t ≥ 2∕n, fn (tγ) = at + b, t ∈ [1∕n, 2∕n], a = n(f(2γ∕n) − f(γ)), b = 2f(γ) − f(2γ∕n). In the zero-dimensional case (being the base of the induction) we apply the same construction as in the induction step. the first assumption implies that for any point y ∈ P ∖ P′ one can find such ɛ(y) > 0 that Bɛ(y),y = C0(M). Then by considering the continuity modulus of the function f we prove the theorem. ⊳

So later on in this paper we consider only cases in which Bɛ,x ⁄→ Br(G), ɛ → 0. In general, the situation here is rather complicated so we will consider the partial case when there exists a leaf L ∈ F such that L⋂ P¯ ∖ L⋂ P≠∅.

Then the following cases are possible:

1) Let Xh stand for the graph of the map h : P → P. We assume that the graphs Xh, h ∈ HP do not intersect each other (this is possible in case for example there exists a graph of a number of these maps in any neighbourhood of the diagonal graph D = ⋃ x∈M(x,x)).

2) The graphs of the maps generated by elements of HP do intersect in a point (x,x) ∈ D.

The graphs of the maps are uniformly separated in any other case, that is there exists ɛ > 0 such that Uɛ(Xi) ⋂ Uɛ(Xj) = ∅. Let us mention now that in this case we can apply the first part of Statement 1 and there exists a metric on P invariant under the action of the group HP (note that the converse is not true, e.g. any Kronecker foliation on the torus T2 ).

3. The second case.

□ Let us deform a metric ρ′ on P, so that for all n ∈ ℕ, ρ(x,fn(x)) = const(n). Then there exists x ∈ P such that ρ(fnk(x),fnk+1(x)) → 0, (k →∞), this implies that the graphs Γ(fn) are dense in the space P × P. Thus we will consider only the situation in which any leaf is everywhere dense because otherwise we can apply Statement 1.

Then we can consider only the case in which for any n ∈ ℤ, and for all x ∈ P, fn (x)≠x, otherwise for some fixed x ∈ P we can find a natural number k = inf{n ∈ ℕ∣fn(x) = x}. Fix the set of points X = {fn(x)} n∈ℤ. On the one hand X¯ = P but on the other hand fn(x) = flk+j(x) = fj(x), 0 ≤ j ≤ k − 1 is a finite set.

Now for a compact manifold P we take the following form as ρ:

here ρ′ is a nonsingular metric on P. Let us turn to the proof of metric axioms. Let us prove first that ρ(x,y) ≥ 0 for x≠y ∈ P. Assume the contrary. Then there exists a subset J of density 0 of the set ℕ such that the sequence ρ(fn(x),fn(y)) → 0 as n →∞,n ∈ ℕ ∖ J [2]. But then the compactness of P implies that there exists the point z ∈ P fnk (x),fnk(y) → z as k →∞, this contradicts the third condition of the assumption. The other metric axioms follow from the construction of ρ′ .

Let us prove now that the metric defined above is equivalent to the first one. Assume the contrary, i.e. there exist a ɛ > 0 and x ∈ P such that for all σ > 0 one can find y ∈ P ρ′ (x,y) < σ ρ(x,y) ≥ ɛ. Then again there exists a sequence fnk(x),fnk(y) such that ρ′(fnk(x),fnk(y)) ≥ ɛ. In case σ is sufficiently small and ɛ∕σ > δ we again arrive to the contradiction with the third assumption of the statement. The same considerations show us that the convergence yn →ρx implies that yn →ρ′x as n →∞. Let us again assume the contrary, i.e. there exists a sequence (yn )n∈ℕ yn →ρx but yn ⁄→ρ′x. We infer from the first convergence that for any ɛ > 0 there exist n(ɛ), l(ɛ) ∈ ℕ, ρ′ (fk(y n),fk(x)) < ɛ for ∀n ≥ n(ɛ) and k ∣k∣ > l(ɛ), of the density 1. Then again by the third assumption ρ′(y n,x) = ρ′(f−k ∘ fk(y n),f−k ∘ fk(x)) ≤ Dρ′(fk(y n),fk(x)) + o(ɛ) = o(ɛ) , this contradiction completes the proof.

Now let P be a noncompact manifold and the set of images fn (p) be dense in P , then any two points from A = (P∕ ∼), (here x ∼ y if and only if there exists L ∈ F such that the points x,y ∈ L) can not be separated. Let us construct the invariant metric on P as follows: Let ρ′ be a fixed metric on P . Fix a neighbourhood Uɛ (x) of a point x ∈ P. Let us now define the metric inside Uɛ(x) by putting for fn+k(x),fn(x) ∈ U ɛ(x)

Now since the set fn(p) is dense the sequence fn(U ɛ(x)) defines a locally finite atlas on P. So we define the metric on P as the image of the metric on Uɛ(x) under the actions of fn. Let us then glue metrics on the images of Uɛ (x) as it was done in the first statement for a Hausdorff factor P∕HP . Let us then make ɛ → 0 and consider the limit metric ρl. Note that ρl ≤ ρɛ ≤ ρɛ′ for 0 < ɛ < ɛ′. This makes the definition correct one.

Equivalence of the constructed metric to the given one holds true due to the third condition and the construction algorithm. The limit length for y → x is bounded from both sides: δρ(x,y) < ρl(x,y) < Δρ(x,y). ⊳

□ Let x ∈ P be such a point that (fn(x)) n∈ℤ is dense. Let there exist y ∈ P such that the set ⋃ n∈ℤfn(y) is not everywhere dense. Let fnk(x) → y be a subsequence of fn(x) then for arbitrary l ∈ ℤ fl ∘ fnk(x) → fl(y) ( nk →∞), moreover the union of all shifted subsequences coincides with the sequence fn (x). Then the limit point set of fn(x) must coincide with the same set of the sequence fn (y). This contradiction completes the proof. ⊳

In order to make condition 3) more evident we consider the following example:

□ First let us prove this for a 1-dimensional manifold P with a Riemannian metric. Let us consider a groupoid of geodesics on P . γ : [0, 1] → P, γ(0) = x, γ(1) = y, γ ∘ γ′ ≡ γ′′, for γ′ (0) = γ(1), γ′′ being the geodesic from γ(0) to γ′(1). In this case these rays are uniquely defined by source and range, so we get a groupoid structure on P × P.

Let us point out now that a metric ρ on P gives rise to a cocycle D : P˜ ×P˜ → ℝ [9, 11] defined on the product of the universal covering spaces of P . Since there exists a direction on the universal covering space of P we can correctly define D(x,y) = ρ(x,y) in case x < y and D(x,y) = −ρ(x,y) otherwise. Now, since ρ is not a singular metric, the cocycle D gives rise to an invariant measure on P˜ [9]. Since this measure is unique D must be equal to K for a K ∈ ℝ. This proves the statement in this case.

Now we turn to the general case. Fix the set of complete geodesic lines for any x ∈ P and any direction v ∈ TxX, γ : ℝ → P, γ(0) = x, dγ dt = v. Next we can construct for fk : P → P, k ∈ ℤ a pair of maps γ±′−1 ∘ f ∘ γ : ℝ → ℝ for any γ′ with γ±′(±t(k)) = f(x), where x = γ(0), and t(k) = ρ(x,fk(x)). This is true since the set of directions is compact. The set ⋃ k∈ℤfk(x) is everywhere dense on P, hence ⋃ k∈ℤt(k) is dense on ℝ. Thus we reduce the general case to the 1-dimensional one. From this follows the statement for the manifolds of dimension greater than 1. ⊳

Now the statements above provide us with the set of almost periodic transformations [3].

This homeomorphism can be constructed as follows. Let us take the set of graphs Γfn(P) ⊂ P × P. Let us deform all these graphs simultaneously. Let φ(fnk−nk−1(x)) ≡ y(n k − nk−1) ∈ P such that ρ(y,x) = ρ(fnk(x),y) (there is a wide range of choice of points that satisfy this property). Then we define φ(fn+nk−nk−1(x)) := z(n + n k − nk−1) such that ρ(z,x) = ρ(fn+nk−nk−1(x),fn(x)). The open problem here is whether there exists a map φ, which at the same time is a homeomorphism.

With the help of this homeomorphism one can improve the situation from the first example. Nevertheless the second example can not be improved in the same way.

So we consider only the possibilities given below:

1) There exists a metric ρ : P × P → ℝ+ and ɛ > 0 such that for all x ∈ P and k,l ∈ ℤ, k≠l, ρ(fl(x),f(k)(x)) ≥ ɛ. This situation was explored in [4] where P∕HP is a Hausdorff space. One can apply Statement 1.

2) There exists at least one point x ∈ P such that (fl(x)) l∈ℤ is not closed. Then we can apply Note 4 and — in partial cases — Statement 2 (namely the case of almost periodic transformations [3]). Here nevertheless we can find ourselves in the situation similar to one described in Example 2.

3) For any ɛ > 0 there exist p ∈ P and n,l ∈ ℤ such that

So we must find out what can be said about Br (G) under the specified restrictions. As was already mentioned the only interesting situations are 2) and 3).

□ This is a consequence of the almost periodicity of the considered set of functions on the leaves and the existence of the integrable Ehresmann connection on (M,F).

We must point out first that for any x ∈ P, limn→∞∑ i=1nf(a ix)∕n≠0 for f ∈ Br(M), μ(supp(f))≠0 by Poincaré theorem [2]. Thus we obtain the limit function fl (x) = lim n→∞∑ i=1nf(a ix)∕n and note that if fl(x)≠0, x ∈ satF′(supp(f)) = HP ⋅ supp(f) then the mean of the function M(fl) = lim k,l→∞,xl∈δ(S ⋂supp(f))1∕(kl) ∑ j,l=1j=k,l=mf(a jxl)≠0. ⊳

Hence we get the map M : P → (Br(G)∣L)′ which does not depend on the point p ∈ P. Moreover, almost periodicity provides us with the following

□ By general assumption there exists a sequence (fnk(x)) ⊂ P, fnk (x) → x, ( k →∞). Then, since g ∈ C0(M) is a continuous function, g(fnk(x)) → g(x), k →∞. The only thing left to prove is that these sequences coincide for all y ∈ P.

Since any leaf is dense and there exists an invariant metric, these sequences coincide for the everywhere dense set of points. Let fnk (y) ⁄→ y, k →∞. Then there exists fnk(y) → y ( k →∞) because otherwise Dv fnk are not bounded from above. ⊳

Let us describe the algebra C0(M)∣L in this case.

□ Let x ∈ P then for all ɛ > 0 there exists y(ɛ) ∈ Uɛ(x) ∖{x}. For hm (ɛ) = min{h ∈ HP ∣y = hx} we can cover a manifold M by varieties {hUɛ(x)∣h ∈ H,h ≥ hm}. It is clear that hm(ɛ) →∞ as ɛ → 0. Consider the continuity modulus Δf of the function f of C0 (M)∣L. Then by general assumption for all x ∈ M, ∣f(hx) − f(x)∣ < Δf(ɛ) if d(x,hx) < ɛ. For any Δ > 0, there exist ɛ > 0 and h ∈ HP . ⊳

The last case is then the worst to explore: there is no leaf with the property similar to one described above, since here we have only sequence of leaves such that there are leaves with points infinitely close one to another.

4. The third case.

Let there exist a point x ∈ P such that Bɛ,x ⁄→ Br, and the leaf passing through x is compact. Assume also the following: for any p ∈ P, p≠x there exists a neighbourhood U(p) ⊂ P of the point p such that HU(p) = HL(p). Thus we assume that the equivalence relation induced on P i by HPi is topological [13]. Let us suppose also that for any point p ∈ P there exists a basis (a1,…,an) of HP such that aik(p) → x as k → +∞. Thus the set HP is a contraction [8] (each ai, i = 1,n¯ is a contraction of P with the common accumulation point x).

Thus we consider the situation in which the accumulation point belongs to the diagonal D = {(x,x)∣x ∈ X}⊂ X × X and is separated from the other similar points.

□ The inclusion C0,F (M) ⊂ CLx is evident. The converse follows from the consideration of the modulus of continuity.

The ideal property follows from the definition of the C0,F (M). ⊳

Thus we get C0(M)∕C0,F (M)≅C(Lx). Let us take into consideration the following subspace of the almost periodic functions in Bohr sense: V τ = {f ∈ C(X)∣∣f(nτ + x) − f((n + 1)τ + x)∣→ 0(n → +∞)}. We must span the following set of vectors fn = 0, if x < 0; xeinτx,if 0 ≤ x < 1; einτx, if x ≥ 1 n ∈ ℕas a subspace of V τ . For the mean we take the limit Mmn(f(x)) = ∑ i=1n ∑ j=1mf(a iγj)∕mn ( ai ∈ Hx, γj ∈ Uj). One can induce the scalar product on V τ with the help of this mean. This scalar product will be correctly defined on V ′ = span(⋃ n∈ℕfn). Then the following statement holds true on any leaf of the foliation:

□ For any ɛ > 0 let us find an element g ∈ V ′ such that M(∣f(x) − g(x))∣) < ɛ. To do this, first one must find h(x) = lim n→∞f(x + nτ) which exists by the general assumption on the structure of (M,F). Let us then construct the function g(x + kτ) = h(x). Since g is a periodic function, it is enough to approximate it on one period by elements from V ′ ∣ [0,τ], but this is possible since V ′ is dense in C([0,τ]). ⊳

□ To construct the homomorphism φ : C0(M)∕C0,F (M) → C0(M) one can take the following function:

Here t = ∑ i=1dim(L)t i, ti are the natural coordinates on H ≃ ℝn, θ is the radius coordinate on Δ = P ∖{x}∕HP ≃ Sdim(P)−1 × [0, 1], and p is the coordinate on Sdim(P)−1. The last diffeomorphism exists due to the assumptions on the equivalence relation on P. ⊳

Thus V τ = C0(X) ⊕ C(S).

Fix any f ∈ Br ∖ B0.

Let Hx be an isotropy subgroup of the leaf passing through the point x and {Uj} be a ɛ-scattering of L defined by f [10].

□ The proof follows from the fact that in this case there exists at least one element h ∈ HP that gives rise to a contraction in a neighbourhood of the singular point x ∈ P. Moreover, the assumptions of this statement imply that there are no periodic or almost periodic functions in the considered class, this gives us the second part of the proposition. So any continuous function must satisfy: for all y ∈ P limn→∞f(hny) = f(x) implies that for all g ∈ S = H∕HP , limn→∞f(hngy) = f(gx).

Let us point out that the mean defined above coincides with the mean defined in [10] for the set of continuous uniform almost periodic functions. This is a consequence of

for ai, aij , i = 1,n¯, j = 1,k¯, being elements from an ɛ-scattering of the leaf L, i.e. for all j ∈{1,…,k}, i ∈{1,…,n} ∣fi − f(aij)∣≤ ɛ, ∣fi − f(ai)∣≤ ɛ [10]. Then we must pass to the limit as k, n →∞ to obtain the result. ⊳

5. Measures and means.

Let us point out the natural connection between our problem and the so-called Radon-Nikodym problem on the leaves of the foliation. We can classify the equivalent, in general non-Borel invariant probability measures. Assume that the equivalence relation defined on M is generalized ( γ ≡ alγ) and extended (γ ≡ γ′ ↔ s(γ) = hs(γ′), r(γ) = h′r(γ′)) to the trivial groupoid G = L × L of the leaf L. Let us then assume that the Radon-Nikodym derivative D(x,y) = D(x,y) + D(y,z) of the given measure belongs to the set of functions on the trivial groupoids of the leaves and the corresponding measure is quasiinvariant. So in the first case D(x,y) → 0 as x,y →∞. D(ankx,anky) → D(x,y) ( nk →∞) in the second case, and in the third case there exists c ∈ ℝ, D(anx,any) → c ( n →∞). The measure can be a Borel probability measure only in the first case and in the third case when c = 0. In the first case we can naturally induce a measure μ(E) on the manifold M as an integral of the measures νp(E ⋂ Lp) over P∕HP .

Since the first case is trivial let us turn to the other cases.

The second case. Fix the cocycle D such that

As D(x,ankx) → D(x,x) = 1, this D is a uniform almost periodic function with respect to each variable.

□ 1) Fix y ∈ H and take My(D(x,y)) as p(x). The correctness of this definition follows from the properties of D:

2) By assumption, the potential constructed in the first part is bounded from above and below by constants, from this follows the result. ⊳

So it seems possible to define a measure on the manifold μ(E) as an integral over P of the mean E ⋂ Lp, here Lp is a leaf passing through p ∈ P. Later on we will show that it suffices in some cases to take only one leaf. Conversely, any measure on M invariant under HP gives rise to a quasiinvariant measure on any leaf L ∈ F: Let us consider λ(E1) = μ(E)∕ν(E2) for an HP -invariant transverse measure ν on P and E = E1 × E2, E1 ⊂ S, E2 ⊂ S in the decomposition φ : M ↔ S × P [14].

□ First let us prove this fact for the sets E1 × E2, where E1 ⊂ P, E2 ⊂ σ(H∕HP ) ⊂ H. The main problem here is to show that M(E) > 0 in case μ(E) > 0. Let us consider lim N→∞1∕(2N) ∫ −NNI Edλ = lim N→∞KN 2N IE, where KN is the number of returns from a point x ∈ E1 to E1 on the set [−N,N] and IE is the restriction of the characteristic function of E to the arbitrary leaf L. Since HP generates an ergodic transformation on P, the Kats theorem [2] implies that lim N→∞KN 2N →μ(E1) μ(P) . This completes the proof since we get a measure, which is correctly defined for simple functions on M. ⊳

To get a measure of any measurable subset E of M one can’t consider an arbitrary leaf as a support of the restriction of the characteristic function IE , nevertheless, since M ≃ P × S almost every leaf will serve as the target one.

It seems that this statement is true in general case. To prove it one should find a generalization of Kats theorem.

So, to get a measure on the closed subsets of M one can take only an HP -invariant measure on the leaf and consider the standard mean with respect to this measure.

The third case. Any f ∈ V τ defines two measures: the periodic measure on the summand Cp (S) and the ordinary measure on the algebra C0(L) for all L ∈ F.

Both of them are rather well investigated. The classification problem for each of them was solved by the Radon-Nikodym theorem. The problem here is to classify the sum of these objects. We must turn again to the set of functions on the groupoid of the foliation. Since we must consider ordinary measures with the multiplicative law of substantiation the Radon-Nikodym derivatives of the quasiinvariant measures on the equivalence relation [11] must satisfy the condition limx,y→+∞D(x,y) = 1. Then as in the previous statement we can prove the existence of a potential ρ of the probability measure (sum of the objects defined above) by passing to the mean p(x) = My′(D(x,y)) = lim T→+∞∫ 0T D(x,t)dt. Note that this potential satisfies the following condition: limn→∞1∕n∑ i=1nρ(aix) = 1, a ∈ HP . Let us define the conditional expectation ED : C(L∕HP ) → ℝ corresponding to ρ as ED(f)(ω) = lim n→∞1∕n∑ i=1nρ(aiσ(ω))f(aiσ(ω)).