Transverse Hausdorff dimension of codim-1 C 2-foliations

149 (1996)

Transverse Hausdorff dimension of codim-1 C ² -foliations

by

Takashi I n a b a (Chiba) and Paweł W a l c z a k (Łódź)

Abstract. The Hausdorff dimension of the holonomy pseudogroup of a codimension- one foliation F is shown to coincide with the Hausdorff dimension of the space of compact leaves (traced on a complete transversal) when F is non-minimal, and to be equal to zero when F is minimal with non-trivial leaf holonomy.

1. Introduction. In [Wa], the second author introduced the notion of a Hausdorff dimension dim H G for finitely generated locally Lipschitz pseu- dogroups G acting on compact metric spaces X. Recall that

(1) dim _H G = inf{s > 0 : H ^s (G) = 0} = sup{s > 0 : H ^s (G) = ∞}, where

H ^s (G) = lim

ε→0 H _ε ^s (G), (2)

H _ε ^s (G) = inf{H _s (A) : A ∈ A(ε)}, (3)

H _s (A) = X

g∈A

(diam D _g ) ^s , (4)

D _g stands for the domain of the map g ∈ G and A(ε) denotes the family of all finite sets generating G and consisting of maps with domains of diameter less than ε.

Note that our definitions are analogous to those involved in defining the Hausdorff dimension of metric spaces (see [Ed], for example). In particular,

(5) dim H X = dim H G(id X ),

where G(f ₁ , . . . , f _n ) is the pseudogroup generated by the maps f ₁ , . . . , f _n . The dimension dim H has the following properties (see [Wa]):

(i) dim H G ≤ dim H X,

1991 Mathematics Subject Classification: Primary 57R30.

The second author was partially supported by the KBN grant no 2P30103604.

[239]

(ii) dim H G 1 = dim H G 2 when the pseudogroups G 1 and G 2 are Lipschitz equivalent,

(iii) dim H G ⁰ ≥ dim H G when G ⁰ is a subpseudogroup of G,

(iv) dim H (G|Y ) ≤ dim H G when Y is a closed G-invariant subset of X and G|Y denotes the pseudogroup generated by the maps g|Y , g ∈ G,

(v) dim _H G ≥ s if G preserves a Borel probability measure µ on X which is s-continuous, i.e. satisfies the condition

(6) µ(Z) ≤ c(diam Z) ^s

for all Borel subsets Z of X and a positive constant c independent of Z, (vi) dim H G = dim H X when X is a Riemannian manifold and G a pseu- dogroup of local isometries of X.

Property (ii) implies that the Hausdorff dimension of the holonomy pseu- dogroup H T of a C ¹ -foliation F of a compact manifold M is independent of the choice of a compact complete transversal T . Therefore, the transverse Hausdorff dimension dim ^t _H F of F can be defined by

(7) dim ^t _H F = dim _H H _T ,

T being any transversal as above.

In this article, we compute the transverse Hausdorff dimension for codim- ension-one C ² -foliations of compact manifolds (Section 2) and collect some examples which show that the assumptions of our Theorem are essential (Section 3).

2. Main results. Let F be a transversely oriented codimension-1 C ² - foliation of a compact manifold M .

Theorem. (i) If F is not minimal, then dim ^t _H F = dim H (T ∩ C(F)), where T is a compact complete transversal and C(F) is the union of all the compact leaves.

(ii) If F is minimal and has non-trivial holonomy, then dim ^t _H F = 0.

Let us explain that F is minimal if its leaves are dense in M ; its holonomy is non-trivial when the germ holonomy group of some leaf L is non-trivial.

P r o o f. (i) The inequality

dim ^t _H F ≥ dim H (T ∩ C(F))

follows directly from Property (iv) (Section 1) and the fact that the holon- omy of F|C(F) is trivial.

To prove the converse, fix s > dim H (T ∩C(F)), η > 0 and ε > 0. Observe

that the center Z(F) = C(F)∪E ₁ ∪. . .∪E _m is compact and contains finitely

many exceptional minimal sets E i ([La], [HH], etc.). Moreover, every leaf L

of F satisfies L ∩ Z(F) 6= ∅.

First, the Sacksteder Theorem [Sa] allows us to choose points x i ∈ T ∩E i

(i = 1, . . . , m) and holonomy maps h _i contracting some neighbourhoods J _i ⊂ T of x _i .

Next, since C(F) is compact, we can cover it by a finite number of mutually disjoint foliated I-bundles (possibly reducing to single isolated leaves) C ₁ , . . . , C _n bounded by closed semi-isolated leaves L ⁺ _i and L ⁻ _i . Let I j = [x j , y j ] ⊂ T , j = 1, . . . , n, be the fibres of C j . Extend each of the intervals I _j slightly to get larger intervals I _j ⁰ = [x ⁰ _j , y _j ⁰ ] such that the segments [x ⁰ _j , x _j ] and [y _j , y ⁰ _j ] are attracted to x _j and y _j by the global holonomy groups Γ j of the foliated bundles F|C j . More precisely, for any j = 1, . . . , n and δ > 0 there should exist h, h ⁰ ∈ Γ _j which extend to holonomy maps (denoted by h and h ⁰ again) defined on I _j ⁰ and bringing x ⁰ _j (resp., y _j ⁰ ) to within distance δ of x j (resp., y j ).

Set

T ⁰ = [ m i=1

J i ∪ [ n j=1

I _j ⁰⁰ ,

where I _j ⁰⁰ = [x ⁰⁰ _j , y ⁰⁰ _j ] for some x ⁰⁰ _j ∈ (x ⁰ _j , x j ) and y _j ⁰⁰ ∈ (y j , y _j ⁰ ) such that the intervals J _i and I _j ⁰⁰ remain mutually disjoint. Obviously, T ⁰ is a complete transversal for F.

For any j, fix a finite symmetric set Γ _j ⁰ generating Γ j . Shrinking the intervals I _j ⁰ if necessary we may assume that all the maps of Γ _j ⁰ extend to holonomy maps defined on I _j ⁰ . Let H ⁰ be the subpseudogroup of H _T

generated by the contractions h i , i = 1, . . . , m, and the (extended to I _j ⁰⁰ ) holonomy maps of Γ _j ⁰ , j = 1, . . . , n.

Now, cover T ⁰ ∩ C(F) by intervals K 1 , . . . , K N with endpoints in C(F) and lengths l(K _i ) < ε, and such that

X

i

l(K _i ) ^s < η.

(This is possible since s > dim _H (T ⁰ ∩ C(F)).)

Put δ = ¹ ₂ ε min{1, l(K ₁ ), . . . , l(K _N )}. For any j choose holonomy maps f j , f _j ⁰ ∈ Γ j which extend to I _j ⁰ and bring x ⁰ _j (resp., y _j ⁰ ) to within distance δ of x _j (resp., y _j ). Also, for any gap (i.e., a connected component) U = (a, b) of I _j \ S

i K _i choose points c, d ∈ U , c < d, and holonomy maps f _U , f _U ⁰ ∈ Γ _j which bring c (resp., d) to within distance δ of b (resp., a).

Now, if K i = [α, β] ⊂ (x j , y j ), take the gaps U and U ⁰ for which α ∈ U and β ∈ U ⁰ and let

A i = {g|K2 i , g ◦ f _U ⁻¹ |V, g ◦ (f _U ⁰

) ⁻¹ |V ⁰ : g ∈ Γ _j ⁰ },

where V and V ⁰ are the δ-neighbourhoods of α and β, respectively. If K _i =

[x j , β] ⊂ I j , choose U ⁰ , f _U ⁰

and V ⁰ as before, and let

A i = {g|K2 i , g ◦ (f _U ⁰

) ⁻¹ |V ⁰ , g ◦ f _j ⁻¹ |W j : g ∈ Γ _j ⁰ },

where W j is the δ-neighbourhood of x j . Define A i similarly in the case when K _i = [α, y _j ].

Finally, for any i = 1, . . . , m choose an exponent n _i ∈ N such that the image of J i under h ⁿ _i

has diameter less than ε and let

A = {h ⁻ⁿ ₁

, h ⁻⁽ⁿ ₁

⁺¹⁾ , . . . , h ⁻ⁿ _m

, h ⁻⁽ⁿ _m

⁺¹⁾ } ∪ A ₁ ∪ . . . ∪ A _N .

The set A generates H ⁰ , the diameters of the domains of maps in A are bounded by ε and

H _ε ^s (H ⁰ ) ≤ H s (A) ≤ 2mε ^s + max

j #Γ _j ⁰ · (1 + 2ε ^s )η.

Consequently, H ^s (H ⁰ ) < ∞, s ≥ dim _H H ⁰ and dim _H H ⁰ ≤ dim _H (T ∩ C(F)). Moreover, by Property (iii), dim ^t _H F = dim H H T

≤ dim H H ⁰ .

(ii) The proof is essentially the same. One has to take as T ⁰ any segment transverse to F and short enough to be attracted to a point x 0 by the holonomy of the leaf L x

. For any symmetric set A 0 generating H T

and two holonomy maps h ₁ and h ₂ attracting the endpoints of T ⁰ to within distance ε of x 0 and satisfying the condition diam R h

< 2ε, the set

{g ◦ h ⁻¹ ₁ , g ◦ h ⁻¹ ₂ : g ∈ A 0 } generates H T

and satisfies the inequality

H s (A) ≤ #A 0 · (2ε) ^s for any s and ε > 0.

3. Examples. First, we will discuss the case of minimal foliations with- out holonomy.

Example 1. If F is the suspension of an irrational rotation of S ¹ , then F is a Riemannian foliation of the 2-dimensional torus T ² and therefore (Property (vi) of Section 1) dim ^t _H F = 1. Obviously, F is minimal and has no holonomy.

Example 2. In [Ar], one can find a construction of a sequence (f _k ) of analytic diffeomorphisms of S ¹ with the following properties:

(1) |f k (z) − f k+1 (z)| < δ k ,

(2) the rotation numbers %(f _k ) are rational, %(f _k ) = p _k /q _k , and satisfy the inequalities

    p k

q k − p k+1

q k+1

  < 1 (k − 1) ² (max

l<k q l ) ² ,

(3) any map f k is forward semi-stable and has a unique cycle z k,1 , . . . . . . , z _k,q

(of length q _k ),

(4) there exist exponents N (k) ∈ N such that f _k ^{N (k)} (S ¹ \ U _k ) ⊂ U _k ,

where U k = U _k ¹ ∪ . . . ∪ U _k ^q

and U _k ^j is the ε k -neighbourhood of z k,j .

In the above, (δ _k ) and (ε _k ) are sequences of positive numbers converging to 0 sufficiently fast as k → ∞, so we may assume that

(8) ε k ≤ q ^−1/s _k

,

where s _k is an a priori given sequence of positive reals converging to 0.

From properties (1)–(4) above it follows that the limit f = lim _k→∞ f _k exists, is analytic and has an irrational rotation number %(f ). Therefore, the suspension F of f provides us with an analytic minimal foliation without holonomy. Its holonomy pseudogroup is isomorphic to G = G(f ), the pseu- dogroup of local diffeomorphisms of S ¹ generated by f . Lemma γ of [Ar]

shows that if δ _k ’s are small enough, then

(9) f ^{N (k)} (S ¹ \ e U k ) ⊂ e U k , k = 1, 2, . . . , where e U _k = S

j U e _k ^j and e U _k ^j is the ε _k -neighbourhood of U _k ^j , j = 1, . . . , q _k . From (9) it follows that each of the sets

A _k = {f ^{−N (k)} | e U _k ^j , f ^{−(N (k)+1)} | e U _k ^j , f | e U _k ^j : j = 1, . . . , q _k }, k = 1, 2, . . . , generates G. Obviously, A _k ∈ A(4ε _k ) and H _s

(A _k ) = 3q _k (4ε _k ) ^s

. From (8) it follows that for any k > k 0 (k 0 ∈ N) we have

H _4ε ^s

≤ 3q _k (4ε _k ) ^s

≤ 3 · 4 ^s

.

Consequently, H ^s

(G) < ∞ and dim H G ≤ s k

for any k 0 . Finally, dim ^t _H F = dim _H G = 0.

Examples 1 and 2 provide minimal foliations without holonomy with transverse Hausdorff dimension equal to, respectively, 0 and 1. It seems to us that Arnold’s construction cannot be modified to get a similar foliation with dim ^t _H ∈ (0, 1). So, one could search either for other examples of this sort or for the proof of the following: If F is minimal, C ² -differentiable and has no holonomy, then either dim ^t _H F = 0 or dim ^t _H F = 1.

The following example shows that the assumption of C ² -differentiability in our Theorem is essential.

Example 3. Let f : S ¹ → S ¹ be the classical Denjoy C ¹ -diffeomorphism

constructed as in, for instance, [Ta]. Then S ¹ contains a minimal closed

invariant set X such that f |X preserves the 1-dimensional Lebesgue measure

λ (f ⁰ = 1 on X) and λ(X) > 0. Moreover, λ is 1-continuous, and therefore, by Properties (iv) and (v) of Section 1, we get

1 ≥ dim H G(f ) ≥ dim H G(f |X 0 ) ≥ dim H X 0 = 1.

Suspending f we arrive at a non-minimal C ¹ -foliation F which has no compact leaves but has transverse Hausdorff dimension 1.

R e m a r k. The arguments in Example 3 do not work when λ(X) = 0, X being the minimal set of a Denjoy diffeomorphism f . Examples of such diffeomorphisms are provided in [He], Section X.3. It would be interesting to calculate (or estimate) dim _H G(f ) for such an f . Is it possible to find a Denjoy diffeomorphism f for which 0 < dim _H G(f ) < 1?

References

[Ar] V. I. A r n o l d, Small denominators. I. Mappings of the circumference onto itself , Izv. Akad. Nauk SSSR Ser. Mat. 25 (1962), 21–86 (in Russian); English transl.:

Amer. Math. Soc. Transl. 46 (1965), 213–284.

[Ed] G. A. E d g a r, Measure, Topology and Fractal Geometry, Undergrad. Texts in Math., Springer, New York, 1990.

[HH] G. H e c t o r and U. H i r s c h, Introduction to the Geometry of Foliations, Part B, Vieweg, Braunschweig, 1983.

[He] M. H e r m a n, Sur la conjugaison diff´erentiable des diff´eomorphismes du cercle `a des rotations, Publ. Math. IHES 49 (1979), 1–233.

[La] C. L a m o u r e u x, Quelques conditions d’existence de feuilles compactes, Ann. Inst.

Fourier (Grenoble) 24 (4) (1974), 229–240.

[Sa] R. S a c k s t e d e r, Foliations and pseudogroups, Amer. J. Math. 87 (1965), 79–102.

[Ta] I. T a m u r a, Topology of Foliations: An Introduction, Amer. Math. Soc., Provi- dence, 1992.

[Wa] P. W a l c z a k, Losing Hausdorff dimension while generating pseudogroups, this is- sue, 211–237.

Department of Mathematics and Informatics Institute of Mathematics

Chiba University Łódź University

Chiba, Japan Banacha 22

E-mail: inaba@math.s.chiba-u.ac.jp 90-238 Łódź, Poland

E-mail: pawelwal@krysia.uni.lodz.pl Received 28 February 1995;

in revised form 16 October 1995