© 2004, Sociedade Brasileira de Matemática
On the dynamics of generic non-Abelian free actions
Julio C. Rebelo
Abstract. We investigate some global generic properties of the dynamics associated to non-Abelian free actions in certain special cases. The main properties considered in this paper are related to the existence of dense orbits, to ergodicity and to topological rigidity.
We first deal with them in the case of conservative homeomorphisms of a manifold and C1-diffeomorphisms of a surface. Groups of analytic diffeomorphisms of a manifold which, in addition, contain a Morse-Smale element and possess a generating set close to the identity are considered as well. From our discussion we also derive the existence of a rigidity phenomenon for groups of skew-products which is opposed to the phenomenon present in Furstenberg’s celebrated example of a minimal diffeomorphism that is not ergodic (cf. [Ma]).
Keywords: free groups, dense orbits, vector fields.
Mathematical subject classification: 37A15, 37B05, 37C20.
1 Introduction
The dynamics generated by one “generic” diffeomorphisms on a compact mani- foldMis an intensively studied subject. This study is basically divided into the case of generalCr-diffeomorphisms (r =0,1, . . . ,∞, ω) and the case of dif- feomorphisms preserving a fixed volume onM(some works are also devoted to symplectic diffeomorphisms). Whereas this is undoubtedly an important topic, it cannot reflect the typical behavior of more general dynamics as already pointed out by Gromov [Gr1] and explained below.
A dynamical system should be understood as agroup actioni.e. it is obtained through a faithful representation of an abstract groupGinto Diffr(M). Therefore the traditional case of the dynamics generated by one diffeomorphisms corre- sponds to aZ-action. In this broader sense, it soon becomes clear that there
Received 12 November 2003.
are many more dynamical systems (i.e. group actions) than those associated to Z-actions which suggests that the typical features ofZ-actions may be different from the features of group actions “chosen at random”. The present work is a first attempt of making sense and investigating some of the generic properties of a “dynamics” in this general sense.
In this paper we focus attention on non-Abelian free group actions which we believe are likely to lead to a fair idea of typical dynamics in our sense. In fact, there are at least three reasons to consider these actions. First, as observed in Section 5, a collection of diffeomorphisms “randomly chosen” generates a non-Abelian free group. Secondly, considering the space of abstract groups, one can wonder which are the “generic properties” of a group, so as to try to find good representatives for this space. This question was first considered in [Gr2]
(see also [Ch]) and it turned out that, in a reasonable sense, a typical group is word-hyperbolic (but it is not a finite extension ofZwhich also confirms that Z is too particular as a group). It is then natural to consider a non-Abelian free group as a good representative of the space of groups which can act on a manifold since these hyperbolic groups always contain non-Abelian free groups on (say) two generators (conversely a free group on two or more generators is clearly word-hyperbolic, cf. [Gr2]). At last, dealing with free groups, we do not become involved with “functional identities” coming from relations in the group which would quickly put the problem out of reach.
In view of the preceding, we shall be concerned with non-Abelian free actions.
Among the several dynamical features which could be analysed, we concentrate our efforts on 3 aspects of global nature: density of orbits, ergodicity and struc- tural stability. As it will be clear after Section 2, the investigation of these questions depend on the nature of the generators (i.e. they depend on specific open sets of Diffr(M)to which the generators are supposed to belong).
This work consists of 2 parts which are rather different in their settings. How- ever these two settings are both natural, at least for the beginning of such discus- sion. The first part deals with volume-preserving diffeomorphisms and requires low-regularity for the diffeomorphisms involved. The second part requires real analycity of these diffeomorphisms, is not volume-preserving and the genera- tors of the group in question are supposed to be “close to the identity”. For these analytic actions, we shall establish a phenomenon of rigidity which may be thought of as being “opposite” to the structural stability. Actually this phe- nomenon seems to appear with significative frequency for actions of free groups while it does not take place forZ-actions. Curiously enough key arguments of these parts have some similarity between them; in fact they are both based on
controlling the dynamics on a small region by comparing it with a vector field.
We then try to globalize this “local” dynamics as far as it is possible. At the end of this article we also raise a few questions related to our main results.
2 Statement of results
Let us begin by describing an example of a free action which will imply that most assumptions in our statements cannot be dropped. Letbe the free group gener- ated by two M ¨œbius transformationsM1,M2where the projective action ofM1
(resp.M2) on the Riemann sphereCP (1)S2is given by a “south pole - north pole” (resp. “west pole - east pole”) diffeomorphism. WhenM1,M2are chosen so as to define a Schottky group, the action ofonS2(CP (1)) possesses an invariant Cantor set (which coincides with the Limit set of). Furthermore, this action is structurally stable: iff1(resp. f2) is aC1-diffeomorphism ofS2which isC1-close to the diffeomorphism induced byM1(resp. M2), then there is a homeomorphismh:S2→S2such thatf1◦h=h◦f1andf2◦h=h◦f2. In other words, the group generated byf1, f2is topologically conjugate to.
In our example, the diffeomorphisms induced byM1,M2are Morse-Smale diffeomorphisms whose individual dynamics is very simple: apart from the repelling fixed point, all points converge to the attracting fixed point. In particular the dynamics ofM1(resp. M2) is “wandering” i.e. it does not have any non- trivial recurrence. Also these diffeomorphisms possess attractors and do not preserve any volume measure.
Now assume that M is a compact manifold endowed with a finite volume measureµ (i.e. µ is obtained through a volume form). In the case of home- omorphisms, a volume measure means a measure topologically conjugate to the measure associated to a volume form. These measures are characterized by a theorem due to Oxtobi and Ulam which asserts that a Borel measureµis topologically conjugate to a volume measure if and only if it has no atoms and is positive on open sets (cf. [O-U]). Furthermore let Homeoµ(M)denote the group of homeomorphisms of M preserving µ (in other words a homeomor- phismh:M →Mbelongs to Homeoµ(M)if and only ifµ(B)=µ(h(B))for every Borel setB⊆M). Also consider the (abstract) free groupF (a1, . . . , ar) generated by the symbolsa1, . . . , ar. Similarly we define the group Diff1µ(S) ofC1diffeomorphisms ofMpreservingµ(note that we also use the word “con- servative” to refer to the tranformations which preserveµ).
It is easy to see that there cannot exist a faithful representation ρ from F (a1, . . . , ar) (r ≥ 2) to Diff1µ(S) which is structurally stable: note that Poincaré’s Recurrence Lemma implies that the non-wandering set of eachρ(ai)
(i = 1, . . . , r) coincides with the entire M. Thus the conservative version of Pugh’sC1-Closing Lemma (cf. [P-R]) allows us to approximate the original action (induced byρ) by an action such that all theρ(ai)’s (i =1, . . . , r) share a common periodic point. On the other hand, the usual transversality arguments show that the original action may also be approximated by actions for which the stabilizer of every point is either trivial or infinite cyclic. It follows the non-existence of the structurally stable representation in question.
However there are more subtle questions about the “generic” dynamics of these groups. A classical problem in Ergodic Theory is to decide whether or not ergodic diffeomorphisms (resp. homeomorphisms) are generic in Diffrµ(M) (resp. Homeoµ(M)). At this level of generality, very little is known about this question. One of the by-products of KAM theory (Rüssmann’s theorem [Ru]) asserts that ergodic diffeomorphisms of a surface are not generic in class Cr, r ≥4. On the other hand, a deep theorem due to Oxtobi and Ulam establishes that, in an arbitrary compact manifold, ergodic homeomorphisms are generic in classC0. For non-Abelian free subgroups, we can obtain a slight improvement of the latter theorem namely
Theorem A. Assume that the dimension ofMis different from4. There exists a residual (i.e. denseGδ) setU ⊂ Homeoµ(M)×Homeoµ(M)such that, if (f1, f2) is inU, then the action of F (a1, a2)defined by ai → fi (i = 1,2,) possesses the following properties:
a) It is minimal (i.e. the orbit of any pointp∈Mis dense).
b) µis the unique measure simultaneously preserved byf1, f2.
Remark. The assumption dimM=4 is required by the “Approximation The- orem” (cf. Section 3), but this is an inessential difficulty: it is actually possible to verify the statement even for 4-dimensional manifolds (see comments in [M-P-V]
or cf. [O-U]).
The case of Cr (r ≥ 1) diffeomorphisms is much harder. Recall that KAM theory implies, in particular, that conservative ergodic diffeomorphisms of a sur- face are not dense forr ≥3. This is, indeed, a consequence of the existence of invariant curves for the twist map. However, in the case of non-Abelian actions, the preceding construction does not allow us to conclude that ergodic diffeomor- phisms are not dense since the mentioned curves are not simultaneously invariant by a pair of generic diffeomorphisms. On the other hand, the “topological ana- logue” of ergodicity, namely the existence of dense orbits, can be verified for
surfaces. It is also possible to establish the existence of an ergodic component with positive measure. More precisely, using a result due to Newhouse, we shall prove the following theorem:
Theorem B. LetSbe a compact surface endowed with aC1-area measureµ.
Denote byDiff1µ(S) the group ofC1-diffeomorphisms ofS which preserve µ.
Assume that we are given three diffeomorphismsf1, f2, f3∈Diff1µ(S)such that at least one among them is not Anosov. Then, arbitrarilyC1-close tof1, f2, f3, there exist diffeomorphismsfˆ1,fˆ2,fˆ3∈ Diff1µ(S)which generate a groupG⊂ Diff1µ(S)having the following property: there exists an open dense setV ⊂ S invariant underGand such that the action ofGrestricted toV is minimal and ergodic (with respect the normalized measure onV).
Of course the main defficiency of Theorem B above is the fact that we do not know whether or not one can always takeV =S. Actually it is not even clear if µ(V )=µ(S).
Since Anosov diffeomorphisms have dense orbits, we deduce Corollary C.
Corollary C. There exists a residual setU⊂Diff1µ(S)×Diff1µ(S)such that, if(f1, f2)belongs toU, then the action ofF (a1, a2)defined by sendingai →fi
(i =1,2) has a dense orbit.1
On the other hand, one can ask if the property of having a dense orbit is
“generic” to non-Abelian free actions whether or not they preserve a volume.
This is however not the case as shown by the example of the Schottky group discussed above. In fact, since the action of a Kleinian group on the complement of its limit set is properly discontinuous, this action has no dense orbit. Further- more the fact that this action is structural stable implies that no dense orbit can be produced by a perturbation of the generators.
As mentioned in the Introduction, the second part of this paper is devoted to actions admitting a finite set of generators which are close to the identity.
Here we leave the low-regularity setting required by Theorems A and B and work in the world of real analytic diffeomorphisms. For dimensions greater than one the main available tool to study these actions is, to the best of my knowledge, Proposition (2.1) below which was obtained with F. Loray in [L-R]
(Proposition 4.6 of [L-R]).
1added in the proofs: recently Bonatti and Crovisier [B-C] have proved the existence of a dense orbit for a single generic conservative diffeomorphism of a manifold; their methods however do not yield the existence of an ergodic component with positive measure.
Denote byBn⊂Cnthe unit ball ofCn. Consider a pseudogroupconsisting of holomorphic mapsfrom open subset ofCntoCnwhich satisfies the assumptions below.
a) There exists a sequence {hi} ⊂ , hi = id for every i ∈ N, whose elements are defined on the entireBnand, in addition, converge uniformly to the identity onBn.
b) contains a homothetyF :Bn→Bnwhich is given byF (z1, . . . , zn)= (λ1z1, . . . , λnzn)whereλi ∈Rand 0<|λ1|≤ · · · ≤|λn |<1.
Proposition 2.1. ([L-R]) Assume we are givenas above. Then there exists ε >0and a non-trivial real analytic vector fieldXdefined onB(ε)(the ball of radiusεand center at the origin) which possesses the following property: given a relatively compact open setV ⊂B(ε)andt0∈R+such thattXis defined on V whenever0≤t ≤t0, the maptX0 :V →tX0(V )is uniformly approximated onV by elements in.
Note that the vector fieldX above can, in fact, be defined on the wholeBn thanks to the existence of a homothety in the pseudogroup. We also point out that, ifV ⊂ B(ε) andt0 > 0 are as in the statement, then the local flow tX : V → tX(V ) is uniformly approximated on V by elements in for every 0 ≤ t < t0. We shall say that a vector fieldX possessing the property stated in Proposition (2.1) isin the closure of relative toV. In practice the pseudogroup is generated by the restrictions of elements in G to an open set ofM. This proposition becomes very effective when the groupGcontains a Morse-Smale (or gradient-like) diffeomorphism. In particular, denoting by V ⊂Diffω(M)×Diffω(M)the set of the pairsf1, f2in Diffω(M)such that the group generated byf1, f2contains a Morse-Smale diffeomorphism, we have:
Theorem D. LetMbe an analytic manifold and consider the groupDiffω(M) consisting of the real analytic diffeomorphisms ofM equipped with its natural analytic topology (cf. Section 4). There exists a neighborhoodUof the identity inDiffω(M)and a residual (i.e. denseGδ) setKofV ∩(U×U)such that, if (f1, f2)belongs toK, then the dynamics associated to the groupGgenerated byf1, f2has the following properties:
1. it is minimal (i.e. all orbits are dense);
2. it is ergodic (i.e. every borelian invariant under G has zero or total Lebesgue measure);
3. it is topologically rigid (i.e. ifG is generated by(f1, f2)∈Kis conju- gated to Gby a homeomorphism h : M → M then h is, in fact, an element ofDiffω(M)).
Perturbations of analytic diffeomorphisms and natural topologies on their group will be discussed in Section 5 so as to make sense of the notion of “generic”
action. We should also point out that a very similar but a slightly stronger re- sult was independently obtained by M. Belliart in [Be] by elaborating on the proof of Proposition (2.1) given in [L-R]. The present paper is indeed a revised version of my 2001-preprint [Reb3] which has an overlap with [Be]. However the “generic character” of actions as above, which constitutes the point of view of this article, is not developed in [Be] (for instance, it is not proved in [Be]
that a group generated by randomly chosen diffeomorphisms is free). An extra reason to include Theorem D in our dicussion is the fact that it can immediately be derived from a short and self-contained proof of Proposition (2.1) which is provided in Section 6. This proof is already contained in the long discussion of [L-R] but here we give a clearer presentation which makes it promptly accessible to the reader. Compared to [Be], the present proof has two advantages. First it is rather simpler and more “down-to-earth” than the treatment of [Be]. Secondly it is necessary to derive Theorem E below which cannot be obtained with the arguments of [Be]. Actually the main virtue of the version of Proposition (2.1) given here is to single out the essential difficulty to generalize this type of result to other pseudogroups (which might be overlooked in [L-R]). It is the precise understanding of this difficulty that allows us to derive Theorem E as a further application.
To state Theorem E, we consider the group of cocyles (or skew-products) G (resp. GC) acting in S1×S1 (resp. S2 ×S1) which is constituted by the diffeomorphisms of the form
F (x, y)=(Ax, y+u(x)) ,
whereA represents the action of an element of PSL(2,R) (resp. PSL(2,C)) onS1 (resp. S2) anduis an analytic function onS1(resp. S2). This class of diffeomorphisms constitutes a classical and interesting object of study in Ergodic Theory.
Given a subgroupGofG, denote byGπ the subgroup of PSL(2,R) (resp.
PSL(2,C)) corresponding to the natural projection onto the first component of the elements inG.
Theorem E. Assume thatG ⊂ G (orGC) is not Solvable. Assume also that Gπ ⊂PSL(2,R)(resp. PSL(2,C)) is not discrete. ThenGis ergodic and has all orbits dense inS1×S1(resp. S2×S1).
From Theorem E we can derive additional dynamical properties of groups as above. Particularly interesting is a topological/measurable rigidity phenomenon whose proof would follow the lines of [Reb2] (cf. Corollary F below). Such rigidity contrasts with Furstenberg’s celebrated example of a skew-product of S1×S1which is measurably conjugate to a translation but has all orbits denses (so that, in particular, it is not topologically conjugate to a translation, cf. [Ma]).
It is surprising that these rigidity statements seem to have been missed in the vast literature about skew-products. Whereas we shall not provide a detailed proof of Corollary F here, since this would force us to do a long detour from the goals of this article, we give a precise (and simplified) statement. In any case the proof will follow from the combination of Theorem E with the technique introduced in Sections 4 of [Reb2]. Several additional details on this kind of argument are provided in the appendix.
Corollary F. Assume thatG1,G2 ⊂G are as above. Then the following are equivalent:
1. G1,G2are differentiably conjugate.
2. G1,G2are topologically conjugate.
3. G1,G2are measurably conjugate.
3 Proof of Theorem A
The proof of Theorem A presented here naturally depends on different aspects of Oxtobi-Ulam work [O-U], nonetheless it was inspired by ideas of S. Alpern which appear in [M-P-V]. Precisely the proof of Proposition (3.1) is an easy adaptation of the argument employed in the proof of the “Approximation Theorem” in [M-P-V], lecture 18.
Let Ik ⊂ Rk be the unit cube spanned by the vectors (1,0, . . . ,0), . . . , (0, . . . ,0,1) and denote by SnIk the 4−adic subdivision of Ik with order n.
This means the following. By definition,S1Ikis obtained by dividing each edge ofIk into 4 parts of same length and then forming the obviousk-dimensional cubes with the resulting segments. More generally,Sn+1Ikis obtained fromSnIk by dividing the edges of each cube belonging toSnIkinto 4 parts of same length
and then proceeding as before. It is clear thatSnIk consists of 4nk cubes In,jk (j =1, . . . ,4nk) whose union is the wholeIk. Clearly the length of the edges ofIn,jk is 4−n. Fixed a cubeIn,jk andα ∈ R,α >1, letIn,j,αk be the open cube, concentric toIn,jk , but having edges of length 4−n/α.
For each s ∈ N sufficiently large, we choose a finite “covering” Bs ofM consisting of coordinate neighborhoods{(Bi, φi)}(i =1, . . . , s) which satisfies the following conditions:
1. φi(Bi)=Ik ⊂Rkwherekis the dimension ofM;
2. eachBi is strictly contained in someBiso thatφi(Bi)is (defined and) a neighborhood ofIk inRk;
3. the direct image of the restriction ofµtoBibyφi is the Lebesgue measure onφi(Bi);
4. µ(B1)=µ(B2)= · · · =µ(Bs)=1/s;
5. s
i=1Bi =MwhereBi stands for the topological closure ofBi; 6. Bi∩Bj =∂Bi∩∂Bj fori=j (∂Bi =Bi \Bi).
Note that the collection of the neighborhoodsBi’s does not constitute a cov- ering ofMsince it misses the union of the boundaries of theBi’s. The union of these boundaries can, however, be thougth of as a finite union of hypersurfaces so that it will play no role in our discussion. In view of this we shall refer to the union of theBi’s as forming a “covering” (using quotes) forM.
Now using the “covering” above, we are able to define the 4−adic subdivisions ofMas follows. The 4−adic subdivision ofMwith ordern,SnM, consists of the cells (“cubes”)φi−1(In,jk )(i =1, . . . , ls; j =1, . . . ,4nk) whereIn,jk are the cells (cubes) of the 4−adic subdivision ofIk with ordern(again this procedure misses the union of the boundaries∂Bi ofBi, however this will be irrelevant to our discussion). Similarly we define the setsφ−i 1(In,j,αk ).
On the other hand, fixed s and the covering Bs, consider the set Us ⊂ Homeoµ(M)×Homeoµ(M)=(Homeoµ(M))2formed by the pairs(f1, f2)such that, for any pointp∈M, theG-orbit ofpintersects all theBi’s (i=1, . . . , ls; whereGstands for the group generated byf1, f2).
The setUsis clearly open for theC0−topology. Moreover one has:
Proposition 3.1. Fixeds ∈ N∗ andBs, the subsetUs ⊂ (Homeoµ(M))2 is dense.
Consider the setRnM =M\
i,jφi−1(In,jk )(recall that the setsIn,jk are open by definition). In other words, RnM consists of the union of the boundaries of the Bi’s and the union of the pre-images by the corresponding φi’s of the complementsIk\
jIn,jk . Lemma (3.2) below is very elementary.
Lemma 3.2. Assume thatn0 ∈ N∗ is fixed. The set of homeomorphismsh ∈ Homeoµ(M) whose dynamics admits no orbit entirely contained in Rn0M is open and dense.
Proof. Note thathhas no orbit contained inRn0Mif and only if the orbit of every pointp ∈Rn0MintersectsM\Rn0M. SinceRn0Mcan be thougth of as a finite union of “smooth hypersurfaces” (modulo using good coordinates asφi)
the statement follows at once.
The next result is the main ingredient of the proof of Proposition (3.1). As mentioned the Approximation Theorem of [M-P-V], lecture 18 is stated for α = √k
2 but a quick look at the proof makes it clear that it works also for any α >1.
Theorem 3.3. (Alpern in [M-P-V], lecture 18). Assume we are given h ∈ Homeoµ(M),α ∈ Rsatisfying α < 1andε > 0. Then there existsn0 ∈ N depending only onε(and not on α) andhˆ ∈ Homeoµ(M),ε-close toh, which cyclically permutes the cubes (cells)φi−1(Ink0,j,α)ofSn0M(i =1, . . . , ls;j = 1, . . . ,4nk). In other words, chosenn0large enough, we can findh ε-close toˆ h and satisfying the desired condition for anyα >1.
Proof of Proposition (3.1). Recall thatBs is fixed and assume we are given a pair of homeomorphisms(f1, f2)in(Homeoµ(M))2. Givenε >0, we have to check the existence of(fˆ1,fˆ2)∈Us ⊂(Homeoµ(M))2satisfying|| ˆf1−f1||<
ε(resp.|| ˆf2−f2||< ε).
According to Lemma (3.2), we can perturbf2intofˆ2so thatRn0M contains no non-trivial minimal set offˆ2. In other words, thefˆ2-orbit of every point q∈Rn0intersects the complement ofRn0. Thus there is an open neighborhood V ⊂MofRn0 with the same property, namely we can associate to each pointq ofV an iteratefˆ2mq offˆ2so thatfˆ2mq(q)lies onM\V.
Finally we takeα <1 so that the unionV∪
φi−1(Ink0,j,α)covers the whole of M(whereφ−i 1(Ink0,j,α)are contained in the cells ofSn0M). Using Theorem (3.3) we obtainfˆ1cyclically permuting the cubesφi−1(Ink0,j,α)andε-close tof1.
To finish the proof it is enough to verify that(fˆ1,fˆ2)as above belongs toUs. This is however obvious: ifp ∈ M belongs to
φi−1(Ink0,j,α)then itsfˆ1-orbit intersects all the sets in Bs; if p ∈ M belongs to V, then there is a point of itsfˆ2-orbit lying inM\V and therefore itsG-orbit intersects all the sets inBs
(whereGstands for the group generated byfˆ1,fˆ2). The proposition is proved.
Notice that Proposition (3.1) enables us to prove the genericity of groupsG⊂ Homeoµ(M)as in the statement of Theorem A which have minimal dynamics.
In fact, it is enough to consider a sequence of coveringsB1,B2, . . ., satisfying the conditions in the beginning of this section, and such that the diameters of the open sets inBsconverge uniformly tozero. For such a sequence letU1,U2, . . . be the corresponding open dense sets given by Proposition (3.1). Clearly any element in the intersection∞
s=1Us is minimal and, on the other hand, Baire’s theorem asserts that this intersection is “generic”.
To complete the proof of Theorem A, it remains to analyse the structure of invariant measures for generic subgroups of Homeoµ(M). For the rest of this section we suppose fixed the coveringBs (which was defined at the beginning of the section). We begin with the lemma below which is still an elementary generalization of Lemma (3.2).
Givenn0∈Nandε >0, denote byWεn0the subset of Homeoµ(M)consisting of those homeomorphismsf satisfying the following condition: ifνis a Borel probability invariant underf thenν(Rn0) < ε.
Lemma 3.4. Assume n0 ∈ Nis fixed. Givenε > 0, the set Wεn0 is dense in Homeoµ(M).
Proof. Let us prove that the complement of Wεn0 is closed and has empty interior. Assume that h ∈ Homeoµ(M) has an invariant probability measure satisfyingν(Rn0) ≥ ε. Notice that, in this case, Rn0 cannot have more than 1/ε disjoint images underh. In other words, ifm > 1/ε, then for somei ∈ {1, . . . , m}, hi(Rn0)∩Rn0 = ∅. Therefore the idea is to use an argument of general position to show thatf can be approximated by a homeomorphism for whichRn0 has “many disjoint” images. Let us make this idea more precise.
First suppose that M is a surface so that Rn0 consists of a union of lines.
Choosem >1/ε. By a “transversality argument”, it is possible to aproximateh byh1so that thehi1(Rn0)∩hj1(Rn0)is reduced to a finite number of points for everyi, j ∈ {1, . . . , m},i=j. Now we can approximateh1byh2so that
hi1(Rn0)∩hj1(Rn0)=hi2(Rn0)∩hj2(Rn0)
and, furthermore, theh2-orbit of any point inhi2(Rn0)∩hj2(Rn0),i=jis infinite.
Thus, ifν2is a probability preserved byh2, one hasν2(hi2(Rn0)∩hj2(Rn0))=0 wheneveri=j. Hence
ν2
m
i=1
hi2(Rn0)
=m . ν2(Rn0)≤1. We conclude thatν2(Rn0)≤1/m < ε.
For a general manifoldM we proceed in a recurrent way on the intersections hi(Rn0)∩hj(Rn0). Precisely, by the transversality argument, the dimension of these intersections will be smaller than dim(M)−1 (where dim(M)stands for the dimension ofM). So we can approximatehbyh1so that
ν1(hi1(Rn0)∩hj1(Rn0)) < δ ,
for every probability preserved byh1and anyδ > 0. Ifδis small enough, we then conclude that
m . ν1((Rn0))≤1−m(m−1)δ < ε .
The proof of the lemma is over.
Remark 3.5. Assume thath belongs toWεn0 ⊂ Homeoµ(M). We claim the existence of a relatively compact open neighborhood V of Rn0M such that ν (V ) <2εfor every Borel probabilityνpreserved byh. In fact suppose for a contradiction the claim is false. Thus there is a sequence of relatively compact neighborhoodsVi and a sequence of measuresνi preserved byhwhich satisfy the conditions below:
• V1⊇V2⊇V3⊇ · · · ⊇Vi ⊇ · · · ;
• ∞
i=1Vi =Rn0M;
• νi(Vi) >2εfor everyi.
Since the space of Borel probabilities is compact (cf. below), we can suppose that νi converges toνwhich is automatically preserved byh. Sinceν (Rn0M) < ε, there is a neighborhoodU ofRn0M such thatν (U ) ≤ 3ε/2. For sufficiently largei, one hasVi ⊂U, thusνi(U )−ν (U )≥ε/2>0 which is a contradiction with the fact thatνi →ν. This proves our claim.
It is well-known that the space of all Borel probabilities onM is a compact metric space (cf. for instance [Ma]). Denote byd(.)the associated metric. To
constructd(.), we consider a countable set of open balls {Bi}i∈N defining the topology ofM. Ifν1, ν2are probability measures onM, we set
d(ν1, ν2)= ∞
i=1
||ν1(Bi)−ν2(Bi)||
2i .
Consider the setMn ⊂(Homeoµ(M))2of pairs(f1, f2)which do not preserve any borelian probabilityνwhose distance toµis greater or equal 1/n.
Lemma 3.6. Mn⊂(Homeoµ(M))2is an open set.
Proof. Suppose for a contradiction that the statement is false. Then there is (f1, f2)∈Mnand a sequence{(fˆ1l,fˆ2l)} ⊂(Homeoµ(M))2\Mnsuch that each coordinate{ ˆfil}converges uniformly tofi. For eachl, we consider a measure νl preserved by(fˆ1l,fˆ2l)and having distance toµat least equal to 1/n. Clearly {νl}can be supposed to converge towards some measureν which also satisfies d(ν, µ)≥1/n, in particularνis not preserved by(f1, f2).
Becauseν is not preserved by (f1, f2), there is an elementf in the group
< f1, f2 > generated by f1, f2 and an open set U ⊂ M such that ν(U ) <
ν(f(U )). Next consider the elementsfˆl ∈<fˆ1l,fˆ2l >corresponding tof in the obvious way (in particular{ˆfl} → f uniformly). In addition note that we can find a relatively compact setV ⊂f(U )such thatν(V ) > ν(U ). However, since {ˆfl} →f, forllarge enoughV is enclosed infˆl(U )so that
lim inf
l→∞ νl(fˆl(U ))≥lim inf
l→∞ νl(V )=ν(V ) .
On the other handνl(U )converges toν(U )which is strictly less thanν(V ), so we finally obtainνl(U ) < ν(V )≤ νl(fˆl(U ))forllarge enough. The resulting
contradiction establishes the lemma.
We are ready to complete the proof of Theorem A.
Proof of Theorem A. Since we have seen that there is a residual subset of (Homeoµ(M))2 whose elements have all orbits dense, we just need to verify the existence of another residual subset of(Homeoµ(M))2whose elements have onlyµas common invariant measure. After Lemma (3.6), we just need to check that the setsMn ⊂(Homeoµ(M))2are dense.
Assume we are given δ > 0 and consider the collection{B1,B2, . . . ,Bm} of the balls involved in the definition of the metricd(.). In order to prove that Mn is dense, we need to prove that a pair (f1, f2) ∈ (Homeoµ(M))2 can be
approximated by another pair (fˆ1,fˆ2) ∈ (Homeoµ(M))2 with the following property: ifνis a borelian probability preserved byfˆ1,fˆ2, then
||ν(Bi)−µ(Bi)||< δ for every i =1, . . . , m .
Indeed, iffˆ1,fˆ2are as above for appropriateδandm, it follows from the definition ofd(.)that the pair(fˆ1,fˆ2)belongs toMn.
Thus assume from now on thatδ > 0 andm∈Nare fixed. Givenε >0, we need to findfˆ1(resp. fˆ2)ε-close tof1(resp.f2) satisfying the above condition for everyBi i =1, . . . , m. Using the notation of Theorem (3.3), choosen0so large that Theorem (3.3) does apply with respect toεand
K2i −K1i Sn0M < δ
10, for everyi=1, . . . , m, where:
1. Sn0Mdenotes the number of cells inSn0M(i.e. the cardinality ofSn0M which is equal tols4n0k);
2. K1i (i = 1, . . . , m) is the maximum number of cellsσi ofSn0M whose union is contained inBi;
3. K2i (i = 1, . . . , m) is the minimum number of cellsσi ofSn0M whose union containsBi \Rn0M(by a small abuse of notation, in this case we simply say that the union of theσi’scontainsBi).
Using Lemma (3.4) we approximatef2byfˆ2belonging toWδ/20n0 . Next letV be an open neighborhood ofRn0Msatisfying the following condition:
(∗) – ifνis a Borel probability invariant underfˆ2, thenν (V ) < δ/10.
The existence ofV is ensured by Remark (3.5). Now we takeα >1 so close to 1 thatV ∪
φi−1(Ink0,j,α)covers the whole ofM. Finally we approximatef1
byfˆ1so thatfˆ1permutes the cubesφi−1(Ink
0,j,α)as in Theorem (3.3).
To finish the proof, it is enough to verify that(fˆ1,fˆ2)satisfy the above condition onδandBi. Thus consider a probabilityνsimultaneously invariant underfˆ1,fˆ2. Note that all the cubesφi−1(Ink0,j,α) have the sameν-measure since they are permuted byfˆ1. Furthermore the estimate below does hold for the measure of the union of all these cubes:
ν
Sn0M
φi−1(Ink0,j,α)
<1−ν (V ) <1−δ/10.
Denote byσα one of these cubes. Because they are also pairwise disjoint, one obtains
1
Sn0M ≥ν (σα)≥ 1−δ/10 Sn0M . It follows that
K2i
Sn0M ≥ν (Bi)≥ K−1i(1−δ/10) Sn0M . On the other hand
K2i
Sn0M ≥µ(Bi)≥ K1i Sn0M. Thus
||µ(Bi)−ν(Bi)||≤ K2i −K1i
Sn0M + δK1i
10Sn0M ≤ δ 10 + δ
10 < δ .
The proof of the theorem is completed.
4 Proof of Theorem B
This section is devoted to the proof of Theorem B. Thus Sdenotes a compact surface,µis aC1-area measure andf1, f2are elements of Diff1µ(S). Without loss of generality, we can suppose thatf1is not of Anosov type.
Recall that a fixed pointpof a diffeomorphismf ofSis calledellipticif the eigenvalues off atpare non-real numbers of modulus 1. More generally, ifp is periodic forf with periodk, thenpis said elliptic forf if the eigenvalues of fk atpare non-real and have modulus 1.
The non-wandering set of a conservative diffeomorphism coincides with the whole ambient manifold by virtue of Poincaré Recurrence Lemma. Given a point p ∈ Sand a diffeomorphismf ∈ Diff1µ(S), the Closing Lemma (conservative version, [P-R]) then states thatf can beC1-approximated by a diffeomorphism f˜∈Diff1µ(S)for whichpis a periodic point. Furthermore, unlessfis of Anosov type, a theorem due to Newhouse [Ne] asserts that we can, in fact, assume that pis elliptic forf˜.
Definition 4.1.A periodic pointpof periodkforh∈Diff1µ(S)will be called a local rotation (“Siegel disk”) if and only if there is a relatively compact neigh- borhoodU ofpsatisfying the two conditions below:
1. The boundary∂U ofUis invariant underhk.
2. There is a local coordinateφtaking the closure ofU,U, to the closed unit ballBofCin which the restriction ofhk toUhas the form
hk(z)=exp(iθ ).z , whereθ/2π is irrational.
The proof of our first lemma relies heavily on the theorem of Newhouse men- tioned above.
Lemma 4.2.Assume thath∈Diff1µ(S)is not of Anosov type. Then, givenε >0 and a pointp∈S, there is a homeomorphismhˆ ∈Diff1µ(S)havingC1-distance tohless thanεfor whichpis a local rotation.
Proof. First we approximate hby hˆ1 ∈ Diff1µ(S)for which p is a periodic elliptic point with periodk∈N∗. Next letBδ(p)be the ball of radiusδcentered atp. Choose δ > 0 so that the sets Bδ(p),hˆ1(Bδ(p)), . . . ,hˆk1−1(Bδ(p))are pairwise disjoint. Now observe that the boundary ofhˆk1(Bδ(p)), ∂hˆk1(Bδ(p)), converges in theC1 topology to∂Bδ(p)(the boundary ofBδ(p)) whenδgoes tozero even after a dilation of ratio 1/δ of Bδ(p). Choosingδ very small, it follows thathˆ1can be approximated byhˆ2so thathˆk2leaves∂Bδ(p)invariant i.e.
hˆk2(∂Bδ(p))=∂Bδ(p).
Since the mapping from∂Bδ(p)to∂Bδ(p)induced by hˆk2 is isotopic to the local map given by the derivative ofhˆk2atp, it follows thathˆk2can beC1-deformed insidehˆk2−1(Bδ(p))intohˆ so thatpbecomes a local rotation of periodkforh.ˆ
This proves the lemma.
We now introduce the definition of vector fields in the closure of a group following previous works as [Na] and [Reb1].
Definition 4.3. LetU be an open set on a manifoldM equipped with a vector fieldX. Denote byXthe local flow ofXonUand assume that there is a group Gacting onM. We say thatXis in theC1-closure (resp. Ck, C∞-closure) of Gif and only if the following holds: given a relatively compact subsetV ⊂U andt0 ∈ Rsuch thattX is defined onV whenever0 ≤ t ≤ t0, the mapping tX0 :V →tX0(V )is theC1-limit (resp. Ck, C∞-limit) onV of the restriction toV of a sequence of elements inG.
Note that the above definition is natural under the action ofG in the sense that, ifXdefined onU is in theC1-closure ofGandhbelongs toG, thenh∗X defined onh(U )is in theC1-closure ofGas well. The next lemma contains the main idea in the proof of Theorem B.
Lemma 4.4.LetG=< f1, f2>⊂Diff1µ(S)be as before. Assume we are given ε >0and a pointpinS. Then there existsfˆ1∈Diff1µ(S)(resp.fˆ2∈Diff1µ(S)), which isε-close tof1(resp. f2) with respect to theC1-distance, such that the conditions below hold.
1. There is a neighborhood U ofpinS equipped with2vector fieldsX, Y which are linearly independent at every point ofU;
2. X, Y are contained in theC1-closure ofG, the group generated by fˆ1,fˆ2. Proof. Using Lemma (4.2), we can suppose thatp is a local rotation with periodk1∈Nforfˆ1. Letψ:V ⊂S→D⊂Cbe the coordinate in whichfˆ1k1 becomes
fˆ1k1(z)=eiθz , where ||z||<1 and θ/2π is irrational.
Consider the vector fieldZdefined onD and given byZ(z)=|| z || (∂/∂x −
∂/∂y). Becauseθ is irrational, the powers offˆ1k1 approximate the flow ofZ. In particular the vector fieldZ =ψ∗Zdefined onV belongs to the closure of the cyclic group generated byfˆ1k1.
We can suppose the existence of a neighborhoodV1⊂V ofpand a number k2 ∈ N∗ such thatU = V1∩ ˆf2k2(V1) is still a neighborhood ofp. Indeed, it would be enough to apply the Closing Lemma tof2. Obviously we can also have fˆ2k2(p) =p. Hence the vector fieldX =(fˆ2k2)∗Zis defined onU and verifies X(p) = 0. Furthermore X belongs to the closure of the group generated by fˆ1,fˆ2. Finally recalling that the derivative offˆ1k1 atpis an irrational rotation, it results the the vector fieldY =(fˆ1k1)∗X is such thatX(p), Y (p)are linearly independent atp. Therefore they remain linearly independent in a neighborhood Uofp(modulo reducingU). Since they are both contained in the closure of the group generated byfˆ1,fˆ2, the proof of the lemma is finished.
Assume thatG⊂Diff1µ(S)acts onSand letU ⊂Sbe an open set equipped with vector fieldsX, Y in the closure ofGwhich are in addition linearly inde- pendent at every point ofU. Observe that theG-orbit of any point inUis dense inU: in fact, ifp, qare distinct points ofU, then they can be “joined following the flows ofX, Y”. BecauseX, Yare in the closure ofG, we can find a sequence of elements{hi} ⊂ G such that hi(p)converges to q. This shows that these orbits are dense inU. Furthermore a similar argument holds in the sense of
“local ergodicity”, it suffices to take into account the fact that the sequence{hi} actually convergesC1onUto the composition of local flows ofX, Y.
We are now able to prove the Theorem B.
