A Wong-Rosay type theorem for proper holomorphic self-maps
aa r X i v : . [ m a t h . C V ] S e p A Wong-Rosay type theorem for proper holomorphi self-maps.Emmanuel OpshteinAbstra tIn this short paper, we show that the only proper holomorphi self-maps of boundeddomains in C k whose dynami s es ape to a stri tly pseudo onvex point of the boundaryare automorphisms of the eu lidean ball. This is a Wong-Rosay type theorem for asequen e of maps whose degrees are a priori unbounded.Introdu tion.In 1977, Wong proved that the only stri tly pseudo onvex domain with non- ompa t au-tomorphism group is the ball [16℄. This result was generalized by Rosay [12℄ (see also[11℄).Theorem (Wong-Rosay). Let Ω be a bounded domain in C k and ( f n ) a sequen e of itsautomorphisms. Assume that the orbit of a point of Ω under ( f n ) a umulates a smoothstri tly pseudo onvex point of b Ω . Then Ω is biholomorphi to the eu lidean ball.This theorem remains valid for a sequen e of orresponden es provided that their de-grees remain bounded [10℄. In this paper, we prove that the theorem above also holds truein presen e of unbounded degree, when the sequen e of automorphisms is repla ed by theiterates of a proper holomorphi self-map.Theorem 1. Let Ω be a bounded domain in C k with a proper holomorphi self-map f . Ifthere is a point y of Ω whose orbit under the iterates of f a umulates a smooth stri tlypseudo onvex point a of b Ω (that is f n k ( y ) −→ a ), then Ω is biholomorphi to the eu lideanball and f is an automorphism.In [9℄, the question of wether a proper holomorphi self-map of a smoothly boundeddomain in C k has to be an automorphism of the domain was onsidered. In C for instan e,it was proved that non-inje tive proper self-maps of su h domains has a non- ompa tdynami s (all the limit maps of the dynami s have value on the boundary of Ω ). Theorem1 goes one step further in this dire tion : the limit maps even take values in the weaklypseudo onvex part of the boundary.The main ingedient for this result is a lo al version of Wong-Rosay's theorem on erningsequen es of CR-maps. It was (cid:28)rst obtained by Webster in the wake of Chern-Moser'stheory of stri tly pseudo onvex hypersurfa es [15℄.Theorem 2 (Webster). Let (Σ , a ) and S be two germs of stri tly pseudo onvex hypersur-fa es. Assume there is a sequen e of CR-embeddings of S into Σ whose images onverge to a . Then S is spheri al, i.e. lo ally CR-di(cid:27)eomorphi to the eu lidean sphere.The idea behind the proof of theorem 1 is to onsider the CR-maps indu ed by f n onthe boundary rather than the maps f n themselves. Using te hniques developped in [9℄, westudy the way these CR-maps degenerate and he k that theorem 2 applies : around a ,the boundary of Ω is spheri al. The lo al biholomorphism between our domain and theball then extends to the whole of Ω thanks to the dynami al situation.1he paper is organised as follows. We (cid:28)rst olle t some trivial dynami al fa ts about f and the automorphisms of the ball whi h will allow us to propagate the lo al spheri ityto the whole domain. In se tion 2, we prove theorem 1 modulo the entral question of thelo al spheri ity around a . In se tion 3, we (cid:28)nally turn ba k to this problem.1 Preliminary remarks.Surprisingly enough, the onvergen e hypothesis on f n k ( y ) in theorem 1 has very strong(though very lassi al) impli ations in the holomorphi ontext. The aim of this se tion isto larify some of them, as well as pointing out the well-known properties of the dynami sof the automorphisms of the ball whi h will be usefull to us. Hen eforth, Ω , f and a areas in theorem 1.First of all, this hypothesis may seem weaker than it a tually is. Indeed, the ontra tingproperty of holomorphi maps for the Kobayashi distan e (whi h is a genuine distan e onbounded domains) leads to the following lassi al fa t :Lemma 1.1. Any sequen e of holomorphi maps between bounded domains Ω and Ω ′ ,whi h takes a point y in Ω to a sequen e onverging to a stri tly pseudo onvex point of theboundary of Ω ′ onverges lo ally uniformly to this point on Ω . For instan e, the sequen e f n k onverges lo ally uniformly to a on Ω .Corollary 1.2. The map f extends smoothly to a neighbourhood of a in b Ω and f ( a ) = a .Moreover, f is a lo al biholomorphism (resp. CR-automorphism) in a neighbourhood of a (resp. in b Ω ).Proof : Call z k := f n k ( y ) and w k := f ( z k ) . Sin e w k = f ( f n k ( y )) = f n k ( f ( y )) , both z k and w k tend to a be ause of the previous lemma. Sin e a is a stri tly pseudo onvex point,an observation of Berteloot ensures that f extends ontinuously to a neighbourhood of a in b Ω [3℄, with f ( a ) = lim f ( z k ) = lim w k = a . Su h an extension is automati ally smoothbe ause a is a stri tly pseudo onvex smooth point of b Ω [2℄. Sin e we are lose to a stri tlypseudo onvex point of the boundary, bran hing is prohibited and f must be one-to-one(see [5℄). (cid:3) Let us now dis uss the dynami al type of the (cid:28)xed point a . Although it attra ts part ofthe dynami s, it is not obvious at (cid:28)rst glan e that a is not a repulsive (cid:28)xed point. The orbitof y ould in prin iple jump lose to a from time to time, then get expelled away from a . The following lemma shows that su h a behaviour does not o ur in our holomorphi ontext.Lemma 1.3. The point a is a non-repulsive (cid:28)xed point of f .Proof : Assume by ontradi tion that f is repulsive at a . By de(cid:28)nition, there is an openneighbourhood U of a on whi h the inverse f − of f is well de(cid:28)ned, takes values in U , andis even ontra ting : d ( f − | U ( z ) , a ) < d ( z, a ) for any z ∈ U . By assumption, there is a point y ∈ Ω su h that f n k ( y ) ∈ U as soon as k is large enough. De(cid:28)ne then n ′ k := min { n | f i ( y ) ∈ U, ∀ i ∈ [ n, n k ] } , so that f n ′ k − ( y ) / ∈ U . Sin e f − | U is ontra ting, the point f n ′ k ( y ) is loser to a than f n k ( y ) , so it tends to a (in parti ular ( n ′ k ) is an extra tion). Equivalently f n ′ k − ( f ( y )) a , so f n ′ k − onverges lo ally uniformly to a by lemma 1.1. This is in ontradi tionwith f n ′ k − ( y ) / ∈ U . (cid:3) Let us (cid:28)nally dis uss the dynami s of the automorphisms of the ball. Sin e there arevery few of them (they form a (cid:28)nite dimensional group), their dynami s is rather poorand any small pie e of information on it may give rise to strong restri tions. Re all thefollowing well-known lassi(cid:28) ation (see [13℄, se tion 2.4).Proposition 1.4. Let g be an automorphism of the unit ball in C n . Then the dynami sof g is • either hyperboli (North-South) : there exist exa tly two (cid:28)xed points N, S ∈ bB of g and g n onverges lo ally uniformly to S on B \{ N } . • or paraboli (South-South) : there exists a unique (cid:28)xed point S ∈ bB of g and g n onverges lo ally uniformly to S on B \{ S } . • or re urrent ( ompa t) : The g -orbits remain at (cid:28)xed distan e from bB . If g has a(cid:28)xed point on bB then it has a whole omplex pointwise (cid:28)xed line through this point(see also [6℄).What will be of interest for us in this lassi(cid:28) ation is ontained in the following lemma,whose proof is straightforward from the lassi(cid:28) ation.Lemma 1.5. Let g be a ball automorphism whi h has a non-repulsive (cid:28)xed point p on bB ,and no interior (cid:28)xed point near p . Then the dynami s of g is either hyperboli or paraboli ,with south pole p (meaning that S is p in the previous lassi(cid:28) ation). Moreover, given anyneighbourhood U of p , there is a point z in U whose orbit remains in U and onverges to p .2 Proof of theorem 1.In this se tion, we prove theorem 1 leaving aside the entral question of the spheri ity of b Ω around a , whi h will be dealt with in the next se tion. Let us (cid:28)rst (cid:28)x the notation. Let (Ω , f, a ) be a triple as in theorem 1. By a global hange of oordinates in C k , we an take a to the origin, the tangent plane of b Ω at a to { Re z = 0 } , and make Ω stri tly onvexlo ally near a . For α small enough, de(cid:28)ne U α and Ω α as being the onne ted omponentsof a in b Ω ∩ { Re z < α } and Ω ∩ { Re z < α } .The (cid:28)rst step of the proof, postponed to the following se tion, onsists in showing that b Ω is spheri al around a .Lemma 2.1. A neighbourhood of a in b Ω is spheri al.This means that there exists a CR-di(cid:27)eomorphism Φ : U ε −→ V ⊂ bB . A lassi alextension theorem even shows that Φ extends to a biholomorphism Φ : Ω ε −→ D where D is an open set of B whose boundary ontains V (see [4℄). This biholomorphism allows totransport f to a lo al automorphism of B , de(cid:28)ned by g : Φ(Ω ε ∩ f − (Ω ε )) −→ Φ(Ω ε ) x Φ ◦ f ◦ Φ − ( x ) . g uniquely extends to a global automorphism of the ball, again denoted by g .The se ond step onsists in using the dynami s of f and the inje tivity of g (whi h wegot for free thanks to Alexander's theorem) to propagate the lo al spheri ity, and produ ea biholomorphism between Ω and B . Let us (cid:28)rst dis uss the possible dynami s of g . Bylemma 1.3, a is not a repulsive (cid:28)xed point of f so Φ( a ) is neither one for g . Moreover,sin e f has no (cid:28)xed point inside Ω (be ause of lemma 1.1), g has also no (cid:28)xed point in V . By lemma 1.5, g is either hyperboli with attra tive (cid:28)xed point Φ( a ) or paraboli withonly (cid:28)xed point Φ( a ) . From now on, we will denote S := Φ( a ) . The same lemma alsoguarantees that there are points in D whose (positive) orbits under g remain in D and tendto S . Sin e their whole orbits remain in D , the onjuga y thus allows to get the followinginformations on f in return.Lemma 2.2. The whole sequen e of iterates ( f n ) (rather than only a subsequen e) on-verges to a on Ω . Moreover, the set Ω ′ ε := { z ∈ Ω ε , f n ( z ) ∈ Ω ε ∀ n ∈ N } is a non-empty open invariant set of f .Proof : From the dis ussion above, we on lude that there is a point y in Ω ε su h that f n ( y ) remains in Ω ε (thus Ω ′ ε is not empty). Its orbit also onverges to a . By lemma 1.1, f n must therefore onverge to a lo ally uniformly on Ω . The set Ω ′ ε is obviously invariant by f .Finally, it is open be ause the Kobayashi metri de reases under f . Indeed, if z is in Ω ′ ε , sois a Kobayashi δ -neighbourhood of this point (take δ := d K (cid:0) Ω ∩{ Re z = ε } , Orbit ( z ) (cid:1) ). (cid:3) Corollary 2.3. The map Φ extends to a holomorphi map from Ω to B .Proof : Let O i denote f − i (Ω ′ ε ) . Be ause of the invarian e of Ω ′ ε by f and sin e f n onvergesto a on Ω , we on lude that ( O i ) i is a growing sequen e of open sets whi h exhausts Ω .De(cid:28)ne therefore Φ : Ω = ∪ O i −→ Bz ∈ O i g − i ◦ Φ | Ω ′ ε ◦ f i ( z ) . This map is obviously holomorphi (be ause Ω ′ ε is open), and oin ides with Φ on Ω ′ ε . Itis therefore an extension of Φ itself. (cid:3) The remaining point to prove is that Φ is a biholomorphism. Let us (cid:28)rst prove that it isproper.Lemma 2.4. The map Φ is a proper map from Ω to B .Proof : Re all that the dynami s of g is either hyperboli or paraboli . Moreover, Φ − ◦ g ( w ) = f ◦ Φ − ( w ) for any w ∈ D su h that g ( w ) belongs to D (re all that Φ : Ω ε −→ D is a biholomorphism). A basi onsequen e of these two fa ts is that Φ( O n \ O n − ) goes to bB with n . Indeed, the f -orbit of a preimage by Φ of a point w in this set rea hes O = Ω ′ ε only at time n , so the g -orbit of w annot remain in D before the same time (if g k ( w ) ∈ D for k ≥ N , then f k (Φ − ( w )) = Φ − ( g k ( w )) is in Ω ε for k ≥ N also). If n is large, Φ( z ) hasto be very lose to some pole of the dynami s whi h is either S if g is paraboli or anotherpoint of bB if g is hyperboli . Anyway Φ( z ) is lose to the boundary of B .4or an arbitrary sequen e ( z i ) i ∈ N ∈ Ω onverging to b Ω , we must show that Φ( z i ) tendsto the boundary of B . For this, (cid:28)x a positive real number δ and an integer n su h that d (Φ( O n \ O n − ) , bB ) ≤ δ for all n ≥ n , meaning that Φ(Ω \ O n ) is δ - lose from bB . Splitthen ( z i ) into two subsequen es, one ontaining all the elements whi h belong to O n , theother one those whi h es ape from O n : ( z i ) := { z n i ∈ { z n } | z n i / ∈ O n } , ( z i ) := { z n i ∈ { z n } | z n i ∈ O n } . By onstru tion, d (Φ( z i ) , bB ) ≤ δ . Sin e f n ( z i ) ⊂ O and sin e f n , Φ | O and g areproper maps, Φ( z i ) = g − n ◦ Φ | O ◦ f n ( z i ) is also δ - lose to bB for i large enough. (cid:3) Finally, we need to show that Φ is a biholomorphism. It is not yet lear sin e there existholomorphi overings of the ball. Anyway we know that any proper map to a boundeddomain has a (cid:28)nite degree (see [13℄, hap. 15). In parti ular, there is an integer d whi hbounds the numbers of preimages of Φ : − ( z ) ≤ d, ∀ z ∈ B. Noti e now that the degree of Φ bounds this of f n for all n be ause Φ = g − n ◦ Φ ◦ f n . Thedegree of f n is thus bounded on one hand and equal to ( deg f ) n on the other. So f is anautomorphism of Ω . The inje tivity of Φ is now immediate sin e Φ | O i = g − i ◦ Φ | O ◦ f i isa omposition of inje tive maps for all (cid:28)xed i . (cid:3) a in b Ω is spheri al.We re all that all the results proved in the previous se tion used this fa t, so we have to goba k to the general situation of theorem 1. Nevertheless, remind that we an speak of thea tion of f on b Ω , at least lose to a , thanks to lemma 1.2. The idea behind this te hni alpart of the proof is based on previous results on erning behaviours of sequen es of CR-maps (see [9, 8℄). Unformally speaking, they explain that non-equi ontinuous sequen esof CR-maps on stri tly pseudo onvex hypersurfa es dilate a ertain (anisotropi ) distan e.The proof of the spheri ity then goes as follows. Either f n k onverges to a on SPC ( b Ω) andtheorem 2 gives the spheri ity. Or f n k is not equi ontinuous on SPC ( b Ω) and it is dilating.Then the inverse bran hes of f n k are ontra ting CR-di(cid:27)eomorphisms and theorem 2 givesthe spheri ity. Let us (cid:28)rst (cid:28)x the easy situation where f n k onverges to a on SPC ( b Ω) .Proposition 3.1. Assume f n k onverges lo ally uniformly to a on a neighbourhood of a in b Ω . Then b Ω is spheri al near a .Proof : Theorem 2 explains that it is enough to (cid:28)nd a ontra ting sequen e of CR-automorphisms on a neighbourhood of a . We are assuming here that ( f n k ) is a sequen eof ontra ting CR-maps on a pie e of SPC ( b Ω) . Also, orollary 1.2 shows that f is a lo aldi(cid:27)eomorphism at a . We thus only need to prove that there is a (cid:28)xed neighbourhood of a on whi h all f n k are inje tive. To see this, (cid:28)rst assume that f n , and not only f n k , onvergesto a . Fix then a neighbourhood U of a on whi h f is inje tive. Sin e f n onverges to a on U , f n ( U ) ⊂ U for all large enough integers n ≥ n . Consider now a neighbourhood U ′ of a in U whose images U ′ , f ( U ′ ) , . . . , f n ( U ′ ) are all ontained in U . Su h a set exists be ause f is ontinuous and a is a (cid:28)xed point of f . By onstru tion f n ( U ′ ) ⊂ U for all n ∈ N , andthe restri tion of f n to U ′ is inje tive as a omposition of inje tive maps.5n the general setting, let us (cid:28)rst he k that in fa t, the onvergen e of the subsequen e f n k to a implies the onvergen e of the whole dynami s of an iterate h = f p to a . Pi k againa small neighbourhood U of a in SPC ( b Ω) and an integer p = n k su h that f p ( U ) ⊂ U .The map h := f p restri ts to U to a lo al di(cid:27)eomorphism from U to itself, whose sequen eof images h n ( U ) is obviously de reasing (i.e. h i ( U ) ⊃ h i +1 ( U ) ). Observe then that thesubsequen e ( h n ′ k ) de(cid:28)ned by n ′ k := E ( n k /p ) + 1 onverges uniformly to a on U . Indeed, h n ′ k = f pn ′ k = f n k + i with i < p , so h n ′ k ( U ) ⊂ ∪ i ≤ p f i ( f n k ( U )) . Sin e f n k ( U ) is lose to a by hypothesis (for k large enough) and a is a (cid:28)xed point of f , the ontinuity of f impliesthat h n ′ k ( U ) is also lose to a . Sin e the sequen e h n ( U ) de reases, it thus onverges to a .Repla ing f by h , we an therefore apply the above argument, so a neighbourhood of a isindeed spheri al. (cid:3) Consider now the situation when f n k does not onverge to a on a neighbourhood of a .Let us (cid:28)rst des ribe the (cid:28)gure and notation. As in the previous se tion, we assume that Ω is strongly onvex in a neighbourhood O of a , that a is the origin and that Ω ∩ O is ontained in { Re z ≥ } . We put Ω ε := Ω ∩ O ∩ { Re z ≤ ε } , U ε := b Ω ∩ O ∩ { Re z ≤ ε } and we assume without loss of generality that Ω ⋐ O . Also sin e all the argumentsto ome are purely lo al and o ur in O , we will onsider in the sequel that f extendssmoothly to the boundary (lemma 1.2), without expli itly mentionning any further thene essary restri tion of f to O . The non- onvergen e of f n k means the existen e of asequen e of points z i ∈ b Ω tending to a , and integers k i su h that the points f n ki ( z i ) layout of a (cid:28)xed neighbourhood of a , say U . Sin e a is (cid:28)xed by f n ki , we an even assumethat f n ki ( z i ) ∈ bU = b Ω ∩ O ∩ { Re z = 1 } by moving z i loser to a . Finally, put f i := f n ki and de(cid:28)ne ε i by z i ∈ { Re z = ε i } . Ω z i g i { Re z = 1 }{ Re z = ε i }{ Re z = 0 } a = (0 , O Ω g i ( z i ) The main point of this se tion is that f n k has a strong expanding behaviour.Proposition 3.2. (see also [8℄) For all ε there exists an integer k = k ( ε ) su h that f k ( U ε ) ⊃ U \ U ε .The spheri ity near a is a dire t onsequen e of this proposition :Corollary 3.3. If ( f n k ) does not onverge to a in a neighbourhood of a then b Ω is spheri alnear a .Proof : Fix an open ontra tible set V ompa t in U \{ a } . For ε small enough, V ⊂ U \ U ε and there is an integer k ε su h that f k ε ( U ε ) ⊃ V . Moreover, there are no riti al valueof f k ε | U ε inside V be ause both U ε and V are stri tly pseudo onvex (see [5℄). Sin e V is simply onne ted, there exists an inverse bran h of f k ε | U ε on V , whi h means a CR-di(cid:27)eomorphism h ε : V −→ U ε with f k ε ◦ h ε = Id. The sequen e h ε is therefore ontra tingon V , and theorem 2 implies that V is spheri al. We have thus proved the lo al spheri ityof U \{ a } , whi h even proves the spheri ity of U be ause a is a stri tly pseudo onvex6oint. Indeed, Chern-Moser's theory expresses the spheri ity of an open stri tly pseudo- onvex hypersurfa e by the vanishing of a ontinuous invariant tensor. Sin e this tensorvanishes on U \{ a } , it also vanishes on the whole of U so U itself is spheri al. In thespirit of [8℄, It would be pleasant to get a more down-to-earth argument for this last point. (cid:3) The proof of proposition 3.2 relies on the following lemma.Lemma 3.4. For all ε there exists a diverging sequen e c i −→ + ∞ su h that for all p ∈ U with f i ( p ) / ∈ U ε we have : k f ′ i ( p ) u k ≥ c i k u k ∀ u ∈ T C p b Ω . Proof : The idea is that Hopf's lemma gives estimates on the normal derivative of f i , whi htransfer automati ally to omplex tangential estimates in stri tly pseudo onvex geometry.For p ∈ U , let ~N ( p ) be the unit ve tor normal to b Ω pointing inside Ω and B + δ ( p ) := B ( p + δ ~N ( p ) , δ ) ∩ {h ~N ( p ) , ·i ≥ δ } . When δ is small enough but (cid:28)xed, B + δ ( p ) is in Ω and its image by f i for i large is in Ω ε ( f i onverges to a inside Ω ). Thus if f i ( p ) / ∈ U ε , the non-positive p.s.h fun tion ϕ := −h ~N ( f i ( p )) , f i ( · ) − f i ( p ) i vanishes at p while it is less than − cε on B + ε ( p ) ( c is a onstant depending only on the urvature of b Ω at a ). Hopf's lemma then asserts that n i ( p ) := h f ′ i ( p ) ~N ( p ) , ~N ( f i ( p )) i = k ~ ∇ ϕ ( p ) k ≥ c ′ ε δ . Sin e δ was arbitrary, we ould take it mu h smaller than ε , so that the radial es ape rate n i ( p ) is large. To transfer this radial estimate on the derivatives of f i to omplex tangentialones, onsider the Levi form of b Ω de(cid:28)ned by ( p, u ) := h [ u, iu ] , i ~N ( p ) i , u ∈ T C p b Ω , where u stands for the ve tor in T C p b Ω as well as any extension of it to a ve tor (cid:28)eld of T C b Ω . The smoothness and stri t pseudo onvexity of U implies the existen e of geometri onstants c , c su h that c k u k ≤ ( p, u ) ≤ c k u k ∀ p ∈ U , ∀ u ∈ T C p b Ω . Easy omputations show that : c k f ′ i ( p ) u k ≥ ( f i ( p ) , f ′ i ( p ) u ) = n i ( p )( p, u ) ≥ c n i ( p ) k u k . Sin e n i ( p ) is large when i is, this serie of inequalities implies lemma 3.4. (cid:3) The previous lemma asserts that f i dilates the omplex tangential dire tions of b Ω if f i ( p ) is not lose to a . The last observation we need to make in order to prove proposition 3.2is that this (cid:16) omplex tangential dilation(cid:17) property implies a genuine dilation.A path γ in b Ω will be alled a omplex path if ˙ γ ( t ) ∈ T C γ ( t ) b Ω for all t . Its eu lideanlength will be denoted by ℓ ( γ ) . For x, y ∈ U , de(cid:28)ne the CR-distan e d CR ( x, y ) between x and y as the in(cid:28)mum of the lengths of omplex paths joining x to y . The point is that thestri t pseudo onvexity ondition means that the omplex tangential distribution is onta tso omplex paths an join any two points. Even more, the open set U \ U ε is d CR-bounded7see theorem 4 of [7℄, or [9℄).Proof of proposition 3.2 : Fix τ > su h that B CR ( z i , τ ) ⊂ U ε for all i large enough. Sin e U \ U ε is d CR-bounded, it is enough to prove that bf i ( B CR ( z i , τ )) ∩ B CR ( f i ( z i ) , c i τ ) ∩ (cid:0) U \ U ε (cid:1) = ∅ be ause c i τ an be made greater than the CR-diameter of U \ U ε . Take a point x ∈ bf i ( B CR ( z i , τ )) ∩ U \ U ε and let us prove that d CR ( f i ( z i ) , x ) ≥ c i τ. (1)Consider an ar -length parameterized omplex path γ in U \ U ε joining f i ( z i ) to x . Sin e f i is a lo al CR-di(cid:27)eomorphism at ea h point of B CR ( z i , τ ) whose image lies in the stri tlypseudo onvex part of b Ω , the onne ted omponent of f i ( z i ) in γ ∩ f i ( B CR ( z i , τ )) an belifted to a omplex path e γ through f i . Thus there exists l ≤ ℓ ( γ ) and e γ : [0 , l ] −→ B CR ( z i , τ ) joining z i to bB CR ( z i , τ ) su h that f i ◦ e γ ( t ) = γ ( t ) for all t ∈ [0 , l ] . Sin e e γ ( t ) ∈ U and f i ( e γ ( t )) ∈ U \ U ε for all t , the estimates obtained in lemma 3.4 yield : ℓ ( γ ) ≥ l = Z l k ˙ γ ( t ) k dt = Z l k f ′ i ( e γ ( t )) ˙ e γ ( t ) k dt ≥ c i Z l k ˙ e γ ( t ) k dt ≥ c i ℓ ( e γ ) . This proves (1) sin e e γ joins z i to bB CR ( z i , τ ) (so ℓ ( e γ ) ≥ τ ) and γ is any omplex pathjoining f i ( z i ) to x . (cid:3) Referen es[1℄ H. 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