Berger domains and Kolmogorov typicality of infinitely many invariant circles
BBERGER DOMAINS AND KOLMOGOROV TYPICALITY OF INFINITELY MANYINVARIANT CIRCLES
PABLO G. BARRIENTOS AND ARTEM RAIBEKASA bstract . Using the novel notion of parablender, P. Berger proved that the existence of finitelymany attractors is not Kolmogorov typical in parametric families of di ff eomorphisms. Here,motivated by the concept of Newhouse domains we define Berger domains for families ofdi ff eomorphisms. As an application, we show that the coexistence of infinitely many attrac-ting invariant smooth circles is Kolmogorov typical in certain non-sectionally dissipativeBerger domains of parametric families in dimension three or greater.
1. I ntroduction
Many dynamical properties, such as hyperbolicity, are robust in C r -topology of di ff eo-morphisms. That is, the property holds under any appropriate small perturbation of thedynamical system. However, many others interesting phenomena, non-hyperbolic strangeattractors for instance, are not stable in that sense. Hence, the question that arises is whethersuch dynamical properties could be survive if not for all perturbations but, at least, formost. For one-dimensional dynamics the Malliavin-Shavgulidze measure has been recentlyproposed as a good analogy to the Lebesgue measure in order to quantify this abundancein a probabilistic sense [Tri14]. However, in higher dimensions, it is not known how tointroduce a good notion of a measure in the space of dynamical systems. Kolmogorov inhis plenary talk ending the ICM 1954 proposed to consider finite dimensional parametricfamilies taking into account the Lebesgue measure in the parameter space (see [HK10]). Aparametric family ( f a ) a exhibits persistently a property P if it is observed for f a in a set ofparameter values a with positive Lebesgue measure. Furthermore, the property P is called typical (in the sense of Kolmogorov) if there is a Baire (local) generic set of parametric fami-lies exhibiting the property P persistently with full Lebesgue measure. In this direction, amilestone in recent history of dynamical systems has been the paper of Berger [Ber16] (seealso [Ber17]) where it was proven that the coexistence of infinitely many periodic sinks isKolmogorov typical in parametric families of endomorphisms in dimension two and di ff eo-morphisms in higher dimensions. The work of Berger extends, in a measurable sense accord-ing to Kolmogorov, the important results in the 70’s due to Newhouse [New74, New79] (seealso [Rob83, PT93, PV94, GTS93]) on the local genericity of the coexistence of infinitely manyhyperbolic attractors (sinks) in C r -topology. This celebrated result was coined as Newhousephenomena . Mimicking this terminology we will refer to the Kolmorov typical coexistence ofinfinitely many attractors as
Berger phenomena . a r X i v : . [ m a t h . D S ] F e b BARRIENTOS AND RAIBEKAS
Newhouse phenomena has been showen to occur in open sets of di ff eomorphisms havinga dense subset of systems displaying homoclinic tangencies associated with saddle periodicpoints. Such an open set of dynamical systems is called a Newhouse domain . In certaincases, these open sets are also the support of many other interesting phenomena such as thecoexistence of infinitely many attracting invariant circles [GST08] and infinitely many strangeattractors [Col98, Lea08], or wandering domains [KS17] among others. Berger phenomenaalso occurs with respect to some open set but now in the topology of parametric families.Namely, in open sets where the families having persistent homoclinic tangencies are dense.As before, mimicking the terminology, we will refer to these open sets of parametric familiesas
Berger domains . In the original paper of Berger [Ber16, Ber17], these open sets wereimplicitly constructed for sectional dissipative dynamics. In this paper, we will introduceformally the notion of a Berger domain and construct new examples, not necessarily forsectional dissipative dynamics. As an application, we will prove Berger phenomena for acertain type of non-sectional dissipative Berger domains and obtain that the coexistence ofinfinitely many attracting invariant circles is also Kolmogorov typical.1.1.
Degenerate unfoldings. A C r -di ff eomorphism f of a manifold M has a homoclinic tan-gency if there is a pair of points P and Q , in the same transitive hyperbolic set, so that theunstable invariant manifold of P and the stable invariant manifold of Q have a non-transverseintersection at a point Y . The tangency is said to be of codimension c > c = c Y ( W u ( P ) , W s ( Q )) def = dim M − dim( T Y W u ( P ) + T Y W s ( Q )) . This number measures how far from being transverse is the intersection between the invari-ant manifolds at Y . Since the codimension of W u ( P ) coincides with the dimension of W s ( Q )we have, in this case, that the codimension c at Y coincides with dim T Y W u ( P ) ∩ T Y W s ( Q ).A homoclinic tangency can be unfolded by considering a k -parameter family in the C d , r -topology with 1 ≤ d ≤ r . That is, a C d -family ( f a ) a of C r -di ff eomorphisms parameterizedby a ∈ I k with f a = f where I = [ − ,
1] and k ≥ Y of codimension c is said to be C d -degenerate at a = a if thereare points p a ∈ W u ( P a ), q a ∈ W s ( Q a ) and c -dimensional subspaces E a , F a of T p a W u ( P a ) and T q a W s ( Q a ) respectively such that d ( p a , q a ) = o ( (cid:107) a − a (cid:107) d ) and d ( E a , F a ) = o ( (cid:107) a − a (cid:107) d ) at a = a . Here P a and Q a are the continuations of P a = P and Q a = Q for f a . Also p a = q a = Y and ( p a , E a ), ( q a , F a ) vary C d -continuously with respect to the parameter a ∈ I k . Observe thatin this case it is necessary to assume that d < r because the above definition involves thedynamics of the family ( f a ) a in the tangent bundle (in fact, in certain Grassmannian bundles).In [Ber16], the notion of C d -degenerate unfoldings of homoclinic tangencies were introducedfor short under the name of C d -paratangencies .1.2. Berger domains.
Let us remind the reader the notion of Newhouse domains. Follow-ing [BD12], we say that a C r -open set N of di ff eomorphisms is a C r -Newhouse domain (oftangencies of codimension c >
0) if there exists a dense set D in N such that every g ∈ D ERGER DOMAINS 3 has a homoclinic tangency (of codimension c >
0) associated with some hyperbolic periodicsaddle. A C r -Newhouse domain N ( r ≥
1) of homoclinic tangencies (of codimension one)associated with sectional dissipative periodic points gives rise to the C r -Newhouse phenom-enon. Namely, there exists a residual subset R of N where every g ∈ R has infinitely manyhyperbolic periodic attractors [New74, New79, PT93, GTS93, PV94, Rom95, GST08]. AsBerger showed in [Ber16], open sets of families displaying degenerate unfoldings play thesame role for parametric families as Newhouse domains do for the case free of parameters.For this reason mimicking the above terminology, one could say that: An open set U of k-parameter C d -families of C r -di ff eomorphisms is called a C d , r -Bergerdomain of paratangencies (of codimension c > ) if the following holds. There exists adense set D ⊂ I k × U such that for any ( a , f ) ∈ D , the family f = ( f a ) a displays a C d -degenerate unfolding at a = a of a homoclinic tangency (of codimension c > ) associatedwith a hyperbolic periodic saddle. For codimension c =
1, this definition appears implicitly in [Ber16] where it is proven that thecoexistence of infinitely many hyperbolic periodic attractors is Kolmogorov typical. Actually,by modifying the initial construction Berger showed a stronger result in [Ber16, Ber17] thatwe will refer to as C d , r -Berger phenomena : the existence of a residual set in a C d , r -open setof parametric families where each family has infinitely many sinks at any parameter. Thefollowing stronger version of the above tentative definition allowed Berger to prove such aresult: Definition 1.1.
An open set U of k-parameter C d -families of C r -di ff eomorphisms is called C d , r -Berger domain of persistent homoclinic tangencies (of codimension c > ) if there exists a densesubset D of U such that for any f = ( f a ) a ∈ D there is a covering of I k by open balls J i having thefollowing property: there is a continuation of a saddle periodic point Q a having a homoclinic tangencyY a (of codimension c > ) which depends C d -continuously on the parameter a ∈ J i . Observe that the first tentative definition above requires d < r because of the notion ofthe C d -paratangency. However, definition 1.1 admits d ≤ r since it deals with the notionof a C d -persistent homoclinic tangency. The following result shows the existence of Bergerdomains of large codimension for families of di ff eomorphisms: Theorem A.
Any manifold of dimension m > c + c admits an open set U of k-parameter C d -familiesof C r -di ff eomorphisms with < d < r − , so that U is a C d , r -Berger domain of persistent homoclinictangencies of codimension c > . The proof of Theorem A is based on the notion of a C d -degenerate unfolding of homoclinictangencies and previous results from [BR21]. For this reason, we have only been ableto show the existence of C d , r -Berger for families of di ff eomorphisms with d < r − m ≥
3. Recall that, in the case of codimension c =
1, Berger, inhis original papers [Ber16] and [Ber17], constructed this kind of open sets for C d -families of C r -endomorphisms in any surface with 1 ≤ d ≤ r . Afterwards this construction is lifted to BARRIENTOS AND RAIBEKAS C d -families of sectionally dissipative C r -di ff eomorphims in manifolds of dimension m ≥ ff eomorphisms in dimension m = Q ja , having unstable index c and the sametype of multipliers. For instance, we can take these points to be sectionally dissipative (asin the original construction of Berger) but also these saddles can be of type (1 , , ,
2) or (2 ,
2) according to the nomenclature introduced in [GST08]. We remark that in thecodimension one case we may assume the homoclinic tangencies are simple , also in thesense of [GST08].1.3. Berger phenomena:
The C d , r -Berger phenomena was shown in [Ber16, Ber17] for sec-tionally dissipative families in dimension m ≥
3. We will obtain similar results for familiesthat are not sectionally dissipative by working with a C d , r -Berger domain U of type (2 ,
1) withunstable index one. That is, the persistent homoclinic tangencies are simple and associatedwith hyperbolic periodic points having multipliers λ , . . . , λ m − and γ satisfying | λ j | < | λ | < < | γ | and | λ γ | < < | λγ | for j (cid:44) , λ , = λ e ± i ϕ with ϕ (cid:44) , π . (1)In the following result we obtain Berger phenomena with respect to attracting invariantcircles and hyperbolic sinks for these new types of Berger domains. Theorem B.
Let U be a a C d , r -Berger domain whose persistent homoclinic tangencies are simplesand associated with hyperbolic periodic points having multipliers satisfying (1) . Then there exists aresidual set R ⊂ U such that for every family f = ( f a ) a ∈ R and every a ∈ I k , the di ff eomorphism f a has simultaneously- infinitely many normally hyperbolic attracting invariant circles and- infinitely many hyperbolic periodic sinks Topology of families of di ff eomorphisms. Set I = [ − , < d ≤ r ≤ ∞ , k ≥ M we denote by C d , r ( I k , M ) the space of k -parameter C d -families f = ( f a ) a of C r -di ff eomorphisms f a of M parameterized by a ∈ I k such that ∂ ia ∂ jx f a ( x ) exists continuously for all 0 ≤ i ≤ d , 0 ≤ i + j ≤ r and ( a , x ) ∈ I k × M . We endow this space with the topology given by the C d , r -norm (cid:107) f (cid:107) C d , r = max { sup (cid:107) ∂ ia ∂ jx f a ( x ) : 0 ≤ i ≤ d , ≤ i + j ≤ r } where f = ( f a ) a ∈ C d , r ( I k , M ).1.5. Structure of the paper.
Section §2 contains the proof of Theorem A. Independently insection §3 we prove Theorem B. Actually, the proof of Theorem B only requires Definition 1.1. The tangency is called simple if it is quadratic, of codimension one and in the case the ambient manifold hasdimension m >
3, any extended unstable manifold is transverse to the leaf of the strong stable foliation whichpasses through the tangency point. Observe that these conditions are generic.
ERGER DOMAINS 5
2. B erger domains : P roof of T heorem AIn this section we will prove the existence of C d , r -Berger domains of codimension u > ff eomorphisms with d < r − m > u + u ≥ U ⊂ C d , r ( I k , M ) for 0 < d < r − M ≥ f = ( f a ) a ∈ U has a C d -degenerate unfolding of a homoclinic tangency of codimension u at a = a ∈ I k ). The construction of this open set is local and only requires twoingredients: a family of blenders (a certain type of a hyperbolic basic set) Γ = ( Γ a ) a and afamily of folding manifolds ( S a ) a (a certain type of manifold that folds along some direction).We refer to [BR21] for a precise definition of these objets. To be more specific, the main resultcould be stated as follows: Theorem 2.1 ([BR21, Thm. 7.5, Rem. 7.6]) . For any < d < r − and k ≥ , there exists aC d -family Φ = ( Φ a ) a of locally defined C r -di ff eomorphisms of M of dimension m > u + u havinga family of cs-blenders Γ = ( Γ a ) a with unstable dimension u ≥ and a family of folding manifolds S = ( S a ) a of dimension m − u satisfying the following:For any a ∈ I k , any family g = ( g a ) a close enough to f in the C d , r -topology and any C d , r -perturbation L = ( L a ) a of S there exists z = ( z a ) a ∈ C d ( I k , M ) such that(1) z a ∈ Γ g , a , where Γ g , a denotes the continuation for g a of the blender Γ a ,(2) the family of local unstable manifolds W = ( W uloc ( z a ; g a )) a and L have a tangency of dimensionu at a = a which unfolds C d -degenerately. Let us consider the family
Φ = ( Φ a ) a given in the above theorem. Assume in addition thenext hypothesis:(H1) Φ a has a equidimensional cycle between saddle periodic points P a and Q a ,(H2) P a belongs to Γ a and the folding manifold S a is contained in W s ( Q a , Φ a ).Theorem 2.1 implies that the family Φ = ( Φ a ) a under the above assumptions (H1) and (H2)defines a C d , r -open set U = U ( Φ ) of k -parameter C d -families of C r -di ff eomorphisms suchthat any g = ( g a ) a ∈ U is a C d -degenerate unfolding at any parameter a = a of a tangency ofdimension u . The tangency is between W s ( Q a , g a ) and the local unstable manifold of somepoint in the blender Γ a of g a . Since the codimension of W s ( Q a , g a ) and the dimension ofthe local unstable manifolds of Γ a coincide, the tangency also has codimension u . We willprove that the open set U is a C d , r -Berger domain. To do this, we will first need the followingresult, see [Ber16, Lemma 3.7] and [Ber17, Lemma 3.2]. Proposition 2.2 (Parametrized Inclination Lemma) . Let g = ( g a ) a be a C d , r -family of di ff eomor-phisms having a family K = ( K a ) a of transitive hyperbolic sets K a with unstable dimension d u . Let C a be a C r -submanifold of dimension d u that intersects transversally a local stable manifold W sloc ( x a , g a ) with x a ∈ K a at a point z a which we assume depends C d -continuously on a ∈ I k . Then, for any P a ∈ K a BARRIENTOS AND RAIBEKAS there exists a d u -dimensional disc D a ⊂ C a containing z a such that the family of discs D = ( g na ( D a )) a is C d , r -close to W = ( W uloc ( P a , g a )) a , for n su ffi ciently large. Using the parameterized inclination lemma, the following proposition proves that theabove open set U is a Berger domain according to the first tentative (weaker) definitiongiven in §1.2. Proposition 2.3.
For any a ∈ I k and g ∈ U , there is C d , r -arbitrarily close to g a family f = ( f a ) a such that f a = g a for any parameter far from a small neighborhood of a and which displays a C d -degenerate unfolding at a = a of a homoclinic tangency of codimension u associated with the periodicpoint Q a ( f ) .Proof. By construction, any g = ( g a ) a ∈ U is a C d -degenerate unfolding at a = a of a tangencybetween W s ( Q a , g a ) and some local unstable manifold W uloc ( x a , g a ) of a point x a ∈ Γ a . Fromthe assumptions (H1) and (H2) we have that both x a and Q a belongs to the homoclinic class H ( P a , g a ) of P a for g a . Moreover, we get that W u ( Q a , g a ) intersects transversally W sloc ( x a , g a )at a point z a which depends C d -continuously on a ∈ I k . Then Proposition 2.2 implies theexistence of discs D a in W u ( Q a , g a ) containing z a such that the family D n = ( g na ( D a )) a is C d , r -close to W = ( W uloc ( x a , g a )) when n is large. By a small perturbation, we now will find a newfamily C d , r -close to g , which is a C d -degenerate unfolding at a = a of a homoclinic tangencyof codimension u associated with the continuation of the periodic point Q a .We take local coordinates denoted by x in a neighborhood of x a which correspond tothe origin. Also denote by y the tangency point between W s ( Q a , g a ) and the local unstablemanifold W uloc ( x a , g a ). Since the tangency (of dimension u ) unfolds C d -degenerately, wehave (cid:126) p = ( p a , E a ) a , (cid:126) q = ( q a , F a ) a ∈ C d ( I k , G u ( M )) such that q a ∈ W s ( Q a , g a ) , and E a ⊂ T q a W s ( Q a , g a ) , p a ∈ W uloc ( x a , g a ) and F a ⊂ T p a W uloc ( x a , g a ) , p a = q a = y and J ( (cid:126) p ) = J ( (cid:126) q ) . Take δ > δ -neighborhoods of y and its iterationsby g a and g − a are pairwise disjoint. Let U be the 2 δ -neighborhood of y and assume that p a and q a belong to U for all a close enough to a . Call C a the local disc in W s ( Q a , g a ) containingthe point q a and we may suppose that U is such that the forward iterates of C a , with respectto g a , are disjoint from each other and from U . Since D n = ( g na ( D a )) a is C d , r -close to W , weobtain a C d , r -family τ n = ( τ n , a ) a of di ff eomorphisms of R m which sends, in local coordinates, g n ( D a ) onto W uloc ( x a , g a ), is equal to the identity outside of U and is C d , r -close to the constantfamily I = (id R m ) a as n → ∞ . Let t a be the point in D n ⊂ W u ( Q a , g a ) so that τ n , a ( t a ) = p a .Consider a C ∞ -bump function φ : R → R with support in [ − ,
1] and equal to 1 on[ − / , / ρ : a = ( a , . . . , a k ) ∈ I k (cid:55)→ φ ( a ) · . . . · φ ( a k ) ∈ R . For a fixed α >
0, define the perturbation g n = ( g n , a ) a of g = ( g a ) a by g n , a = H n , a ◦ g a for all a ∈ I k , ERGER DOMAINS 7 where H n , a in the above local coordinates takes the form H n , a ( x ) = x + θ · ( τ n , a ( x ) − x ) where θ = ρ (cid:18) a − a α (cid:19) φ (cid:32) (cid:107) x − y (cid:107) δ (cid:33) and is the identity otherwise. Observe that if a (cid:60) a + ( − α, α ) k or x (cid:60) U then H n , a ( x ) = x . Inparticular, g n , a ( x ) = g a ( x ) if a (cid:60) a + ( − α, α ) k or x (cid:60) g − a ( U ).On the other hand, if a ∈ a + ( − α, α ) k and x ∈ U , then H n , a ( x ) = τ n , a ( x ). This implies thatfor a ∈ a + ( − α, α ) k , the point g − a ( t a ) that belongs to W u ( Q a , g a ) is sent by g n , a to p a = τ n , a ( t a )and therefore p a ∈ W u ( Q a , g n , a ). Moreover, since τ n , a is a C r -di ff eomorphism ( r ≥
2) we alsohave that F a ⊂ T p a W u ( Q a , g n , a ).At a = a we have that τ n , a ( t a ) = p a = q a ∈ W s ( Q a , g n , a ) and so the stable and unstablemanifolds of Q a for g n , a meet at this point. Moreover, since τ n , a is a C r -di ff eomorphism( r ≥
2) this intersection is still non-transverse, i.e., we have a homoclinic tangency of codi-mension u . Observe that the perturbed family g n , a does not a ff ect the disc C a of the stablemanifold of W s ( Q a , g a ) in U . That is, C a ⊂ W s ( Q a , g n , a ) and E a ⊂ T q a W s ( Q a , g n , a ). Hence,since from the initial hypothesis J ( (cid:126) p ) = J ( (cid:126) q ) we get a C d -degenerate unfolding at a = a of ahomoclinic tangency of codimension u associated with the hyperbolic periodic point Q a ( g n ).Finally, observe (cid:107) f − g n (cid:107) C d , r = (cid:107) ( I − H ) ◦ f (cid:107) C d , r ≤ (cid:107) f (cid:107) C d , r (cid:107) θ ( τ n − I ) (cid:107) C d , r ≤ (cid:107) f (cid:107) C d , r o α → ( α − d ) o δ → ( δ − r ) (cid:107) τ n − I (cid:107) C d , r . Since (cid:107) τ n − I (cid:107) C d , r goes to zero as n → ∞ , we can obtain that for a given (cid:15) >
0, there are n largeenough so that (cid:107) f − g n (cid:107) C d , r < (cid:15) . (cid:3) Remark 2.4.
Notice that the perturbation in the previous proposition, to create the degeneratehomoclinic unfolding at a , is local (in the parameter space and in the manifold). Thus, fixing N ∈ N and a finite number of points a , . . . , a N ∈ I k , we can perform the same type of perturbation inductivelyand obtain a dense set of families in U having degenerate unfoldings at any a = a i for i = , . . . , N. The following proposition is an adaptation to the context of di ff eomorphisms of [Ber16,Lemma 5.4]. Roughly speaking, this proposition explains how it is possible to ”stop” atangency for an interval of parameters using a degenerate unfolding, i.e, how to create apersistent homoclinic tangency in the language of [Ber17]. Proposition 2.5.
Let f = ( f a ) a be a k-parameter C d -family of C r -di ff eomorphisms of a manifold ofdimension m ≥ . Suppose that f is a C d -degenerate unfolding at a = a of a homoclinic tangencyof codimension u > associated with a hyperbolic periodic point Q. Then, for any (cid:15) > there exists α > such that for every < α < α there is a C d -family h = ( h a ) a of C r -di ff eomorphisms such that(1) h is (cid:15) -close to f in the C d , r -topology,(2) h a = f a for every a (cid:60) a + ( − α, α ) k ,(3) h a has a homoclinic tangency Y a of codimension u associated with the continuation Q a of Qfor all a ∈ a + ( − α, α ) k and which depend C d -continuously on the parameter a.Proof. By assumption f is a C d -degenerate unfolding at a = a of a homoclinic tangency Y associated with a hyperbolic periodic point P . By the definition of degenerate unfolding BARRIENTOS AND RAIBEKAS we have points p a ∈ W u ( Q a , f a ), q a ∈ W s ( Q a , f a ) and c -dimensional subspaces E a and F a of T p a W u ( Q a , f a ) and T q a W s ( Q a , f a ) respectively such that d ( p a , q a ) = o ( (cid:107) a − a (cid:107) d ) and d ( E a , F a ) = o ( (cid:107) a − a (cid:107) d ) at a = a . (2)Here Q a is the continuation of Q a = P for f a . Also p a = q a = Y and ( p a , E a ), ( q a , F a ) vary C d -continuously with respect to the parameter a ∈ I k . We take local coordinates x in aneighborhood of Q which correspond to the origin. By considering an iteration if necessary,we assume that the tangency point Y belongs to this neighborhood of local coordinates. Take δ > δ -neighborhoods of Y and its iterations by f a and f − a are pairwise disjoint.Namely, we will denote by U the 2 δ -neighborhood of Y . Assume that p a and q a belong to U for all a close enough to a . From (2) it follows that p a − q a = o ( (cid:107) a − a (cid:107) d ) at a = a . (3)Observe that the Grasmannian distance between E a and F a is given by the norm of I − R a restricted to F a , where I denotes the identity and R a is the orthogonal projection onto E a .Then we obtain from (2) that I − R a = o ( (cid:107) a − a (cid:107) d ) at a = a . (4)We would like that the rotation occurs around the point p a and so consider ˜ R a x = R a ( x − p a ) + p a .Since ( I − ˜ R a ) x = ( I − R a ) x + ( R a − I ) p a then from (4) we still have I − ˜ R a = o ( (cid:107) a − a (cid:107) d ) at a = a . (5)Consider a C ∞ -bump function φ : R → R with support in [ − ,
1] and equal to 1 on[ − / , / ρ : a = ( a , . . . , a k ) ∈ I k (cid:55)→ φ ( a ) · . . . · φ ( a k ) ∈ R . For a fixed α >
0, define the perturbation h = ( h a ) a of f = ( f a ) a by the relation h a = H a ◦ f a for all a ∈ I k . Here H a in the above local coordinates takes the form¯ x = (cid:16) I − θ · ( I − ˜ R a ) (cid:17)(cid:16) x − θ · ( q a − p a ) (cid:17) where θ = ρ (cid:18) a − a α (cid:19) φ (cid:18) (cid:107) x − Y (cid:107) δ (cid:19) and is the identity otherwise. Observe that if a (cid:60) a + ( − α, α ) or x (cid:60) U then H a ( x ) = x . Inparticular, h a ( x ) = f a ( x ) if a (cid:60) a + ( − α, α ) or x (cid:60) f − a ( U ). On the other hand, if a ∈ a + ( − α, α )and x ∈ U then H a ( x ) = ˜ R a (cid:16) x − ( q a − p a ) (cid:17) . Since ˜ R a fixes the point p a , we have that H a ( q a ) = p a . This implies that the point q − a = f − a ( q a )that belongs to W u ( Q a , f a ) is sent by h a to p a which belongs to W sloc ( Q a , f a ). As the orbit of f na ( p a ) for n ≥ f − na ( q − a ) for n > f − a ( U ), then p a ∈ W sloc ( P a , h a ) and q − a ∈ W u ( Q a , h a ). Thus, the stable and unstable manifolds of Q a for h a meet at p a = h a ( q − a ).Moreover, F − a = D f a ( q − a ) F a is a subspace of T q − a W u ( Q a , f a ) = T q − a W u ( Q a , h a ) and Dh a ( q − a ) F − a = DH a ( q a ) D f a ( q − a ) F − a = DH a ( q a ) F a = D ˜ R a ( q a ) F a = R a F a = E a . ERGER DOMAINS 9
Since E a is a subspace of T p a W s ( Q a , f a ) = T p a W s ( Q a , h a ), then the intersection between thestable and unstable manifolds of Q a for h a is tangencial.To conclude the proposition we only need to prove that for a given (cid:15) > α such that for any 0 < α < α the above perturbation h = h ( α ) of f is actually (cid:15) -close in the C d , r -topology. Notice that the C d , r -norm satisfies (cid:107) h − f (cid:107) C d , r = (cid:107) ( H − I ) ◦ f (cid:107) C d , r ≤ (cid:107) I − H (cid:107) C d , r (cid:107) f (cid:107) C d , r . Thus we only need to calculate the C d , r -norm of the family ( I − H a ) a . Since H a = I if a (cid:60) a + ( − α, α ) k or x (cid:60) U then (cid:107) I − H a (cid:107) C d , r ≤ (cid:13)(cid:13)(cid:13) θ · ( I − ˜ R a )( x − θ · ( q a − p a )) + θ · ( q a − p a ) (cid:13)(cid:13)(cid:13) C d , r . Since the C d , r -norm of φ ( (cid:107) x − Y (cid:107) / δ ) is bounded (depending only on δ ), we can disregard thisfunction from the estimate. Then, to bound the C d , r -norms from above it is enough to showthat for a ∈ a + ( − α, α ) k the functions F α ( a ) = ρ ( a − a α )( p a − q a ) and G α ( a ) = ρ ( a − a α )( I − ˜ R a )have C d -norm small when α is small enough. But this is clear from (3) and (5), as having intoaccount that (cid:107) a − a (cid:107) ≤ α , it follows that F a ( a ) = α − d · o a → a ( (cid:107) a − a (cid:107) d ) = o α → (1) and G α ( a ) = α − d · o a → a ( (cid:107) a − a (cid:107) d ) = o α → (1) . This completes the proof. (cid:3)
Remark 2.6.
Observe that the positive constant α in the above proposition depends initially on thefamily f = ( f a ) a , the constants (cid:15) > , δ > and the parameter a ∈ I k . The dependence of a comesfrom the function o a → a ( (cid:107) a − a (cid:107) d ) in (2) . However, one can bound this function by ν ( (cid:107) a − a (cid:107) ) ·(cid:107) a − a (cid:107) d where lim t → ν ( t ) = , controlling the modulus of continuity ν of the derivatives of the unfolding. Inthis form we can get that α does not depend on the parameter a . Also, similarly to what was donein the previous proposition, the surgery using bump functions around a neighborhood of the initialparatangency point Y can actually be done around any point f Na ( Y ) belonging to W sloc ( P a , f a ) . Thisallows us to fix a priori a uniform δ > because we only need to control the distance between oneforward / backward iterate of f Na ( Y ) . Thus, α also does not depend on δ . Finally, if f belongs to U then one can obtain a uniform bound on the continuity modulus using the fact that we are dealingwith compact families of local stable and unstable manifolds. This proves that in this construction α only depends on (cid:15) and U (i.e, on the dynamics of the organizing family Φ ). Remark 2.7.
In the case of codimension u = , the tangency Y a obtained in the previous propositioncould be assumed simple in the sense of [GST08] . Finally, in the next theorem we will show the existence of Berger domains as statedin Definition 1.1. The idea behind the proof is the replication of the arguments comingfrom [Ber16, Sec. 6.1] and [Ber17, Sec. 7].
Theorem 2.8.
There exists a dense subset D of U such that for any h = ( h a ) a ∈ D there is a coveringof I k by open sets J i having a persistent homoclinic tangency of codimension u. That is, U is aC d , r -Berger domain (of paratangencies of codimension u). Proof.
Fora a fixed family g = ( g a ) a ∈ U and (cid:15) >
0, Propositions 2.3, 2.5 and Remarks 2.4, 2.6imply the following. We obtain α = α ( (cid:15) ) > α with 0 < α < α there arepoints a , . . . , a N in I k such that- the open sets a i + ( − α, α ) k for i = , . . . , N are pairwise disjoint;- the union of ( a i + ( − α, α ) k ) ∩ I k for i = , . . . , N is dense in I k ;- there is a C d , r -family h = ( h a ) a , (cid:15) -close to g , having a persistent homoclinic tangencyassociated with the continuation of Q a for all a ∈ ( a i + ( α, α ) k ) ∩ I k and i = , . . . , N .However this result does not provide an open cover of I k . We need to perturbe again h without destroying the persistent homoclinic tangencies associated with Q a and at thesame time provide new persistent homoclinic tangencies in the complement of the union of( a i + ( α, α ) k ) ∩ I k for i = , . . . , N . To do this, we will need a finite set of di ff erent periodicpoints Q ja to replicate the above argument and ensure that each new perturbation does notmodify the previous one (i.e, there exists a common δ > Lemma 2.9.
There exists a set { Q ja } j = ,..., k of k hyperbolic periodic points satisfying the assump-tions (H1) and (H2).Proof. Since the homoclinic class H ( Q a , Φ a ) is not trivial (contains P a ), there exists a horseshoe Λ a containing Q a . Thus, there are infinitely many di ff erent hyperbolic periodic points of Φ a whose stable manifold intersect transversely the unstable manifold of Q a . Then by theinclination lemma, and the robustness of the folding manifold S a , the stable manifold ofthese periodic points also contains a folding manifold. (cid:3) Associated with each point Q ja , as in Proposition 2.5, there is the paratangency point Y j where the size of the perturbation is governed by δ j . Thus, we can obtain a uniform δ bytaking the infimum over δ j . Also corresponding to each Q ja , there exists a lattice of points { a ji } i = ,..., N in I k such that the union of the α -neighborhoods a ji + ( α, α ) k in I k cover I k . That is, I k ⊂ k (cid:91) j = N (cid:91) i = a ji + ( α, α ) k . Then we can apply Proposition 2.5 independently in j to obtain the family with the requiredproperties, concluding the proof the the theorem. (cid:3) Remark 2.10.
The persistent homoclinic tangencies in the open set J i obtained in the above theoremcan be associated with a collection of saddle periodic points, Q ja for j = , . . . , k , where all of themhave the same type of multipliers. For instance, we can take these points being sectionally dissipativeor of type (1 , , (2 , , (1 , or (2 , according to the nomenclature introduced in [GST08] . Also,according to Remark 2.7, in the codimension one case, the persistent homoclinic tangency can beassumed simple in the sense of [GST08] . ERGER DOMAINS 11
3. P roof of T heorem B: periodic sinks and invariant circles In this section we will prove the C d , r -Berger phenomenena of the coexistence of infinitelymany normally hyperbolic attracting invariant circles and also obtain a similar result forhyperbolic periodic sinks. For short, we will refer to both of these types of attractors as periodic attractors .The next proposition claims that every family f = ( f a ) a in U can be approximated by afamily g = ( g a ) a having a periodic attractor for every parameter a in I k . Moreover, the periodof the attractors can be chosen arbitrarily large. To prove this, since U is a Berger domain,it is enough to restrict our attention to the dense set D of U having persistent tangencies. Proposition 3.1.
For any (cid:15) > , and every f = ( f a ) a ∈ D there exists n = n ( ε, f ) ∈ N suchthat for any n ≥ n there is a (cid:15) -close family g = ( g a ) a to f = ( f a ) a in the C d , r -topology satisfyingthat g a has a periodic attractor of period n for all a ∈ I k . Moreover, the attractor obtained for g a isthe continuation of a (hyperbolic or normally hyperbolic) n-periodic attractor obtained for a map g a ,where a belongs to a finite collection of parameters. Before proving the above proposition, we will conclude first from this result the next maintheorem of the section, which in particular proves Theorem B.
Theorem 3.2.
For any m ∈ N , there exists an open and dense set O m in U such that for any familyg = ( g a ) a in O m there exist positive integers n < · · · < n m so that g a has a periodic attractor ofperiod n (cid:96) for all a ∈ I k and (cid:96) = , . . . , m. Moreover, there is a residual subset R of U such that anyg = ( g a ) a ∈ R satisfies that g a has both infinitely many hyperbolic periodic sinks and infinitely manynormally hyperbolic attracting invariant circles for all a ∈ I k .Proof. First of all consider the sequence (cid:15) i = / i for i ≥
1. We will prove the result byinduction. To do this, we are going first to construct O m for m = f = ( f a ) a in D taking a sequences of integers n ( i ) ≥ n ( (cid:15) i , f ) for all i ≥
1, we find a (cid:15) i -close family g = ( g a ) a to f such that g a has a n ( i )-periodicattractor for all a ∈ I k . Actually, for any parameter a , the attractor that we obtain for g a is thecontinuation of a (hyperbolic or normally hyperbolic) n ( i )-periodic attractor obtained for amap g a where a belongs to a finite collection of parameters. Thus, from the hyperbolicityof the attractor, this property persists under perturbations and then we have a sequence ofopen sets O ( (cid:15) i , f ) converging to f where the same conclusion holds for any family in theseopen sets. By taking the union of all these open sets for any f in D and (cid:15) i > i ≥
1, we getan open and dense set O in U where for any g = ( g a ) a ∈ O there exists a positive integer n such that g a has an n -periodic attractor for all parameters a ∈ I k .Now we will assume that O m was constructed and show how to obtain O m + . Since O m is open and dense set in U we can start now by taking f = ( f a ) a ∈ O m ∩ D . Hence,there exist positive integers n < · · · < n m so that f a has a persistent n (cid:96) -periodic attractor(a sink or an invariant circle) for all a ∈ I k for (cid:96) = , . . . , m . As before, these attractors arethe smooth continuation of periodic attractors centered at a finite collection of parameters. Therefore, there exists (cid:15) (cid:48) = (cid:15) (cid:48) ( f ) > (cid:15) (cid:48) -close family g = ( g a ) a still has thesame properties. Then, for any (cid:15) i < (cid:15) (cid:48) /
2, we can apply again Proposition 3.1, taking integers n ( i ) m + > max { n ( (cid:15) i , f ) , n m } in order to obtain an (cid:15) i -perturbation g = ( g a ) a of f such that g a has also a n ( i ) m + -periodic attractor for all a ∈ I k . As before, from the persistence of theseattractors, we have a sequence of open sets O m + ( (cid:15) i , f ) ⊂ O m converging to f where the sameconclusion holds for any family in these open sets. By taking the union of all these open setsfor any f ∈ O m ∩ D and (cid:15) i < (cid:15) (cid:48) ( f ), we get an open and dense set O m + in U where for any g = ( g a ) a ∈ O m + there exist positive integers n < · · · < n m + such that g a has a n (cid:96) -periodicattractor for all a ∈ I k for all (cid:96) = , . . . , m + g = ( g a ) a belongs to the residual set R = ∩ O m then g a has infinitely many of attractors for all a ∈ I k . (cid:3) Now we will prove Proposition 3.1. To do this we need the following lemma.
Lemma 3.3.
Given α > , let g = ( g a ) a be a C d , r -family and assume that g a has a simple homoclinictangency at a point Y a (depending C d -continuously on a) associated with a saddle Q a satisfying (1) for any parameter a ∈ a + ( − α, α ) k . Then there exists a sequence of families g n = ( g na ) a approachingg in the C d , r -topology such that g na = g a if a (cid:60) a + ( − α, α ) k and g na has an n-periodic sink orinvariant circle for all a ∈ a + ( − α, α ) k . Moreover, for n large enough (cid:107) g n − g (cid:107) C d , r = O (cid:32) α − d n (cid:33) . Before proving this result, let us show how to get Proposition 3.1 from the above lemma.
Proof of Proposition 3.1.
Given f = ( f a ) a in D , relabeling and resizing if necessary, we canassume that the cover of I k by the open balls J i that appears in Definition 1.1 is of the form I k ⊂ M (cid:91) j = N j (cid:91) (cid:96) = J j (cid:96) with J j (cid:96) = a j (cid:96) + ( − α j (cid:96) , α j (cid:96) ) k a j (cid:96) ∈ I k and α j (cid:96) > . Moreover, the persistent homoclinic tangency Y a of f a on J j (cid:96) ∩ I k is simple in the senseof [GST08] and is associated with a saddle Q ja for j = , . . . , M , where for each j , the sets J j (cid:96) for (cid:96) = , . . . , N j are pairwise disjoint. To avoid unnecessary complications in the notation,we can assume that α j (cid:96) = α for all j = , . . . , M and (cid:96) = , . . . N j . Moreover, for each j , theintervals 2 J j (cid:96) = a j (cid:96) + ( − α, α ) k are pairwise disjoint with respect to (cid:96) .On the other hand, given (cid:15) >
0, according to Lemma 3.3, we can control the approximationby a function F ( α, n ) of order O ( α − d n − ). We take n = n ( (cid:15), f ) ∈ N where F ( α, n ) = O ( α − d n − ) < (cid:15) . Now, consider an integer n ≥ n . We want to find an (cid:15) -close family g = ( g a ) a having an n -periodic attractor at any parameter a ∈ I k . Having into account that for each j the intervals 2 J j (cid:96) are pairwise disjoint, we can apply Lemma 3.3 inductively to obtain an (cid:15) -close family g = ( g a ) a to f such that g a has an n -periodic attractor for all a ∈ J j (cid:96) . Thisconcludes the proof. (cid:3) ERGER DOMAINS 13
Finally to complete the proof we will show Lemma 3.3. However, in order to understandbetter how periodic sinks and invariant circles appear in the unfolding of homoclinic tan-gencies associated with saddles of the form (1), we need some preliminaries on the theoryof rescaling lemmas from [GST08].3.1.
Rescaling lemma: Generalized Henon map.
Let f be a C r -di ff eomorphism of a mani-fold of dimension m ≥ Q with multipliers satisfying the assumptions (1). We assume that the tangency is sim-ple in the sense of [GST08, sec. 1, pg. 928]. That is, the tangency is quadratic, of codimensionone and, in the case that the dimension m >
3, any extended unstable manifold is transverseto the leaf of the strong stable foliation which passes through the tangency point. We need toconsider a two-parameter unfolding f ε of f = f where ε = ( µ, ϕ − ϕ ) being µ the parameterthat controls the splitting of the tangency and ϕ the parameter related to the eigenvalues of Q (here ϕ is the value of ϕ at ε = T = T ( ε ) denote the local map and in this casethis map corresponds to f q ε defined in a neighborhood W of Q , where q is the period of Q .By T = T ( ε ) we denote the map f n ε defined from a neighborhood Π − of a tangent point Y − ∈ W uloc ( Q , f ) ∩ W of f to a neighborhood Π + of Y + = f n ( Y − ) ∈ W sloc ( Q , f ) ∩ W . Then, for n large enough, one defines the first return map T n = T ◦ T n on a subset σ n = T − n ( Π − ) ∩ Π + of Π + where σ n → W sloc ( Q ) as n → ∞ . According to [GST08, Lemma 1 and 3] we have thefollowing result: Lemma 3.4.
There exists a sequence of open sets ∆ n of parameters converging to ε = such thatfor this values the first-return T n has a two-dimensional attracting invariant C r -manifold M n ⊂ σ n so that after a C r -smooth transformation of coordinates, the restriction of the map is given by theGeneralized H´enon map ¯ x = y , ¯ y = M − Bx − y − R n ( xy + o (1)) . (6) The rescaled parameters M, B and R n are functions of ε ∈ ∆ n such that R n converges to zero asn → ∞ and M and B run over asymptotically large regions which, as n → ∞ , cover all finite values.Namely, M ∼ γ n ( µ + O ( γ − n + λ n )) , B ∼ ( λγ ) n cos( n ϕ + O (1)) and R n = J B ( λ γ ) n where J (cid:44) is the Jacobian of the global map T calculated at the homoclinic point Y − for ε = .The o (1) -terms tend to zero as n → ∞ along with all the derivatives up to order r with respect to thecoordinates and up to order r − with respect to the rescaled parameters M and B. Moreover, thelimit family is the Henon map. The dynamics of the following generalized H´enon map¯ x = y , ¯ y = M − Bx − y − R n xy (7)was studied in [GG00, GG04, GKM05] (see also [GGT07]). We present here the main resultsfor the case of small R n with emphasis on the stable dynamics (stable periodic orbits andinvariant circles) in order to apply the corresponding results to the first return maps coming L + n L − n BM HT n BT n L ϕ n L n Figure 1.
Bifurcation curves for the generalized H´enon map (7) with R n > . The case R n = corresponds with the bifurcation diagram of the H´enon map. In that case, L ϕ n collapses with L n atB = . The diagram for the case R n < is similar, changing the position of the curves L ϕ n and L n andthe stability of the invariant circle. from (6). Observe that the di ff erence between equations (7) (generalized H´enon map) and (6)(perturbed map) has order O ( R n ). Then the existence of stable periodic orbits and invariantcircles for (6) can be inferred from the bifurcation diagram of (7). In Figure 1 we show thebifurcation curves for the generalized H´enon map in (7) in the parameter space ( M , B ).The map (7) has in the parameter plane ( M , B ) the following three bifurcation curves L + n : M = − (1 + B ) + R n ) L − n : M =
14 (1 + B ) (3 + R n ) L ϕ n : M = cos ω − cos ω (2 + R n )(1 + R n / , B = + R n cos ω + R n / . These correspond to the existence of fixed points having multipliers on the unit circle: + M , B ) ∈ L + n ; − M , B ) ∈ L − n ; and e ± i ω at ( M , B ) ∈ L ϕ n . Note that the curve L ϕ n is written ina parametric form such that the argument ω (0 < ω < π ) of the complex multipliers is theparameter. We also point out here the pointsBT n : M = − − R n (1 + R n / , B = + R n + R n / n : M = + R n (1 + R n / , B = − R n / + R n / . (8) ERGER DOMAINS 15
KHL − n L ϕ n HT n Figure 2.
Bifurcation diagrams near a Horozov-Takens point for R n < . For R n = (H´enon map)the curves K and H collapse at B = . The diagram for the case R n > is similar but changesthe position of K, H and the stability of both invariant curves. The open domain between K and His rather small, having the size of a finite order along the M-direction and order O ( R n ) along theB-direction. The are called as follows:
Bogdanov-Takens point for BT n and the Horozov-Takens point for HT n .Also denoted by L n is an interesting curve (nonbifurcational) starting at the point BT n , whichcorresponds to the existence of a saddle fixed point of (7) of neutral type (i.e., the fixed pointhas positive multipliers whose product is equal to one). This curve is drawn in Figure 1 as thedotted line and its equation is given by the same expression of L ϕ n replacing cos ω by α > D sn , bounded by the curves L + n , L − n L ϕ n with vertices BT n , HT n (seeFigure 1), such that map (7) has a stable fixed point for parameters in D sn . The bifurcationsof periodic points with multipliers e ± i ω can lead to asymptotically stable or / and unstableinvariant circles. The first return map T n has an invariant circle which is either stable if R n > R n <
0. Observe that the sign of R n actually only depends on J (cid:44) J > J <
0, the existence of a stable closed invariant curve in (7) follows from thebifurcation analysis of the Horozov-Takens point HT n . Some of the elements that appear inthis non-degenerate bifurcation are showen in Figure 2. Actually, when J (cid:44) R n (cid:44) n there are open domains parameter values where stable and unstableclosed invariant curves coexist.Moreover, for any map that is O ( R n )-close to to (7) in the C -topology the correspondingbifurcations still remain non-degenerate and preserve the same stability. Thus, we can obtainthe same type of results for the pertubed map (6). To summarize for future reference, for n large enough we find open sets A n , A n of ( M , B )-parameters accumulating at BT = ( − , = (3 ,
1) respectively as n → ∞ such that the following holds. If ( M , B ) ∈ A n (resp. ( M , B ) ∈ A n ) then T n has a hyperbolic attracting periodic point (resp. a normallyhyperbolic attracting invariant circle).3.2. Proof of Lemma 3.3.
By assumption the map g a has a homoclinic tangency at a point Y a associated with a sectional dissipative periodic point Q a for all a ∈ a + ( − α, α ) k . Actually,the tangency must be smoothly continued until (cid:107) a − a (cid:107) ∞ = α . We consider a two-parameterunfolding g a ,ε of the homoclinic tangency Y a of g a for a ∈ a + [ − α, α ] k , where ε = ( µ, ϕ ). Here µ is the parameter that controls the splitting of the tangency and ϕ is the perturbation ofthe argument of the complex eigenvalues of Q a . We can take local coordinates ( x , u , y ) with x ∈ C , u ∈ R m − and y ∈ R in a neighborhood of Q a which corresponds to the origin, suchthat W sloc ( Q a ) and W uloc ( Q a ) acquire the form { y = } and { x = , u = } respectively. Moreover,the complex eigenvalues related to the stable manifold of Q a correspond to the x -variableand the tangency point Y a is represented by ( x + , u + , C ∞ -bump function φ : R → R with support in [ − ,
1] and equal to 1 on[ − / , / ρ : a = ( a , . . . , a k ) ∈ I k (cid:55)→ φ ( a ) · . . . · φ ( a k ) ∈ R . Take δ > δ -neighborhoods in local coordinates of Q a and g − a ( Y a ) are disjoint.Observe that these two neighborhoods, call U and V respectively, can be taken independentof a . We can write g a ,ε = H a ,ε ◦ g a , where H a ,ε in these local coordinates takes the form¯ x = (cid:32) − ρ (cid:18) a − a α (cid:19) φ (cid:32) (cid:107) ( x , u , y ) (cid:107) δ (cid:33) (1 − e i ϕ ) (cid:33) x ¯ u = u ¯ y = y + ρ (cid:18) a − a α (cid:19) φ (cid:32) (cid:107) ( x , u , y ) − ( x + , u + , (cid:107) δ (cid:33) µ, and is the identity otherwise. Observe that if a (cid:60) a + ( − α, α ) k then g a ,ε = g a and if( x , u , y ) (cid:60) U ∪ V then g a ,ε = g a .Recall in Section 3.1 the definition of the first return map associated with the unfoldingof a simple homoclinic tangency. Since the tangency point Y a depends C d -continuously on a + [ − α, α ] k , we may assume that the first-return map T n = T n ( a , ε ) also depends smoothlyas a function of the parameter a on a + [ − α, α ] k . Lemma 3.5 (Parametrized rescaling lemma) . There exist families of open sets ( ∆ n ( a )) a of param-eters converging to ε = as n → ∞ such that for any ε ∈ ∆ n ( a ) the map T n = T n ( a , ε ) has anattracting C r -manifold M n = M n ( a , ε ) for any a ∈ a + [ − α, α ] k . Moreover, there exists a C d , r -familyof transformations of coordinates which bring the first-return map T n restricted to M n to the formgiven by (6) ¯ x = y , ¯ y = M − Bx − y − R n ( xy + o (1)) . (9) ERGER DOMAINS 17
Here the rescaled parameters M = M n ( a , ε ) and B = B n ( a , ε ) are at least C d -smooth functions (recallthat d ≤ r − ) on ∆ n = (cid:110) ( a , ε ) : a ∈ a + ( − α, α ) k and ε ∈ ∆ n ( a ) (cid:111) . The same property holds for the coe ffi cient R n = R n ( a , ε ) and the o (1) -terms. More specifically,M ∼ γ na ( µ + O ( γ − na + λ na )) , B ∼ ( λ a γ a ) n cos( n ϕ + O (1)) and R n = J a B ( λ a γ a ) n (10) where λ a = λ a ( g ) and γ a = γ a ( g ) are the eigenvalues of Q a = Q a ( g ) satisfying (1) for g a .Proof. Let us analyze the proof of the rescaling lemma in [GST08], more specifically thechange of coordinates for the first return maps given in Section 3.2 [GST08] for the case(2 ,
1) with λγ >
1. From equations [GST08, Eq. (3.12)-(3.16)] we can observe the all thetransformations of coordinates can be performed smoothly on the parameter a ∈ a + [ − α, α ] k .The exponents that will appear in the orders of convergence will not depend on the parameter a but only on n . On the other hand the constants in the O -terms will depend on theparameter but these can be uniformely bounded due to the compactness of the parameterspace. These considerations allow us basically to apply the rescaling Lemma 3.4 smoothlyon the parameter a ∈ a + [ − α, α ] k . (cid:3) Lemma 3.6.
For n large enough, ∆ n ( a ) can be taken in such a way that φ n , a : ∆ n ( a ) → ( − , \ B (0 , r ) , φ n , a ( (cid:15) ) = ( M n ( a , (cid:15) ) , B n ( a , (cid:15) )) is a di ff eomorphism where M n and B n are the functions given in (10) for fixed n and a. Here B (0 , r ) is a closed ball of some small fixed radius r around the origin.Proof. Let us introduce the parameter value ε n ( a ) = ( µ n ( a ) , ϕ n ( a )), which by definition satisfies φ n , a ( ε n ( a )) =
0. It can be seen from the expressions of M and B in (10) that µ n ( a ) = O ( γ − na + λ na )and ϕ n ( a ) = π n + O (1 / n ). We take out B (0 , r ) around the origin in ( − , so that B isbounded away from zero and R n is well-defined. Similar values were considered in [GST08,pg. 946] and as is claimed there, the rescaled functions M and B can take arbitrarily finitevalues when µ varies close to µ n ( a ) and ϕ near ϕ n ( a ). Let us explain this. Although the O -functions in (10) depend on ε , the functions M and B basically only depend on µ and ϕ respectively for n large enough. Actually on [GST08, pg. 946] are giving explicit expressionsfor M n , a and B n , a . Using them one can observe that ∂ µ M n , a ∼ γ na (cid:44) ∂ ϕ B n , a ∼ n ( λ a γ a ) n sin( n ϕ + O (1)) (cid:44) ε = ( µ, ϕ ) close to ε n ( a ). Then the Jacobian of φ n , a converges to infinity, uniformly on a ∈ a + [ − α, α ] k as n → ∞ . The rate of growth is exponential. This implies that φ n , a is aninvertible expanding map with arbitrarily large uniform expansion on a ∈ a + [ − α, α ] k . Onthe other hand, the size of ∆ n ( a ) (coming mainly from considerations on the angle ϕ ) wherethe expanding map φ n , a is defined has decay of order O (1 / n ). Thus, for n large enough weget that a neighborhood of ε n ( a ) can be taken so that its image under φ n , a is ( − , . Inparticular we can take ∆ n ( a ) being di ff eomorphic to ( − , k \ B (0 , r ). (cid:3) Consequently, Φ n ( a , ε ) = ( a , φ n , a ( ε ))is a di ff eomorphism between the set ∆ n defined above and ( a + [ − α, α ] k ) × ( − , \ B (0 , r ).On the other hand, although the coe ffi cient R n depends on B , note that the range of valuesit takes is negligible when B is bounded from zero and n is large enough. Actually from therelations in (10) it follows that R n = o (1). Thus, the bifurcation diagram of (9) can be studiedfrom the results described in Section 3.1 assuming R n = o (1) independent of B .Let us remind the reader of the Bogdanov-Takens BT n ( a ) and the Horozov-Takens HT n ( a )points given in (8), which now also depend on a and accumulate at BT = ( − ,
1) andHT = (3 ,
1) respectively as n → ∞ . Hence, according to Section 3.1 for each n large enoughwe find open subsets A n ( a ), A n ( a ) in the ( M , B )-parameter plane such that if ( M , B ) ∈ A n ( a )(resp. ( M , B ) ∈ A n ( a )), then T n has a hyperbolic attracting periodic point (resp. attractingsmooth invariant circle) for all a ∈ a + [ − α, α ] k . Moreover, we can assume the points BT n ( a )and HT n ( a ) belong to the boundary of A n ( a ) and A n ( a ) respectively. Since these sets vary C d -continuously with respect to the parameter a ∈ [ − α, α ] k , we can choose C d -continuously( M ∗ n ( a ) , B ∗ n ( a )) ∈ A ∗ n ( a ) for ∗ = , n ( a ) and HT n ( a ) respectively.Since Φ n is a di ff eomorphism, we may find a C d -function ε ∗ n ( a ) = ( µ ∗ n ( a ) , ϕ ∗ n ( a )) for a ∈ a + [ − α, α ] k , ∗ = ,
2, defined as Φ − n ( a , ( M ∗ n ( a ) , B ∗ n ( a )) = ( a , ε ∗ n ( a )). In particular, M ∗ n ( a ) = M n , a ( ε ∗ n ( a )) ∼ γ na ( µ ∗ n ( a ) + O ( γ − na + λ na )) B ∗ n ( a ) = B n , a ( ε ∗ n ( a )) ∼ ( λ a γ a ) n cos( n ϕ ∗ n ( a ) + O (1)) . (11)Extending smoothly ε ∗ n ( a ) to I k we can consider the sequence of families ˜ g n = ( ˜ g n , a ) a where˜ g n , a = g a ,ε ∗ n ( a ) for a ∈ I k and n large enough . Observe that ˜ g n , a = g a for a (cid:60) a + ( − α, α ) k and we may assume has an n -periodic attractor(a sink or an invariant circle) for all a ∈ a + ( − α, α ) k .To conclude the proof of the lemma we only need to show that ˜ g n converges to g in the C d , r -topology. To do this, notice that the C d , r -norm satisfies (cid:107) ¯ g n − g (cid:107) = (cid:107) ( I − H a ,ε ∗ n ( a ) ) ◦ g a (cid:107) ≤ (cid:107) I − H a ,ε ∗ n ( a ) (cid:107) (cid:107) g (cid:107) where I denotes the identity and the C d , r -norm of any g = ( g a ) a in the Berger domain U isbounded. Thus, we only need to calculate the C d , r -norm of the family ( I − H a ,ε ∗ n ( a ) ) a . Since H a ,ε ∗ n ( a ) = I if a (cid:60) a + ( − α, α ) k or ( x , u , y ) (cid:60) U ∪ V , therefore (cid:107) I − H a ,ε ∗ n ( a ) (cid:107) is less or equal than (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρ (cid:18) a − a α (cid:19) φ (cid:32) (cid:107) ( x , u , y ) (cid:107) δ (cid:33) (1 − e i ϕ ∗ n ( a ) ) x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρ (cid:18) a − a α (cid:19) φ (cid:32) (cid:107) ( x , u , y ) − ( x + , u + , (cid:107) δ (cid:33) µ ∗ n ( a ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . To estimate the C d , r -norms above it su ffi ces to show that the functions F n ( a ) = ρ ( a − a α )(1 − e i ϕ ∗ n ( a ) ) and G n ( a ) = ρ ( a − a α ) µ ∗ n ( a ) ERGER DOMAINS 19 have C d -norm small when n is large and a ∈ a + ( − α, α ) k . Actually we will prove thefollowing: (cid:107) F n (cid:107) C d = O (cid:32) α − d n (cid:33) and (cid:107) G n (cid:107) C d = O (cid:32) α − d n (cid:33) (12)Observe that this assertion completes the proof. To prove this we will need the followingderivative estimates on the functions µ ∗ n ( a ) and ϕ ∗ n ( a ). Here the symbol ∂ ja is used to denote the j -th partial derivative with respect to the coordinates a i of a using the multi-index notation. Lemma 3.7.
For ≤ | j | ≤ d(i) µ ∗ n ( a ) = O ( γ − na + λ na ) and ∂ ja µ ∗ n ( a ) = O ( n | j | ( γ − na + λ na )) . (ii) ϕ ∗ n ( a ) = O (cid:16) n (cid:17) , ∂ ja ϕ ∗ n ( a ) = O (cid:16) n | j |− ( γ a λ a ) n (cid:17) and ∂ ja ( e i ϕ ∗ n ( a ) ) = O (cid:16) n | j |− ( γ a λ a ) n (cid:17) . Assuming the above lemma let us now prove the estimates in (12), starting with the secondone. To do this, using the Leibniz formula, ∂ (cid:96) a G n ( a ) = (cid:88) j : j ≤ (cid:96) (cid:32) (cid:96) j (cid:33) ∂ (cid:96) − ja ρ (cid:18) a − a α (cid:19) · ∂ ja µ ∗ n ( a ) . Substituting the estimate from Lemma 3.7 (i) in the above expression we obtain that ∂ (cid:96) a G n ,α ( a ) = O ( α − d · n | (cid:96) | ( γ − na + λ na ))which, in fact, implies a better estimate than (12).To prove the first estimate in (12) for (cid:107) F n (cid:107) C d , using again the Leibniz formula ∂ (cid:96) a F n ,α ( a ) = (1 − e i ϕ ∗ n ( a ) ) ∂ (cid:96) a ρ (cid:18) a − a α (cid:19) − (cid:88) j : 0 < j ≤ (cid:96) (cid:32) (cid:96) j (cid:33) ∂ ja ( e i ϕ ∗ n ( a ) ) ∂ (cid:96) − ja ρ (cid:18) a − a α (cid:19) . Applying Lemma 3.7 (ii), ∂ (cid:96) a F n ,α ( a ) = O (cid:32) α −| (cid:96) | n (cid:33) + (cid:88) j : 0 < j ≤ (cid:96) (cid:32) (cid:96) j (cid:33) O (cid:32) n | j |− ( γ a λ a ) n · α − | (cid:96) − j | (cid:33) . Thus, ∂ (cid:96) a F n ,α ( a ) = O (cid:16) α − d n − (cid:17) . To complete the proof of Lemma 3.3 we have to show theestimates of Lemma 3.7. Proof of Lemma 3.7.
First we will prove the estimates on µ ∗ n ( a ) and ∂ ja µ ∗ n ( a ). Let us first observethat M , up to a multiplicative factor, is actually equal to γ na ( µ + O ( γ − na + λ na )) (see the exactexpressions for M in [GST08, Section 3.2]). Although this multiplicative factor depends onthe perturbation ε n , to avoid notational clutter we will assume in what follows and withoutloss of generality that this factor is always 1. Then M ∗ n ( a ) can be written as M ∗ n ( a ) = γ na µ ∗ n ( a ) + h n ( a ) (13)where h n ( a ) is independent of µ ∗ n ( a ) and h n ( a ) = O ( γ − na + λ na ). Again using the explicitexpressions for M in [GST08, Section 3.2], one can see that ∂ ja h n ( a ) = O ( n | j | ( γ − na + λ na )). Claim 3.7.1. M ∗ n = O (1) and ∂ ja M ∗ n = o (1) for all 1 ≤ | j | ≤ d (14) Proof of Claim 3.7.2.
The expressions of BT n ( a ) and HT n ( a ) in (8) imply that these functionshave order O ( R n ) and analogously their respective derivatives have order O ( ∂ ja ( R n )). On theother hand M ∗ n ( a ) is O ( R n )-close to BT n ( a ) or HT n ( a ), see Section 3.1. Also the derivatives of ∂ ja M ∗ n are O ( ∂ ja ( R n ))-close. Therefore, M ∗ n = O (1) + O ( R n ) and ∂ ja M ∗ n = O ( ∂ ja R n ) . Finally, from the equation for R n given in Lemma 3.5, we have that R n = o (1) and ∂ ja R n = o (1).Combining this with the previous estimates proves the claim. (cid:3) In particular, from (13) we obtain that γ na µ ∗ n ( a ) = O (1) + O ( γ − na + λ na ) , which implies µ ∗ n ( a ) = O ( γ − na ) = O ( γ − na + λ na ). To show the estimates on ∂ ja µ ∗ n ( a ) we willproceed by an inductive argument on | j | . In the case | j | =
1, taking the derivate of (13) gives ∂ ja M ∗ n ( a ) = ∂ ja ( γ na ) µ ∗ n ( a ) + γ na ∂ ja µ ∗ n ( a ) + ∂ ja h n ( a ) . We have that µ ∗ n ( a ) = O ( γ − na + λ na ), ∂ ja M ∗ n ( a ) = o (1) from Claim 3.7.1 and ∂ ja h n ( a ) = O ( n ( γ na + λ na )).Also ∂ ja ( γ na ) = n γ n − a · ∂ ja ( γ a ) = O ( n γ na ). Combining all these estimates with the previousequation implies γ na ∂ ja µ ∗ n ( a ) = O ( n γ na ) · O ( γ − na + λ na ) + O ( n ( γ − na + λ na )) + o (1) . Then ∂ ja µ ∗ n ( a ) = O ( n ( γ − na + λ na )), which proves the formula for | j | =
1. To prove the necessaryexpression for any multi-index (cid:96) , we will proceed by induction assuming that ∂ ja µ ∗ n ( a ) = O ( n | j | ( γ − na + λ na )) for any j with 1 ≤ | j | < | (cid:96) | and will show the same estimate for (cid:96) . From (13)and using the Leibniz formula we obtain ∂ (cid:96) a M ∗ n ( a ) = (cid:88) j : j ≤ (cid:96) (cid:32) (cid:96) j (cid:33) ∂ (cid:96) − ja ( γ na ) · ∂ ja µ ∗ n ( a ) + ∂ (cid:96) a h n ( a ) . (15)Now ∂ (cid:96) − ja ( γ na ) = O ( n | (cid:96) − j | γ na ) and so the order of ∂ (cid:96) − ja ( γ na ) · ∂ ja µ ∗ n ( a ) is O ( n | (cid:96) − j | γ na ) · O ( n | j | ( γ − na + λ na )) = O ( n | (cid:96) | γ na ( γ − na + λ na )) . Also ∂ (cid:96) a M ∗ n ( a ) = o (1) by Claim 3.7.1 and ∂ (cid:96) a h n ( a ) = O ( n | (cid:96) | ( γ na + λ na )). Then isolating the termwith j = (cid:96) from the sum in (15) we get that γ na · ∂ (cid:96) a µ ∗ n ( a ) = O ( n | (cid:96) | γ na ( γ − na + λ na )) + O ( n | (cid:96) | ( γ − na + λ na )) + o (1) . Then we may conclude that ∂ (cid:96) a µ ∗ n ( a ) = O ( n | (cid:96) | ( γ − na + λ na )) proving item (i) of Lemma 3.7.Now we will prove the second part of the lemma, that is, the estimates on ϕ ∗ n ( a ) and ∂ ja ϕ ∗ n ( a ).As was mentioned before in Lemma 3.6 because of the expressions in [GST08, pg. 946], wehave that ϕ n ( a ) = π n + O (1 / n ) and the size of ∆ n ( a ) has decay of order O (1 / n ). Thus, we may ERGER DOMAINS 21 also conclude that ϕ ∗ n ( a ) = O ( n − ). Now assuming ∂ ja ϕ ∗ n ( a ) = O (cid:16) n | j |− ( γ a λ a ) − n (cid:17) , it is not hardto see by an inductive argument on the derivatives that also ∂ ja (cid:16) e i ϕ ∗ n ( a ) (cid:17) = O (cid:16) n | j |− ( γ a λ a ) − n (cid:17) .In what follows we will prove that ∂ ja ϕ ∗ n ( a ) = O ( n | j |− ( γ a λ a ) − n ) for j ≥
1. This will bedone by induction in | j | , starting with | j | =
1. To do this, according to (11) and writting h n ( a ) = n ϕ ∗ n ( a ) + O (1), we have that B ∗ n ( a ) ∼ ( λ a γ a ) n cos( h n ( a )). Having in mind the exactexpression for the function B given in [GST08, pg. 946], B n ( a ) di ff ers from ( λ a γ a ) n cos( h n ( a ))by a multiplicative constant b , that depends on ε n . Similarly to what was done for M ∗ n ( a ) andto avoid unnecessary notational complications we will assume, without loss of generality,that this factor is always 1. Then, B ∗ n ( a ) = ( λ a γ a ) n cos( h n ( a )) (16)and derivating both sides of the equation with respect to some index j , | j | =
1, we obtain ∂ ja B ∗ n ( a ) = n · B ∗ n ( a ) · ∂ ja (log λ a γ a ) + ( λ a γ a ) n sin( h n ( a )) · ∂ ja h n ( a ) . (17)Now to estimate the order of ∂ ja B ∗ n ( a ), we have the following claim whose proof we omit asit is analogous to that of Claim 3.7.1. Claim 3.7.2. B ∗ n = O (1) and ∂ ja B ∗ n = o (1) for all 1 ≤ | j | ≤ d (18)Since the norms of the functions B ∗ n ( a ) , ∂ ja (log λ a γ a ) are bounded from above and usingthat ∂ ja B ∗ n = o (1) due to Claim 3.7.2 we get from (17) that( λ a γ a ) n sin( h n ( a )) · ∂ ja h n ( a ) = O ( n ) + o (1) . (19)Since sin( h n ( a )) is bounded away from zero by the choice of ϕ n , resolving the previousequation for ∂ ja h n ( a ) we obtain that ∂ ja h n ( a ) = O ( n ( λ a γ a ) − n ). This implies that ∂ ja ϕ ∗ n ( a ) = O (( λ a γ a ) − n ) and proves the formula for | j | =
1. To prove the expression for any index (cid:96) , wewill proceed by induction assuming that ∂ ja h n ( a ) = O ( n | j | ( γ a λ a ) − n ) for any j with 1 ≤ | j | < | (cid:96) | and will show that ∂ (cid:96) a h n ( a ) = O ( n | (cid:96) | ( γ a λ a ) − n ). This will imply that ∂ (cid:96) a ϕ ∗ n ( a ) = O ( n | (cid:96) |− ( λ a γ a ) − n ),as it is required to complete the proof of the lemma.Take (cid:96) with | (cid:96) | >
1, and fix ι < (cid:96) such that | ι | =
1. Since ∂ (cid:96) a B ∗ n ( a ) = ∂ (cid:96) − ι a ( ∂ ι a B ∗ n ( a )) we getfrom (16), (17) and by using the Leibniz formula, ∂ (cid:96) a B ∗ n ( a ) = n · ∂ (cid:96) − ι a ( B ∗ n ( a ) · ∂ a (log λ a γ a )) + (cid:88) j : j ≤ ( (cid:96) − ι ) (cid:32) (cid:96) − ι j (cid:33) ∂ ( (cid:96) − ι ) − ja (( λ a γ a ) n sin( h n ( a ))) · ∂ j + ι a h n ( a ) . (20)Now we will determine the order of the terms in the above equation. Claim 3.7.3. ∂ ja (( λ a γ a ) n sin( h n ( a ))) = O ( n | j | ( λ a γ a ) n ) for all 1 ≤ | j | ≤ d (21) Proof of Claim 3.7.3.
Observe from (16) that [( λ a γ a ) n sin( h n ( a ))] = ( λ a γ a ) n − ( B ∗ n ) . Since ∂ ja B ∗ n = o (1) from Claim 3.7.2, we get that ∂ ja (( λ a γ a ) n sin( h n ( a ))) = O ( ∂ ja ( λ a γ a ) n ) . On the other hand it can be seen by an inductive argument that ∂ ja ( λ a γ a ) n = O ( n | j | ( λ a γ a ) n ),which gives the required estimate. (cid:3) For j < ( (cid:96) − ι ), by the induction hypothesis ∂ j + ι a h n ( a ) = O ( n | j + ι | ( γ a λ a ) − n ). Combining thiswith Claim 3.7.3 we obtain that for j < ( (cid:96) − ι ) ∂ ( (cid:96) − ι ) − ja (( λ a γ a ) n sin( h n ( a ))) · ∂ j + ι a h n ( a ) = O ( n | ( (cid:96) − ι ) − j | ( λ a γ a ) n ) · O ( n | j + ι | ( γ a λ a ) − n ) = O ( n (cid:96) ) . On the other hand ∂ (cid:96) a B ∗ n ( a ) = o (1) and n · ∂ (cid:96) − ι a ( B ∗ n ( a ) · ∂ a (log λ a γ a )) = O ( n ). Thus, putting allthese estimates together in (20) and isolating the term corresponding with the index j = (cid:96) − ι we obtain ( λ a γ a ) n sin( h n ( a ))) · ∂ (cid:96) a h n ( a ) = O ( n ) + O ( n | (cid:96) | ) + o (1) . This implies that ∂ (cid:96) a h n ( a ) = O ( n | (cid:96) | ( γ a λ a ) − n ) concluding the proof. (cid:3) R eferences [BD12] C. Bonatti and L. J. D´ıaz. Abundance of C -homoclinic tangencies. Transactions of the American Mathe-matical Society , 264:5111–5148, 2012.[Ber16] P. Berger. Generic family with robustly infinitely many sinks.
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