Typical coexistence of infinitely many strange attractors
aa r X i v : . [ m a t h . D S ] F e b TYPICAL COEXISTENCE OF INFINITELY MANY STRANGE ATTRACTORS
PABLO G. BARRIENTOS AND JUAN DAVI ROJASA bstract . We prove that the coexistence of infinitely many prevalent H´enon-like phenomenais Kolmogorov typical in sectional dissipative C d , r -Berger domains of parameter families ofdi ff eomorphisms of dimension m ≥ d < r −
1. In particular, we get the coexistenceinfinitely many non-hyperbolic strange attractors for 3 ≤ d < r −
1. I ntroduction
Homoclinic bifurcation are one of the main mechanisms to create complicated dynamicalbehavior in the evolution of parametric families of discrete systems. A C r -di ff eomorphismhas a homoclinic tangency if there is a pair of points P and Q in the same transitive hyperbolicset such that the unstable invariant manifold of P and the stable invariant manifold of Q have a non-transverse intersection at a point Y . A homoclinic tangency can be unfolded byconsidering a C d -family ( f a ) a of C r -di ff eomorphisms parameterized by a ∈ I k with f = f , I = [ − , k ≥ d ≤ r . From the pioneer work of Newhouse [New70], it is well knownthat the set of C surface di ff eomorphisms exhibiting homoclinic tangencies has not emptyinterior. See also [PV94, GTS93b, Rom95, BD12, BR17] for higher dimensional dynamics.In a Newhouse domain , i.e, in an open set of di ff eomorphisms (or in the correspondingparameters space) where dynamics with homoclinic tangencies associated with periodicpoints is dense, generic di ff eomorphisms exhibit coexistence of infinitely many sinks [New79,Rob83, GTS93a, GST08]. This result was coined under the name of Newhouse phenomena . Butalso the unfolding of homoclinic tangencies in Newhouse domains brings the presence andcoexistence of more complicate chaotic dynamics as (non-hyperbolic) strange attractors . Thatis, a compact invariant set having a dense orbit with at least one positive Lyapunov exponentand whose stable set has non-empty interior.Obviously, strange attractors are always non-trivial (i.e., they are not reduced to a periodicorbit). However, they could be still hyperbolic as for instance the Plykin attractor or theSmale solenoid. One of the first examples of a non-hyperbolic strange attractors was given(numerically) by H´enon [H´en76] for the two-parameter family given by H a , b ( x , y ) = (1 − ax + y , bx ) . The limit family ( b =
0) is given by the quadratic maps T a ( x ) = − ax . Benedicks andCarleson proved in [BC85] that there exists a positive Lebesgue measure set of parameters a ∈ (1 ,
2] such that the compact interval [1 − a ,
1] is a strange attractor of T a . These results,see also [Jak81], were key to prove the existence of strange attractors for the family of H´enonmaps in [BC91]. It was quickly observed that an extension of conclusions in [BC91] could bepossible for some other family F a , b whose family of limit maps was also the quadratic family f a . Families of this type will be called H´enon-like families . Namely, Mora and Viana in [MV93]and [Via93] showed that any H´enon-like family of di ff eomorphisms has a set of parameters BARRIENTOS AND ROJAS with positive Lebesgue measure for which exhibits a strange attractor. Such families appearin generic unfoldings of homoclinic tangencies associated with sectional dissipative periodicpoints [PT93, Via93, GTS93a, GST08]. That is, periodic points which have the product of anypair of multipliers less than one in absolute value.H´enon-like maps are (strongly) area-contracting everywhere in their domain, and so, theycannot have non-trivial hyperbolic attractors. This lack of hyperbolicity prevents stabilityunder perturbations, and thus, the classical arguments (see [PT93]) to provide coexistence ofinfinitely many of such attractors do not work. This di ffi culty was overcome by Colli [Col98]and Leal [Lea08] who proved that, in Newhouse domains associated with homoclinic tan-gencies to sectional dissipative periodic points, there exists a locally dense set of di ff eomor-phisms (or corresponding parameters in the parameter space) exhibiting the coexistence ofinfinitely many non-hyperbolic strange attractors.The above results say nothing about persistence of the coexistence of infinitely manyattractors. Recall that a parametric family f = ( f a ) a of dynamics exhibits persistently aproperty P if P is observed for f a in a set of parameter values a with positive Lebesguemeasure. It was an open question (see [Col98]) whether for ”most” or there exists a k -parameter family ( f a ) a which has infinitely many attractors (strange or not) for values of theparameter a in a positive Lebesgue measure set E ( f ) ⊂ I k . The abundance of such familiesmust be understood in the sense of typicality introduced by Kolmogorov (see [HK10]).That is, a property P is called typical (in the sense of Kolmogorov) if there is a Baire(local) generic set of parameter families of dynamics exhibiting the property P persistentlywith full Lebesgue measure. On the other hand, Palis claimed that the measure of theset E ( f ) is generically zero for families f = ( f a ) a of one-dimensional dynamics and surfacedi ff eomorphism [PT93, Pal00].Pumari ˜no and Rodr´ıguez in [PR01, Thm. B] (see also [PR97]) provided a first example of anon-generic family of dynamical systems with persistent coexistence of infinitely many non-hyperbolic strange attractors. Although Palis’ conjecture remains still open, some advancesin the opposite direction has been made by Berger in [Ber16, Ber17] for families of surfaceendomorphisms (in fact, local di ff eomorphisms) and higher dimensional di ff eomorphisms.Namely, Berger constructed open sets U of k -parameter families of the above describeddynamics in the C d , r -topology (see §2.1 for more details) such that residually in these opensets any family exhibits simultaneously infinitely many hyperbolic periodic attractors (sinks) for all parameter value. Mimicking previously introduced terminology, this result was coinedin [BR21a] under the name of Berger phenomena and the open sets U as Berger domains . In fact,in [BR21a], Raibekas and the first author of this work provide the following more specificdefinition of such domains:
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 if there exists a dense subset D of U such thatfor any f = ( f a ) a ∈ D there is a covering of I k by open balls J i having the following property: thereis a continuation of a saddle periodic point Q a having a homoclinic tangency Y a which dependsC d -continuously on the parameter a ∈ J i . TRANGE ATTRACTORS 3
This definition appears implicitly in the constructions of Berger in [Ber16, Ber17]. Ac-tually, he constructed C d , r -Berger domains of persistent homoclinic tangencies associatedwith sectional dissipative periodic points Q a with d ≤ r for endomorphisms in dimensiontwo and di ff eomorphisms in higher dimension. New and di ff erent examples of C d , r -Bergerdomains of k -parametric families of di ff eomorphisms in dimension m ≥ d < r − C d , r -Kolmogorovtypical in k -parametric families of di ff eomorphisms in dimension larger than 2. In this paper,we will answer the previous question by Colli by showing that the coexistence of infinitelymany non-hyperbolic strange attractors is typical in the sense of Kolomogorov: Theorem A.
Let U be a C d , r -Berger domain of k-parameter families of di ff eomorphisms of dimensionm ≥ with ≤ d < r − and k ≥ associated with sectional dissipative periodic points. Thenthere exists a residual set R of U such that for every family ( f a ) a ∈ R the di ff eomorphism f a exhibitsinfinitely many non-hyperbolic strange attractors for Lebesgue almost every a ∈ I k . We want to indicate that the above theorem is in fact a consequence of a more generalresult about prevalent phenomena for H´enon-like families. To be more precise we needsome definitions. Let us denote by
Φ = ( Φ M ) M the parabola family given by Φ M ( x , y ) = (0 , M − y ) where ( x , y ) ∈ R m − × R and M ∈ [1 , . (1) Definition 1.2.
A phenomenon P of a dynamics is said to be C s -prevalent for H´enon-like families if there exist ρ > and < c ≤ such that any C d , r -family ϕ = ( ϕ M ) M of di ff eomorphisms withs ≤ d ≤ r which is ρ -C s -close to the parabola family Φ = ( Φ M ) M satisfies P for any M in a subset ofparameters of (1 , with Lebesgue measure at least c. An example of C s -prevalent phenomenon for H´enon-like families is the existence of non-hyperbolic strange attractors [MV93, Via93] (with s =
3) and hyperbolic attractors [PT93,GST08] (with s = Theorem B.
Under the assumption of Theorem A, if P is a C s -prevalent phenomena for H´enon-likefamilies with s ≤ d, then there exists a residual set R of U such that for every family ( f a ) a ∈ R the di ff eomorphism f a exhibits the coexistence of infinitely many phenomena P for Lebesgue almostevery a ∈ I k . Many of the ideas in this paper provide from the Ph.D. thesis of second author [Roj17]where similar results were obtained in the case of endomorphisms on surfaces. In the nextsection we introduce formally the set of families of di ff eomorphisms that we are consideringand the C d , r -topology. After that, we will prove Theorem B.2. T ypical coexistence of infinitely many prevalent phenomena Topology of families of di ff eomorphisms. We introduce the topology of the set offamilies. To do this, set I = [ − , < d ≤ r ≤ ∞ , k ≥ M , BARRIENTOS AND ROJAS 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 in an open neighborhood of 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 given by k f k C d , r = max { sup k ∂ ia ∂ jx f a ( x ) : 0 ≤ i ≤ d , ≤ i + j ≤ r } where f = ( f a ) a ∈ C d , r ( I k , M ).If d = r , we will say that the family is of class C r . Note that a family f = ( f a ) a is of class C r , ifand only if, the map ( a , x ) f a ( x ) is of class C r .2.2. Proof of Theorem B.
Fix a manifold M of dimension m ≥ k ≥
1. Consider aBerger domain U ⊂ C d , r ( I k , M ) with d < r − P be a C s -prevalent phenomenon for H´enon-like families with s ≤ d .Hence, there exist ρ > c > P is a c-prevalent phenomenon for ρ -C s -H´enon-like families . We will prove that the coexistence of infinitely many phenomena P is Kolmogorov typical in U . We need first to introduce an important definition: Definition 2.1.
Let f = ( f a ) a be a family in C d , r ( I k , M ) with d ≤ r and fix α < β and n ≥ .The family f is said to be a ρ -C s -H´enon-like family after renormalization of period n in I = ( α, β ) k if, for each ¯ a ∈ [ α, β ] k − , there is a one-parameter family R ¯ a = ( R ¯ a , b ) b of smooth transformationsR ¯ a , b : [ − , m → M with b ∈ [ α, β ] such that the family F = ( F M ) M given byF M def = R − a ( M ) ◦ f na ( M ) ◦ R a ( M ) where a ( M ) = ( ¯ a , b ( M )) ∈ I with b ( M ) = ( β − α ) M + α − β for M ∈ [1 , , is ρ -C s -close to the parabola family Φ = ( Φ M ) M given in (1) . We observe that to be a ρ - C s -H´enon-like family after renormalization of period n in I is anopen property in the C d , r -topology of parametric families. Moreover, if f = ( f a ) a ∈ C d , r ( I k , M )is a ρ - C s -H´enon-like family after renormalization of period n in I = ( α, β ) k and P is a c -prevalent phenomenon for ρ - C s -H´enon-like families, then f na exhibits P for any a in a subsetof parameters with Lebesgue measure c · | I | . Here | I | = ( β − α ) k denotes the volume of I . Tosimplify notation, we will refer to the Cartesian product of k open intervals with the samelength as an open ball (in the supremum norm in R k ).Recall that D denotes the dense set in U provides by Definition 1.1. Proposition 2.2.
For any ǫ > , ρ > and f = ( f a ) a ∈ D one finds n = n ( ǫ, ρ, f ) ∈ N and a finitecovering of I k by open balls I j = I j ( f ) so that for every n ≥ n there exists an ǫ -close family g = ( g a ) a to f = ( f a ) a in the C d , r -topology such that g is a ρ -C s -H´enon-like family after renormalization ofperiod n in any I j . Before proving the above proposition we will conclude Theorem B:
TRANGE ATTRACTORS 5
Theorem 2.3.
For every m ∈ N and ρ > , there exists an open and dense set O m = O m ( ρ ) in U such that it holds the following:For any family g = ( g a ) a in O m and each ℓ = , . . . , m, there exist a positive integer n ℓ (with n < · · · < n m ) and a finite covering { I ℓ, j } j of I k by open balls I ℓ, j so that g is a ρ -C s -H´enon-like family after renormalization of period n ℓ in any I ℓ, j .Moreover, there is a residual subset R of U such that any family g = ( g a ) a ∈ R satisfies that g a hasthe coexistence of infinitely many phenomenon P for Lebesgue almost every a ∈ I k .Proof. First of all consider the sequence ǫ 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 and ρ >
0, by Proposition 2.2, we find n = n ( ǫ i , ρ, f ) and a finitecovering { I j } j of I k by open balls I j = I j ( f ) (only depends on f ) such that it holds the following:For any n ( i ) ≥ n , we get a ǫ i -close family g i = ( g i , a ) a to f such that g i is a ρ - C s -H´enon-likefamily after renormalization of period n ( i ) in any I j for all i ≥
1. Since this property persistsunder perturbations, we have a sequence { O ( f , ǫ i , ρ ) } i of open sets O ( f , ǫ i , ρ ) converging to f where the same conclusion holds for any family in these open sets. By taking the union ofall these open sets for any f in D and ǫ i > i ≥
1, we get an open and dense set O = O ( ρ )in U where for any g = ( g a ) a ∈ O there exist n and an open covering { I j } j of I k by open balls I j such that g is a ρ - C s -H´enon-like family after renormalization of period n in any I j .Now we will assume O m = O m ( ρ ) constructed and we will show how to obtain O m + .Since O m is an open and dense set in U , we can start now by taking f = ( f a ) a ∈ O m ∩ D .Hence, for each ℓ = , . . . , m , there exist a positive integer n ℓ (with n < · · · < n m ) and afinite covering { I ℓ, j } j of I k by open balls I ℓ, j such that f is a ρ - C s -H´enon-like family afterrenormalization of period n ℓ in any I ℓ, j . As before, from the robustness of this property, thereexists ǫ ′ = ǫ ′ ( f ) > ǫ ′ -close family g = ( g a ) a to f still is a ρ - C s -H´enon-likefamily after renormalization with respect to the same periods and in the same intervals.Then, for any ǫ i < ǫ ′ /
2, we can apply again Proposition 2.2 finding n = n ( ǫ i , ρ, f ) ∈ N and a finite covering { I j } j of I k by open balls I j = I j ( f ). Moreover, by taking an integer n ( i ) m + > max { n , n m } , we get an ǫ i -perturbation g i = ( g i , a ) a of f such that g i is a ρ - C s -H´enon-like family after renormalization of period n ( i ) m + in any I j and i ≥
1. Hence, bythe robustness as before, we have a sequence { O m + ( f , ǫ i , ρ ) } i of open sets O m + ( f , ǫ i , ρ ) ⊂ O m converging to f where the same conclusion holds for any family in these open sets. By takingthe union of all these open sets for any f ∈ O m ∩ D and ǫ i < ǫ ′ ( f ), we get an open and denseset O m + = O m + ( ρ ) in U . In addition, for any g = ( g a ) a ∈ O m + and each ℓ = , . . . , m +
1, thereexist a positive integer n ℓ (with n < · · · < n m + ) and a finite covering { I ℓ, j } j of I k by open balls I ℓ, j such that g is a ρ - C s -H´enon-like family after renormalization of period n ℓ in any I ℓ, j .To conclude the proof of the proposition we need to prove that the coexistence of infinitelymany phenomena P is Kolmogorov typical in U . To do this, let 0 < c ≤ ρ > R of U given by theintersection of O m = O m ( ρ ) for all m ∈ N . Hence, any g = ( g a ) a ∈ R belongs to O m for all m ∈ N and thus we find a strictly increasing sequence of positive integers ( n ℓ ) ℓ and coverings BARRIENTOS AND ROJAS { I ℓ, j } j of I k by open balls I ℓ, j such that g is ρ - C s -H´enon-like renormalizable after period n ℓ inany I ℓ, j for all ℓ ≥
1. Notice that since P is c -prevalent phenomenon for ρ - C s -H´enon-likefamilies we have that the set of parameters J ∗ ℓ, j in I ℓ, j where the phenomenon P holds hasLebesgue measure at least Leb( J ∗ ℓ, j ) ≥ c · | I ℓ, j | for all ℓ ≥ . Since { I ℓ, j } j is a covering of I k by open balls (in the supremum norm in R k ), accordingto [Con12, Lemma 4.9.4], we can extract a subcollection { I ℓ, j i } i of { I ℓ, j } j formed pairwisedisjoint open balls so that X i | I ℓ, j i | > − k . Then, we get that the phenomenon P holds in a subset of parameters J ∗ ℓ in I k havingLebesgue measure at least 3 − k c . Thus, the series P ℓ ≥ Leb( J ∗ ℓ ) diverges and from the secondBorel-Cantelli Lemma, the set of event that occurs for infinitely many ℓ , that is, J ∗ = \ n ≥ [ ℓ ≥ n J ∗ ℓ has full Lebesgue measure in I k . In particular, for any a ∈ J ∗ the map g a has coexistence ofinfinitely many di ff erent phenomena P . This concludes the proof of the theorem. (cid:3) Now we will prove Proposition 2.2. To do this we need the following lemma.
Lemma 2.4.
Let g = ( g a ) a be a C d , r -family and assume that g a has a homoclinic tangency at a point Y a (depending C d -continuously on a) associated with a sectional dissipative saddle Q a for any parametera ∈ a + ( − α, α ) k . Then, for any ρ > , there exists a sequence of families g n = ( g na ) a approaches g inthe C d , r -topology such that g na = g a if a < a + ( − α, α ) k and g n is a ρ -C s -H´enon-like family afterrenormalization of period n in a + ( − α, α ) k for n large enough. Before to prove this result, let us show how to get Proposition 2.2 from the above lemma:
Proof of Proposition 2.2.
Consider a family f = ( f a ) a in D . In order to explain how to applythe above lemma, we introduce some simplifications. Relabeling and resizing, if necessary,we can assume that the cover of I k by the open balls J i that appears in Definition 1.1 is of theform I k ⊂ M [ j = N j [ ℓ = J j ℓ with J j ℓ = a j ℓ + ( − α j ℓ , α j ℓ ) k , a j ℓ ∈ I k and α j ℓ > . Moreover, the persistent homoclinic tangency Y a of f a on J j ℓ is associated with a saddle Q ja for j = , . . . , M and, for each j , the sets J j ℓ for ℓ = , . . . , N j are pairwise disjoint. In fact,to avoid unnecessary complications in the notation, we can assume that α j ℓ = α and, foreach j , the union in ℓ of 2 J j ℓ = a j ℓ + ( − α, α ) k is pairwise disjoint.Fix ǫ > ρ >
0. We want to find n = n ( ǫ, ρ, f ) ∈ N such that for every n ≥ n thereexists an ǫ -close family g = ( g a ) a to f in the C d , r -topology such that g is a ρ - C s -H´enon-likefamily after renormalization in I j ℓ = a j ℓ + ( − α, α ) k . To do this, having into account that for TRANGE ATTRACTORS 7 each j = , . . . , N , 2 J j ℓ are pairwise disjoint sets of the form a j ℓ + ( − α, α ) k for ℓ = , . . . , N j ,applying Lemma 2.4, we get an ǫ -perturbation g n = ( g na ) a of f in the C d , r -topology whichis a ρ - C s -H´enon-like family after renormalization of period n in any a j ℓ + ( − α, α ) k when n istaken large enough (depending on ǫ , ρ and f ). This concludes the proof. (cid:3) Finally, to complete the proof we will show Lemma 2.4.
Proof of Lemma 2.4.
By assumption the map g a has a homoclinic tangency at a point Y a asso-ciated with a sectional dissipative periodic point Q a for all a ∈ a + ( − α, α ) k . Actually, thetangency must be smoothly continued until k a − a k ∞ = α . By means of an arbitrarily small C d , r -perturbation of the family around the tangency point, we can assume that the tangency Y a is simple in the sense of [GST08]. That is, the tangency is quadratic, of codimension oneand, in the case that the dimension m >
3, any extended unstable manifold is transverse tothe leaf of the strong stable foliation which passes through the tangency point. Since Q a is asectional dissipative saddle, if we denote the leading multipliers of this periodic point by λ a and γ a ∈ R we have | λ a | < < | γ a | and | λ a γ a | < . On the other, we can consider a generic one-parameter unfolding g a ,µ of the homoclinictangency of g a . To be more specific, we consider the one-parameter unfolding g a ,µ of g a where µ is the parameter that control the splitting of the tangency. We can take local coordinates( x , y ) with x ∈ 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 = } respectively. Moreover, byconsidering, if necessary, some iterated the tangency point Y a is represented by ( x + ,
0) in thiscoordinates. Let us consider a C ∞ -bump function φ : R → R with support in [ − ,
1] andequal to 1 on [ − / , / ϕ : a = ( a , . . . , a k ) ∈ I k φ ( a ) · . . . · φ ( a k ) ∈ R . Take δ > δ -neighborhoods in local coordinates of Y a , g a ( Y a ) and g − a ( Y a ) aredisjoint. In particular, we denote by U the 2 δ -neighborhood of Y a in this local coordinates.We write g a ,µ = H a ,µ ◦ g a where H a ,µ in this local coordinates takes the form¯ x = x ¯ y = y + ϕ (cid:18) a − a α (cid:19) φ k ( x , y ) − ( x + , k δ ! µ and it is the identity otherwise. Observe that if a < a + ( − α, α ) k then g a ,µ = g a . Also if( x , y ) < g − a ( U ) then g a ,µ = g a .Let us defined the first return map associated with the simple homoclinic tangency of g a ,µ at µ = T = T ( a , µ ) denotes the local mapfor a ∈ a + [ − α, α ] k . In our case, this map corresponds to g k a ,µ at a neighborhood of thecontinuation Q a where k denotes de period of this saddle periodic point. By T = T ( a , µ ) BARRIENTOS AND ROJAS we denote the map g k a ,µ from a neighborhood Π − a of a tangent point Y − a ∈ W uloc ( Q a ) to aneighborhood Π a of Y a = g n a ( Y − a ) ∈ W sloc ( Q a ). Then, one defines the first-return map as T n = T ◦ T n for su ffi ciently large n on σ n ( a ) = Π a ∩ T − n ( Π − a ). Since the tangency point Y a depends C d -continuously on a + [ − α, α ] k , we get that this first-return map T n = T n ( a , µ ) alsodepends smoothly as a function of the parameter a on a + [ − α, α ] k . Lemma 2.5 (Parametrized rescaling lemma) . There are a sequence of open sets ∆ n in the ( a , µ ) -parameter space with ∆ n accumulating on ( a + [ − α, α ] k ) ×{ } such that for any ( a , µ ) ∈ ∆ n thereis a smooth transformation of coordinates which brings the first-return map T n in local coordinateson σ n ( a ) the following form: ¯ x = o (1) and ¯ y = M − y + o (1) where the o (1) -terms tends to zero (uniformly on a) as n → ∞ along with all the derivatives up tothe order r with respect to the coordinates ( x , y ) and up to d ≤ r − with respect to the rescaledparameter M. The domain of definition of T n in these coordinates is an asymptotically large regionwhich, as n → ∞ covers all finite values of ( x , y ) . The rescaled parameter M is at least C d -smoothfunction of ( a , µ ) which for large enough n is given byM ∼ γ na ( µ + O ( γ − na )) . (2) Proof.
A carefully reader of the proof of the rescaling lemmas in [GST08, Lemma 1 andLemma 4] allows us to apply these results smoothly on the parameter a ∈ a + [ − α, α ] k .That is, all the expression and change of variables in [GST08, Lemma 1] can be performedsmoothly on a ∈ a + [ − α, α ] k . This immediately implies the lemma states above. (cid:3) Moreover, M can take arbitrarily finite values when µ varies close to µ n ( a ) = O ( γ − na ). To bemore precise, the parameter µ n ( a ) was introduced in [GST08] so that M n , a ( µ n ( a )) = M n , a is the function given in (2) for fixed n and a . Actually, an explicit expression of M in (2)is provided in [GST08, after Eq. (3.8)] which, up to multiplicative constants, is basically theright hand of (2) where the O -function does not depend on µ and its i -th partial derivativeswith respect to the variable a are of order O ( n i γ − na ). Thus, we can calculate the derivativewith respect to µ of M n , a for n large enough as ∂ µ M n , a ∼ γ na ≫
1. Hence, we get M n , a isan invertible expanding map with arbitrarily large expansion uniform on a ∈ a + [ − α, α ] k .Thus, for n large enough, we can assume that Φ n ( a , µ ) = ( a , M n , a ( µ ))is a di ff eomorphism between the set ∆ n given above in the lemma and ( a + [ − α, α ] k ) × [ − , b ( M ) given in Definition 2.1 takes, on a + [ − α, α ] k , the form b ( M ) = α M − α + π k ( a ) for M ∈ [1 , π k : R k → R is the projection on the k -th coordinate. Consider the inverse map b M ( b ) = b + α − π k ( a )2 α for b ∈ π k (cid:16) a + [ − α, − α ] k (cid:17) . TRANGE ATTRACTORS 9
Now since Φ n is a di ff eomorphism, we find a C d -function µ n on a + [ − α, α ] k defined as Φ − n (cid:16) a , b M ( π k ( a )) (cid:17) = ( a , µ n ( a )) a ∈ a + [ − α, α ] k . In particular, M n , a ( µ n ( a )) = b M ( π k ( a )) for a ∈ a + ( − α, α ) k . (3)Extending smoothly µ n to I k (c.f. [Lee12, Lemma 2.26]) we can consider the sequence offamilies g n = ( g n , a ) a where g n , a = g a ,µ n ( a ) for a ∈ I k and n large enough . Observe that g n , a = g a for a < a + ( − α, α ) k . Moreover, according to Lemma 2.5 andEquation (3), there is a smooth family R = ( R a ) a of smooth transformation of coordinates R a on σ n ( a ) such that bring the first-return map T n = T n ( a , µ n ( a )) of g n , a into R − a ◦ T n ◦ R a whichhas the form ¯ x = o (1) and ¯ y = b M ( π k ( a )) − y + o (1) for a ∈ a + [ − α, α ] k . Substituting a by the linear rescaling a ( M ) = ( ¯ a , b ( M )) ∈ a + [ − α, α ] k and having into accountthat b M ( π k ( a ( M ))) = b M ( b ( M )) = M for all M ∈ [1 ,
2] we arrive into the form¯ x = o (1) and ¯ y = M − y + o (1) for M ∈ [1 , . Since the o (1)-terms above tends to zero as n → ∞ along with all the derivatives up to theorder r with respect to the coordinates ( x , y ) and up to s ≤ d ≤ r − M weget that k T n − Φ k C s , s + = o (1) where Φ = ( Φ M ) M is the parabola family. This proves that for n large enough, g n is a ρ - C s -H´enon-like family after renormalization of period ˜ n = k n + k in a + ( − α, α ) k . For short and simplicity, we can assume that Q a is a fixed point and relabel thesequence of families to just say that the renormalization period of g n is n .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 k g n − g k = k ( I − H a ,µ n ( a ) ) ◦ g a k ≤ k I − H a ,µ n ( a ) k k g k where I denotes the identity. Thus, we only need to calculate the C d , r -norm of the family( I − H a ,µ n ( a ) ) a . Since H a ,µ n ( a ) = I if a < a + ( − α, α ) k or ( x , y ) < U then (cid:13)(cid:13)(cid:13) I − H a ,µ n ( a ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ϕ (cid:18) a − a α (cid:19) φ k ( x , y ) − ( x + , k δ ! µ n ( a ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Then to estimate the C d , r -norm above it su ffi ces to show that the function G n ,α ( a ) = ϕ ( a − a α ) µ n ( a ) for a ∈ a + ( − α, α ) k have C d -norm small when n is large. To do this, we use the multi-index notation for partialderivatives and Leibnitz rule ∂ ℓ a G n ,α ( a ) = X j ≤ ℓ ℓ j ! ∂ ℓ − ja ϕ (cid:18) a − a α (cid:19) · ∂ ja µ n ( a ) ℓ ∈ Z k + with | ℓ | ≤ d . (4) On the other hand, recall that as we have indicated before the µ variable in (2) varies close to µ n ( a ) = O ( γ − na ). Moreover, | µ n ( a ) − µ n ( a ) | → n → ∞ . Thus, we also get that µ n ( a ) = O ( γ − na ).We will show that ∂ ℓ a µ n ( a ) = O ( n | ℓ | γ − na ) for ℓ ∈ Z k + by induction on | ℓ | . To do this, assume that ∂ ja µ n ( a ) = O ( n | j | γ − na ) for all j ∈ Z k + with j ≤ ℓ and | j | < | ℓ | . Notice that (3) can be written as b M ( π k ( a )) ∼ γ na ( µ n ( a ) + O ( γ − na ))where the equivalence is actually an equality up to multiplicative constants (independentof a ) being i -th partial derivatives with respect to the variable a of the O -function of order O ( n i γ − na ) for all i ≥
0. Thus, we get that ∂ ℓ a M ( π k ( a )) = O (1) ∼ O ( n | ℓ | γ na ) + X j ≤ ℓ ℓ j ! ∂ ℓ − ja ( γ na ) · ∂ ja µ n ( a ) = O ( n | ℓ | γ na ) + γ na · ∂ ℓ a µ n ( a ) + X j ≤ ℓ j , ℓ O ( n | ℓ |−| j | γ na ) · O ( n | j | γ − na ) = O ( n | ℓ | γ na ) + γ na · ∂ ℓ a µ n ( a ) . From here it follows that ∂ ℓ a µ n ( a ) = O ( n | ℓ | γ − na ). Indeed, if ∂ ℓ a µ n ( a ) is not a O ( n | ℓ | γ − na ) then,negating the definition of O -function, for all K > n ∈ N there exists n ≥ n such that | ∂ ℓ a µ n ( a ) | > Kn | ℓ | | γ a | − n . In particular ∂ ℓ a µ n ( a ) = Ω ( n | ℓ | γ − na ) where Ω denotes the Big Omegaof Hardy-Littlewood. Hence, we obtain that ∂ ℓ a M ( π k ( a )) which is a O (1)-function is alsoa O ( n | ℓ | γ na ) + γ na · Ω ( n | ℓ | γ − na ) = O ( n | ℓ | γ na ) + Ω ( n | ℓ | γ na ) = Ω ( n | ℓ | γ na ) obtaining a contradiction.Substituting this estimate in (4), we get that ∂ ℓ a G n ,α ( a ) = O ( α − d n | ℓ | γ − na ). In particular, we getthat k G n ,α k C d = O (cid:16) α − d n d γ − n (cid:17) for some 1 < γ ≤ γ a for all a ∈ a + ( − α, α ) k . Observe that this assertion completes the proofof the lemma. (cid:3) Acknowledgements.
We thank E. R. Pujals for his guidance and encourage us to write thispaper providing many ideas to go ahead. The second author also especially thanks hissupervisor E. R. Pujals for his unconditional friendship and enriching talks on mathematicsamong others things during his doctorate. Finally, the first author thanks A. Raibekas forhis tireless patience and friendship during many di ffi cult moments throughout the processof writing and revising the preliminary versions.R eferences [BC85] M. Benedicks and L. Carleson. On iterations of 1 − ax on ( − , Annals of Mathematics , pages 1–25,1985.[BC91] M. Benedicks and L. Carleson. The dynamics of the H´enon map.
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