Heterodimensional tangencies on cycles leading to strange attractors

a rXiv:082.3745v1[mat h.D S]26Fe b28HETERODIMENSIONAL TANGENCIES ON CYCLES LEADING TO STRANGE ATTRACTORS SHIN KIRIKI,YUSUKE NISHIZAW A,AND TERUHIKO SOMA Abstract.In this paper,we study heterodimensional cycles of two-parameter families of 3-dimensional diffeomorphisms some element of which contains non-degenerate heterodimensional tangencies of the stable and unstable manifolds of two saddle points with different indexes,and prove that such diffeomor-phisms can be well approximated by another element which has a quadratic homoclinic tangency associated to one of these saddle points.Moreover,it is shown that the tangency unfolds generically with respect to the family.This result together with some theorem in Viana [14],we detect strange attractors appeared arbitrarily close to the original element with the heterodimensional cycle. 1.Introduction Let ϕbe a diffeomorphism on a smooth manifold M which has two saddle fixed points p and q satisfying index(q )=index(p )+1,where index(·)denotes the di-mension of the unstable manifold of a concerned saddle point.A heteroclinic point r of the stable manifold W s (p )and the unstable manifold W u (q )is called a het-erodimensional tangency of W s (p )and W u (q )if r satisfies •T r W s (p )+T r W u (q )=T r M ,and •dim(T r W s (p ))+dim(T r W u (q ))>dim(M ).When diffeomorphisms act on manifolds of dimension greater than or equal to three,it is well known that nonhyperbolic phenomena (e.g.the coexistence of infin-itely many sinks or sources,see [2,3,11,13]and so on)are caused by the existence of homoclinic tangencies as well as that of heteroclinic cycles containing two saddle points with different indexes,which are called heterodimensional cycles .Moreover,the sets of nonhyperbolic diffeomorphisms of these two types are conjectured in[12,Conjecture 4]to occupy large parts in the space of diffeomorphisms on closed manifolds as the set of hyperbolic diffeomorphisms does.Connected heterodimensional cycles which are non-critical (i.e.cycles without any tangencies or local bifurcations of periodic points)were studied intensively by D´ıaz et al.[8,9,11],see also [5,§6].In this paper,we study 3-dimensional C 2diffeomorphisms which have non-connected critical cycles containing nondegenerate heterodimensional tangencies.Our main theorem is states as follows.2SHIN KIRIKI,YUSUKE NISHIZA WA,AND TERUHIKO SOMATheorem A.Let M be a3-dimensional C2manifold and let{ϕµ,ν}be a two-parameter family of C2diffeomorphismsϕµ,ν:M→M which C2depends on(µ,ν) and has continuations of saddlefixed points pµ,νand qµ,νwith index(pµ,ν)=1and index(qµ,ν)=2.Suppose that the following conditions hold.•Eachϕµ,νis locally C2linearizable in a small neighborhood N(qµ,ν)of qµ,ν.•ϕ=ϕ0,0has a heterodimensional cycle containing thefixed points p= p0,0,q=q0,0,a heterodimensional tangency r as above,a quasi-transverse intersection s∈W s(q)∩W u(p).•{ϕµ,ν}satisfies the generic conditions(C1)-(C3)given in Section2. Then,for a sufficiently smallε>0and anyµin either(0,ε)or(−ε,0),there exist infinitely manyνsuch thatϕµ,νhas a quadratic homoclinic tangency associated to pµ,νwhich unfolds generically with respect to theν-parameter familyϕµ(fixed),ν.Figure1.1illustrates the situation of Theorem A for(µ,ν)=(0,0).The terms and definitions in the statement of this theorem are explained in Section2.RemarkFigure 1.1.A heterodimensional cycle containing the saddlepoints p,q,the heterodimensional tangency r and the quasi-transverse intersection s.that Theorem A holds for a homoclinic tangency associated to qµ,νinstead of pµ,νif we replace the generic conditions in(C3)by appropriate ones.By Theorem A,under certain generic conditions,we have obtained a quadratic homoclinic tangency associated to one of these saddlefixed points which unfolds generically.Viana[14]detected strange attractors in some quadratic homoclinic bifurcations given by C3diffeomorphisms on manifolds of dimension greater than two.Supposing an extra dissipative condition on the saddlefixed points and the C3smoothness on ambient manifolds and diffeomorphisms,we have the following corollary.Note that not just the quadratic property but also the generic unfolding property guaranteed by Theorem A plays an essential role in applying Viana’s result to our heterodimensional bifurcation.Corollary B.Let M be a3-dimensional C3manifold,and let{ϕµ,ν}be a two-parameter family in Diff3(M)satisfying the conditions in Theorem A.Suppose that eachϕµ,νis locally C3linearizable in a small neighborhood of pµ,νandϕ=ϕ0,0HETERODIMENSIONAL TANGENCIES AND STRANGE ATTRACTORS3 is sectionally dissipative at p,i.e.the absolute value of the product of any two eigenvalues of(dϕ)p is less than1.Then there exists a positive Lebesgue measure subset A of theµν-plane such thatϕµ,νhas a strange attractor for any(µ,ν)∈A. Proof.The assertion is derived immediately from our Theorem A together with [14,p.15,Theorem A]. Remark1.1.By using Palis-Viana[13,p.207,Main Theorem],any two-parameter family{ϕµ,ν}satisfying the conditions of Theorem A exhibits the Newhouse phe-nomenon,that is,there exists an open subset B of Diff3(M)with Cl(B)∋ϕand such that generic diffeomorphisms in B have infinitely many sinks.Though the Newhouse phenomena near heterodimensional cycles have been already observed by some authors in the case of C1topology(see for example[3,4]and also[1]), our phenomenon occurs in a mechanism different from theirs.D´ıaz,Nogueira and Pujals[6]also studied some heterodimensional cycles containing heterodimensional tangencies of elliptic type and obtained results concerning robustly non-dominated homoclinic classes inducing coexistence of infinitely many sinks and sources,which are motivation for our investigation in this paper.2.Definitions and generic conditionsIn this section,we present some definitions needed in later sections and generic conditions adopted as hypotheses in Theorem A.2.1.Definitions.Definition2.1.Suppose that M is a3-dimensional C2manifold.Let{lν}ν∈J, {mν}ν∈J be C2families of regular curves in M,and let{Sν}ν∈J,{Yν}ν∈J be C2 families of regular surfaces in M,where J is an open interval.(1)Suppose that lν0and mνintersect at a point s for someν0∈J and some openneighborhood U of s in M has a C2change of coordinates with respect to which mν={(0,0,z)∈U}for anyν∈J nearν0.We say that s is a quasi-transverseintersection of lν0and mνifdim(T s(lν)+T s(mν))=2.Moreover,s unfolds generically atν=ν0with respect to{lν}ν∈J,{mν}ν∈J if there exists a C2continuation sν∈lνwith sν=s and a C2function d:J→R+with d(ν0)=0such that(2.1)T s M=T s(lν0)⊕N⊕T s(mν)and dist(sν,mν)=|ν−ν0|d(ν)for anyνnearν0,where N is the one-dimensional space spanned by the non-zero tangent vector(dsν/dν)|ν=ν.This property corresponds to the conditions(GU1)–(GU3)in[11,§2.2.1].(2)Suppose that lν0and Sνintersect at a pointτfor someν0∈J.We saythatτis a quadratic tangency(or a contact of order1)of lν0and Sνif thereexists some C2change of coordinates on U(τ)with respect to whichτ=(0,0,0), Sν={(x,y,z)∈U(τ);z=0}and lνhas a regular parametrization l(ν,t)= (x(ν,t),y(ν,t),z(ν,t))with l(ν0,0)=(0,0,0)and∂z∂t2(ν0,0)=0,4SHIN KIRIKI,YUSUKE NISHIZA WA,AND TERUHIKO SOMAwhere U(τ)is an open neighborhood ofτin M.The tangencyτis said to unfold generically atν=ν0with respect to{lν}ν∈J and{Sν}ν∈J if∂z∂x (u0,v0)=∂fν∂ν(ν0,u0,v0)=0.Remark2.2.It is easy to see that the property(1)does not depend on the coor-dinates used to set lνin the z-axis.Similarly,the properties(2)and(3)do not depend on the coordinates used to set Sνin the xy-plane.When det(Hfν0)(u0,v0)>0(resp.<0)in Definition2.1(3),we say that thetangency r=(u0,v0,0)is of elliptic(resp.hyperbolic)type.The Taylor expansion of fνaround(u0,v0)isfν0(x,y)=1∂x2(u0,v0)(x−u0)2+∂2fν2∂2fνHETERODIMENSIONAL TANGENCIES AND STRANGE ATTRACTORS5 2.2.Generic conditions.Throughout the remainder of this paper,we suppose thatϕ:M→M is a C2diffeomorphism with saddlefixed points p,q with index(p)=1,index(q)=2and such that W s(p)and W u(q)have a nondegenerate heterodimensional tangency r,W u(p)and W s(q)have a quasi-transverse intersec-tion s.Theϕis locally C2linearizable in a neighborhood U(q)of q if there existsa C2linearizing coordinate(x,y,z)on U(q),that is,(2.4)q=(0,0,0),ϕ(x,y,z)=(αx,βy,γz)for any(x,y,z)∈U(q)withϕ(x,y,z)∈U(q),whereα,βandγare the real eigenvalues of(dϕ)q with0<γ<1<β<α.One can take a local unstable manifold W u loc(q)which is an open disk in the plane {z=0}centered at(x,y)=(0,0).We may assume that the both points r,s are contained in U(q)if necessary replacing r(resp.s)byϕ−n(r)(resp.ϕn(s))with sufficiently large n∈N.We setr=(u0,v0,0)with respect to the linearizing coordinate on U(q).We suppose moreover that{ϕµ,ν}is a two-parameter family in Diff2(M)with ϕ0,0=ϕand satisfying the conditions of Theorem A.In particular,ϕµ,νis locally C2linearizable in a small neighborhood U(qµ,ν)of qµ,νin M and henceϕµ,νhas the form as(2.4)in U(qµ,ν),whereα,β,γare C2functions onµ,ν,i.e.α=αµ,ν,β=βµ,ν,γ=γµ,ν.We will put the following generic conditions(C1)-(C3)as the hypotheses in Theorem A.(C1)(Generic unfolding property for r)The nondegenerate heterodimensional tan-gency r of W u(q)and W s(p)unfolds generically with respect to theµ-parameter families{W u(qµ,0)}and{W s(pµ,0)}.(C2)(Generic unfolding property for s)The quasi-transverse intersection s of W s(q)and W u(p)unfolds generically with respect to theν-parameter families {W s(q0,ν)}and{W u(p0,ν)}.(C3)(Additional generic conditions)The tangency r=(u0,v0,0)is not on the x-axis W uu loc(q),that is,(2.5)v0=0.There exists a regular parametrization l(t)=(x(t),y(t),z(t))(t∈I)of a small curve in W u(p)∩U(q)with respect to the linearizing coordinate(x,y,z) on U(q)with s=l(0)anddx(2.6)(u0,v0)=0.∂x2Note that the condition(2.7)is automatically satisfied when r is of elliptic type.6SHIN KIRIKI,YUSUKE NISHIZA WA,AND TERUHIKO SOMA3.Some lemmasIn this section,we will prove some lemmas needed for the proof of Theorem A.For any(µ,ν)near(0,0),we may assume that U(qµ,ν)is equal toD(δ):=(−δ,δ)3with respect to the linearizing coordinate given in Subsection2.2for some constant δ>0.Since s is a quasi-transverse intersection which unfolds generically with respect to theν-families{W s(q0,ν)}and{W u(p0,ν)}by the condition(C2),there exists a C2continuationˆsν∈W u(p0,ν)∩D(δ)withˆs0=s and such thatˆsνsatisfies the conditions same as those for sνin Definition2.1(1).By(2.6),for anyνnear 0,the component lνof W u(p0,ν)∩D(δ)containingˆsνmeets transversely the yz-plane at a point sνwhich defines a C2continuations{sν}with s0=s,see Fig.3.1. Note that dˆsν/dν(0)=dsν/dν(0)+w for some w∈T s(l0)=T s(W u(p)),whereFigure3.1.dˆsν/dν(0)denotes dˆsν/dν|ν=0.Let y(ν)be the y-coordinate of sν.If dy/dν(0)=0, then dsν/dν(0)would be tangent to the z-axis W s loc(q)at s and hence dˆsν/dν(0)∈T s(W s(q))⊕T s(W u(p)).This contradicts(2.1).Thus,we havedy(3.1)(µ,˜ν(µ))=0∂νfor anyµ∈(−ρ,ρ).This implies that sµ,˜ν(µ)is a quasi-transverse intersection un-folding generically atν=˜ν(µ)with respect to theν-parameter families{W s(qµ,ν)} and{W u(pµ,ν)}.HETERODIMENSIONAL TANGENCIES AND STRANGE ATTRACTORS7A new parametrization.Consider the coordinate(ˆµ,ˆν)on the parameter space defined byˆµ=µ,ˆν=ν−˜ν(µ).For simplicity,we denote the new coordinate again by(µ,ν).Thus,there exists a continuation{sµ,0}µ∈(−ρ,ρ)of quasi-transverse intersections of W s(qµ,0)and W u(pµ,0)such that each sµ,0is a quasi-transverse intersection of W s(qµ,0)and W u(pµ,0)which unfolds generically atν=0with respect to theν-parameter families{W s(qµ,ν)}and{W u(pµ,ν)}.Fixµ0with sufficiently small|µ0|arbitrarily.By the properties(2.4)and(2.6), there exists m0∈N such that,for any m≥m0,one can parametrize the componentl m of W u(pµ0,0)∩D(δ)containingϕmµ0,0(sµ0,0)so that l m(0)=ϕmµ0,0(sµ0,0)and l m(t)=(t,y m(t),z m(t))(t∈(−δ,δ)).Lemma3.2.The sequence{l m}C2converges uniformly to W uu loc(qµ0,0)as m→∞.In particular,for anyε>0,there existsˆm0≥m0such that the curvature at any point of l m is less thanεwith respect to the standard Euclidean metric on U(qµ0,0)=D(δ)if m≥ˆm0.Proof.By(2.4),for any m≥m0,l m(t)=(t,βn y m0(α−n t),γn z m(α−n t)),where n=m−m0,α=αµ0,0,β=βµ0,0,γ=γµ0,0.Thus we havedl mαn dy mαndz mdt2(t)= 0,βn dt2(α−n t),γn dt2(α−n t) uniformly−−−−−−→(0,0,0)(3.2)as m→∞.Since{l m(0)}∞m=m0converges to qµ0,0=(0,0,0),it follows from(3.2)that{l m}C2converges uniformly to the x-axis in D(δ).Lemma3.3.Let S be a regular surface in the Euclidean3-space R3and l a regular curve tangent to S atτ.Suppose that the curvatureκl(τ)of l atτis less than the absolute value of the normal curvatureκS(τ,w)of S atτalong a non-zero vector w tangent to l.Then tangency of S and l atτis quadratic.Proof.By changing the coordinate(x,y,z)on R3by an isometry,we may assume thatτ=(0,0,0),the tangent space of S atτis the xy-plane and w/ w = (1,0,0).Then one can suppose that S(resp.l)is parametrized as(x,y,ψ(x,y)) (resp.(x,f1(x),f2(x))in a small neighborhood of(0,0,0)in R3.Since the graph of z=ψ(x,0)is the cross section of S along the xz-plane,|κS(τ,w)|=|g′′(0)|l of l into the xz-plane,κl(τ)≥κ(f′2(0)2+1)3/2=|f′′2(0)|.It follows from our assumption|κS(τ,w)|>κl(τ)that|g′′(0)|>|f′′2(0)|.This shows that the tangency atτis quadratic.8SHIN KIRIKI,YUSUKE NISHIZA WA,AND TERUHIKO SOMA4.Proof of Theorem A4.1.Existence of quadratic homoclinic tangencies.Let{ϕµ,ν}be the family given Subsection2.2.In particular,r=(u0,v0,0)is a nondegenerate heterodimen-sional tangency of W u(q)and W s(p)which unfolds generically with respect to the µ-parameter families{W u(qµ,0)}and{W s(pµ,0)}.By our settings in Sections2 and3,there exist C2functions fµ,ν:O⊂R2→R C2depending on(µ,ν)with f0,0=f andΣ(µ,ν):={(x,y,fµ,ν(x,y));(x,y)∈O}⊂W s(pµ,ν)∩D(δ)for any(µ,ν)near(0,0).Since det(Hf)(u0,v0)=0,there exists a uniquely deter-mined C2continuation(uµ,ν,vµ,ν)with(u0,0,v0,0)=(u0,v0)and∂fµ,ν∂y(uµ,ν,vµ,ν)=0.Proposition4.1.For a sufficiently smallε>0and anyµin either(0,ε)or (−ε,0),there existsνarbitrarily close to0such thatϕµ,νhas a quadratic homoclinic tangency associated to pµ,ν.By the condition(2.5),r is not in the x-axis.One can take the linearizing coordinate on D(δ)so that s(resp.r)is in the upper half space{z>0}(resp. {x>0}).The Taylor expansion of fµ,νaround(uµ,ν,vµ,ν)has the form:fµ,ν(x,y)=aµ,ν+12dµ,ν(y−vµ,ν)2+o (|x−uµ,ν|+|y−vµ,ν|)2 ,(4.1)where a0,0=0andbµ,ν=∂2fµ,ν∂x∂y(uµ,ν,vµ,ν),dµ,ν=∂2fµ,ν∂µ (µ,ν)=(0,0)=0.If necessary replacingµby−µ,we may assume thatη0>0.By the condition(2.7), b0,0=0and hence bµ,ν=0for any(µ,ν)near(0,0).Proposition4.1is divided to the following two assertions.Assertion4.2(Elliptic case).If r is of elliptic type,then Proposition4.1holds. Proof.First we consider the case of bµ,ν<0for any(µ,ν)near(0,0).By(4.2),for any sufficiently smallµ0>0,the intersection Cµ=Σ(µ0,0)∩{z=0}is a circle disjoint from the x-axis.For a sufficiently small h0>0,A=Σ(µ0,0)∩{0≤z≤h0} is an annulus in D(δ),see Fig.4.1(1).Replacing m0by an integer greater than m0if necessary,we may assume that such that z m(0)<h0/2for any m≥m0.ByLemma3.2,the curve l m⊂W u(pµ0,0)∩D(δ)given in§3is sufficiently C2close tothe x-axis.Thus one can suppose thatπy(l m)∩πy(A)=∅,whereπy:D(δ)−→R is the orthogonal projection defined byπy(x,y,z)=y.For any sufficiently smallν, let l m,νbe the component of W u(pµ0,ν)∩D(δ)such that{l m,ν}is anν-continuationHETERODIMENSIONAL TANGENCIES AND STRANGE ATTRACTORS9Figure4.1.(1)The case ofµ0>0,ν=0,bµ0,0<0.(2)The caseofµ0<0,ν=0,bµ0,0>0.Each shaded region represents A.with l m,0=l m,and set Aν=Σ(µ0,ν)∩{0≤z≤h0}.Moreover,one can suppose that l m,νis parametrized asl m,ν(t)=(t,y m(ν,t),z m(ν,t))(t∈(−δ,δ)).By the condition(C2),one can take¯ν=0with arbitrarily small|¯ν|such that0<πy(l m0,¯ν(0))≤sup{πy(l m0,¯ν)}<min{πy(A¯ν)}.We may assume that¯ν>0if necessary replacingνby−ν.For any integer m sufficiently greater than m0,there exists0<¯νm<¯νsuch the continuation {l m,ν}0≤ν≤¯νm is well defined andmax{πy(A¯νm)}<inf{πy(l m,¯νm)}holds,see Fig.4.2(1).By the Intermediate Value Theorem,there exists0<νm<¯νm such that l m,νm and Aνm have a tangencyτm,see Fig.4.2(2).Since l m,νm⊂Figure4.2.The cross sections.W u(pµ0,νm )and Aνm⊂W s(pµ0,νm),τm is a homoclinic tangency associated topµ0,νm.10SHIN KIRIKI,YUSUKE NISHIZA WA,AND TERUHIKO SOMA When bµ,ν>0for any(µ,ν)near(0,0),one can prove the existence of a homo-clinic tangencyτm near r associated to pµ0,νm by arguments quite similar to thoseas above for anyµ0withµ0<0.It remains to show that the tangencyτm is quadratic.SinceΣ(µ0,νm)is of elliptic type and lim m→∞νm=0,any normal curvature ofΣ(µ0,νm)atτm is greater than some positive constantκ0independent of m.On the other hand,by an argument quite similar to that in Lemma3.2,for any m sufficiently greater than m0,the curvature of l m,νm atτm is less thanκ0.Thus,by Lemma3.3,τm is a quadratic tangency. Assertion4.3(Hyperbolic case).When r is a tangency of hyperbolic type,Propo-sition4.1holds.Proof.Since r is of hyperbolic type,Σ(0,0)∩{z=0}consists of two almost straight curvesα1,α2meeting transversely at r,see Fig.4.3(1).If necessary by reducingFigure4.3.(1)The situation when(µ,ν)=(0,0).(2)The situ-ation when(µ,ν)=(µ0,0).the domain O of fµ,νcontaining(uµ,ν,vµ,ν),we may assume thatΣ(µ,ν)∩{z=0} is disjoint from the x-axis for any(µ,ν)near(0,0).If w i=(ξi,ηi,0)(i=1,2)is a unit vector tangent toαi at r,then b0,0ξ2i+2c0,0ξiηi+d0,0η2i=0.This implies that the normal curvatureκΣ(0,0)(r,w i)ofΣ(0,0)at r along w i is zero.Since b0,0=0 by(2.7),both w1,w2are not parallel to the unit tangent vector v0=(1,0,0). Thus we haveκΣ(0,0)(r,v0)=0.When b0,0<0(resp.b0,0>0),for any sufficiently smallµ0withµ0<0(resp.µ0>0),Σ(µ0,0)∩{z=0}consists of two C2curves β1,β2separated by a line in the xy-plane parallel to x-axis,see Fig.4.3(2),and (4.3)κ0:=|κΣ(µ0,0)(τ,v0)|>0,whereτis a point ofβ1∪β2the tangent line at which is parallel to the x-axis.One can take¯ν>0and h0>0so that Aν=Σ(µ0,ν)∩{0≤z≤h0}is a disjoint union of two curved rectangles for anyνwith0≤ν≤¯ν,see Fig.4.4.Moreover,by(4.3),the¯ν>0can be chosen so that|κΣ(µ0,ν)(˜τ,w)|>κ0/2for any point˜τ∈Aµ0,νsufficiently nearτand any unit vector w∈T˜τ(Aµ0,ν)sufficiently near v0.As in theproof of Assertion4.2,for any integer m sufficiently greater than m0,there existsνm with0<νm<¯νand such that l m,νm⊂W s(pµ0,νm )and Aνm⊂W u(pµ0,νm)have a quadratic tangencyτm,see Fig.4.5.HETERODIMENSIONAL TANGENCIES AND STRANGE ATTRACTORS11Figure4.4.(1)The case ofµ0<0,ν=0,bµ0,0<0.(2)Thecase ofµ0>0,ν=0,bµ0,0>0.Each pair of the shaded regionsrepresents A.Figure4.5.4.2.Generic unfolding of the tangency.For short,set pµ0,ν=pν,fµ0,ν(x,y)=fν(x,y)and(uµ0,ν,vµ0,ν)=(uν,vν).Letτm=(ˆx m,ˆy m,fνm(ˆx m,ˆy m))be the homoclinic tangency of W u(pνm)and W s(pνm)given in Proposition4.1.From(4.1),we have∂fνm∂x(x,y)=b m(x−uνm)+c m(y−vνm)+o1,where b m=bµ0,νm ,c m=cµ0,νm,d m=dµ0,νmand o1=o(|x−uνm|+|y−vνm|).Thus b m∂fνm(x,y)/∂y−c m∂fνm(x,y)/∂x=(b m d m−c2m)(y−vνm)+o1.On the other hand,since there exists a unit vector tangent toΣ(µ0,νm)atτm converges to(1,0,0)as m→∞,lim m→∞∂fνm(ˆx m,ˆy m)/∂x=0.Since lim m→∞b m=bµ0,0=0and lim m→∞b m d m−c2m=det(Hfµ0,0)(uµ0,0,vµ0,0)=0,∂fνmb m ∂fνmb m(ˆy m−vνm)+o1=012SHIN KIRIKI,YUSUKE NISHIZA WA,AND TERUHIKO SOMAfor all sufficiently large m .By the Implicit Function Theorem,there exists a C 2function y =g ν(x,z )=g (ν,x,z )defined in a small neighborhood of (νm ,ˆx m ,f νm (ˆx m ,ˆy m ))in the (ν,x,z )-space with(x,y,f ν(x,y ))=(x,g ν(x,z ),z ).Proposition 4.4.For all sufficiently large m ,the quadratic homoclinic tangency τm of W s (p νm )and W u (p νm )unfolds generically at ν=νm with respect to the ν-parameter families {W s (p ν)}and {W u (p ν)}.Proof.Recall that l m,νhas the parametrization l m,ν(t )=(t,y m (ν,t ),z m (ν,t ))with l m,νm (ˆx m )=τm .By Definition 2.1(2),it suffices to show that(4.4)∂y m∂ν(νm ,ˆx m ,z m (νm ,ˆx m ))for all sufficiently large m .Note thatlim m →∞∂g∂ν(0,ˆx ∞,0),where ˆx ∞is the x -coordinate of a point τin Σ(µ0,0)∩{z =0}the tangent line in xy -plane at which is parallel to (1,0,0),see Fig.4.3(2)in the case that r is of hyperbolic type.If we set ˜x m,ν=α−n νˆx m ,then ϕn ν(l m 0,ν(˜x m,ν))=l m,ν(ˆx m ),wheren =m −m 0and αν=αµ0,ν.As was seen in the proof of Lemma 3.1,(4.5)lim m →∞∂y m 0∂ν(0,0)=0.We denote the ν-function y m 0(ν,˜x m,ν)by h m (ν).Since lim m →∞d ˜x m,ν/dν=0,it follows from (4.5)thatdh m ∂ν(νm ,˜x m,νm )+∂y m 0dν(νm )≥ ∂y m 0∂x (νm ,˜x m,νm )d ˜x m,ν∂ν(νm ,ˆx m )=βn νm dhmdν(νm )h m (νm )=βn νm dh m βνmdβνHETERODIMENSIONAL TANGENCIES AND STRANGE ATTRACTORS13References[1]M.Asaoka,A simple construction of C1-Newhouse domains for higher dimensions,preprint.[2]Ch.Bonatti and L.J.D´ıaz,Connexions h´e t´e roclines et g´e n´e ricit´e d’une infinit´e de puits etde sources,Ann.Sci.´Ecole Norm.Sup.,(4)32(1999),no.1,135–150.[3]Ch.Bonatti,L.J.D´ıaz,and E.R.Pujal,A C1-generic dichotomy for diffeomorphisms:Weakforms of hyperbolicity or infinitely many sinks or sources,Ann.of Math.,158(2003),355–418.[4]Ch.Bonatti and L.J.D´ıaz,Robust heterodimensional cycles and C1-generic dynamics,preprint.[5]Ch.Bonatti,L.J.D´ıaz,and M.Viana,Dynamics beyond uniform hyperbolicity,Ency-clopaedia of Mathematical Sciences(Mathematical Physics)102,Mathematical physics,III, Springer Verlag,2005.[6]L.J.Diaz,A.Nogueira and E.R.Pujals,Heterodimensional tangencies,Nonlinearity19(2006),2543–2566.[7]L.J.D´ıaz and J.Rocha,Non-connected heterodimensional cycles:bifurcation and stability,Nonlinearity5(1992),1315–1341.[8]L.J.D´ıaz,Robust nonhyperbolic dynamics and heterodimensional cycles,Ergod.Th.Dynam.Sys.15(1995),291–315.[9]L.J.D´ıaz and J.Rocha,Large measure of hyperbolic dynamics when unfolding heterocliniccycles,Nonlinearity10(1997),857–884.[10]L.J.D´ıaz and J.Rocha,Noncritical saddle-node cycles and robust nonhyperbolic dynamics,Dynam.Stability Systems,12(2)(1997),109–135.[11]L.J.D´ıaz and J.Rocha,Partially hyperbolic and transitive dynamics generated by hetero-clinic cycles,Ergod.Th.Dynam.Sys.21(2001),25–76.[12]J.Palis,A global perspective for non-conservative dynamics,Ann.I.H.Poincar´e22(2005),485–507.[13]J.Palis and M.Viana,High dimension diffeomorphisms displaying infinitely many periodicattractors,Ann.of Math.(2)140(1994),207–250.[14]M.Viana,Strange attractors in higher dimensions,Bol.Soc.Brasil.Mat.(N.S.)24(1993),13–62.Department of Mathematics,Kyoto University of Education,1Fukakusa-Fujinomori, Fushimi-ku,Kyoto,612-8522,JapanE-mail address:skiriki@kyokyo-u.ac.jpDepartment of Mathematics and Information Sciences,Tokyo Metropolitan Univer-sity,Minami-Ohsawa1-1,Hachioji,Tokyo192-0397,JapanE-mail address:nishizawa-yusuke@ed.tmu.ac.jpDepartment of Mathematics and Information Sciences,Tokyo Metropolitan Univer-sity,Minami-Ohsawa1-1,Hachioji,Tokyo192-0397,JapanE-mail address:tsoma@tmu.ac.jp。