Renormalization of C r Hénon map : Two dimensional embedded map in three dimension
aa r X i v : . [ m a t h . D S ] D ec RENORMALIZATION OF C r H ´ENON MAP : TWO DIMENSIONALEMBEDDED MAP IN THREE DIMENSION
YOUNG WOO NAM
Abstract.
We study renormalization of highly dissipative analytic three dimensional H´enonmaps F ( x, y, z ) = ( f ( x ) − ε ( x, y, z ) , x, δ ( x, y, z ))where ε ( x, y, z ) is a sufficiently small perturbation of ε d ( x, y ). Under certain conditions, C r single invariant surfaces each of which is tangent to the invariant plane field over thecritical Cantor set exist for 2 ≤ r < ∞ . The C r conjugation from an invariant surface tothe xy − plane defines renormalization two dimensional C r H´enon-like map. It also definestwo dimensional embedded C r H´enon-like maps in three dimension. In this class, universalitytheorem is re-constructed by conjugation. Geometric properties on the critical Cantor setin invariant surfaces are the same as those of two dimensional maps — non existence ofthe continuous line field, and unbounded geometry. The set of embedded two dimensionalH´enon-like maps is open and dense subset of the parameter space of average Jacobian, b F d for any given smoothness, 2 ≤ r < ∞ . Contents C r H´enon-like maps 176 Unbounded geometry on the Cantor set 19A Appendix Periodic points and critical Cantor set 22
Date : December 29, 2014.College of Science and Technology, Hongik University at Sejong, Korea.Email : namyoungwoo @ hongik.ac.kr. . Introduction
Renormalization is for the one dimensional maps for a few recent decades by many authorsin various papers. Some of main results and historical facts of renormalization theory of onedimensional maps are in [dFdMP] and references therein. Renormalization of higher dimen-sional maps was started by Coullet, Eckmann and Koch in [CEK]. Period doubling renor-malization of analytic H´enon map with strong dissipativeness was introduced in [dCLM].The average Jacobian b F of infinitely renormalizable H´enon-like map, F , is defined b F = exp Z O F log Jac F dµ where O F is the critical Cantor set and µ is the ergodic measure on O F . Carvalho, Lyubichand Martens in [dCLM] proved Universality Theorem and showed geometric properties ofthe critical Cantor set which are different from those of one dimensional maps. For instance,generic unbounded geometry of the critical Cantor set in the parameter space of the averageJacobian was shown and this geometric property is generalized for the full Lebesgue measureset in [HLM].H´enon renormalization is generalized for three dimensional analytic H´enon-like map in[Nam1]. For instance, the universal asymptotic expression of R n F isJac R n F ( x, y, z ) = b n F a ( x )(1 + O ( ρ n ))where a ( x ) is analytic and positive for 0 < ρ <
1. However, the universal expression ofJacobian determinant of three dimensional renormalized map does not imply the UniversalTheorem because the Jacobian determinant, Jac R n F = ∂ y ε n ∂ z δ n − ∂ z ε n ∂ y δ n contains partialderivatives of both ε and δ . Moreover, infinitely renormalizable H´enon map has maximalLyapunov exponent is zero. Thus ln b is the other exponent for two dimensional map. How-ever, since ln b F for three dimensional map is not an exponent but the sum of Lyapunovexponents. Thus two universal numbers for three dimensional maps would be required inorder to explain geometric properties of O F . One of the universal numbers is a counterpartof the average Jacobian of two dimensional map. The universal numbers, b and b whichrepresent two dimensional H´enon-like map in three dimension and contraction from the thirddimension were found in [Nam1] under certain conditions. For the precise formulation, see § C r surfaces for any natural number 2 ≤ r < ∞ and it is asymptotically slantedplane (Proposition 3.3). The map from invariant surface to xy − plane defines the renor-malization of C r H´enon-like maps and it is the same as the analytic definition of H´enonrenormalization (Proposition 4.1) RF = Λ ◦ H ◦ F ◦ H − ◦ Λ − . Moreover, Universality Theorem for C r H´enon-like map is re-constructed by invariant sur-faces (Theorem 4.3). It defines the embedded two dimensional H´enon-like map in threedimension . Moreover, two dimensional C r H´enon-like map is embedded in three dimensiongenerically in the set of parameter space of average Jacobian (Theorem 5.5). The universalnumbers of three dimensional H´enon-like map, b which is the average Jacobian of two di-mensional C r H´enon-like map and b ≡ b F /b , we would show the unbounded geometry of F for almost everywhere in the parameter space of b of embedded C r H´enon-like maps(Theorem 6.3).
2. Preliminaries
For the given map F , if a set A is related to F , then we denote it to be A ( F ) or A F and F can be skipped if there is no confusion without F . The domain of F isdenoted to be Dom( F ). If F ( B ) ⊂ B , then we call B is an (forward) invariant set under F .The set A in the given topology is called the closure of A . For three dimensional map, letus the projection from R to its x − axis, y − axis and z − axis be π x , π y and π z respectively.Moreover, the projection from R to xy − plane be π xy and so on.Let C r ( X ) be the Banach space of all real functions on X for which the r th derivative iscontinuous. The C r norm of h ∈ C r ( X ) is defined as follows k h k C r = max ≤ k ≤ r (cid:8) k h k , k D k h k (cid:9) . For analytic maps, since C norm bounds C r norm for any r ∈ N , we often use the norm, k · k instead of k · k or k · k C k . For the two sets S and T in R , the minimal distance of twosets is defined as dist min ( S, T ) = inf { dist( p, q ) | p ∈ S and q ∈ T } The set of periodic points of the map F is denoted by Per F . A = O ( B ) means that thereexists a positive number C such that A ≤ CB . Moreover, A ≍ B means that there exists apositive number C which satisfies 1 C B ≤ A ≤ CB . Two di-mensional H´enon-like map is defined as F ( x, y ) = ( f ( x ) − ε ( x, y ) , x )where f is a unimodal map. Assume that the norm of ε is sufficiently small and F isorientation preserving map. Since F is not H´enon-like map, the non linear scaling map forrenormalization of H´enon-like map, F . The horizontal map of F is defined H ( x, y ) = ( f ( x ) − ε ( x, y ) , y ) . The period doubling renormalization of F is defined as RF = Λ ◦ H ◦ F ◦ H − ◦ Λ − where Λ( x, y ) = ( sx, sy ) for the appropriate number s < − F ( x, y, z ) = ( f ( x ) − ε ( x, y, z ) , x, δ ( x, y, z )) . We assume that the norms of both ε and δ are sufficiently small and that the three dimen-sional map F is analytic throughout this paper. The domain of F is cubic box and F has twofixed points and sectionally dissipative at these points. The horizontal-like map is defined H ( x, y, z ) = ( f ( x ) − ε ( x, y, z ) , y, z − δ ( y, f − ( y ) , . hus the (period doubling) renormalization of three dimensional map is the natural extensionof two dimensional H´enon-like map as follows RF = Λ ◦ H ◦ F ◦ H − ◦ Λ − where Λ( x, y, z ) = ( sx, sy, sz ) for the appropriate number s < − Let the set of infinitely renormalizable maps be I (¯ ε ) where the norm k ε k and k δ k (for three dimensional maps) are bounded above by O ( ε ) where ¯ ε is a smallenough positive number. The following definitions and facts are common in both two andthree dimensional H´enon-like maps in I (¯ ε ).If F is n − times renormalizable, then R k F is defined as the renormalization of R k − F for2 ≤ k ≤ n . Denote Dom( F ) to be the box region, B . If the set B is emphasized with therelation of a certain map R k F , for example, then denote this region to be B ( R k F ). F k denotes R k F for each k . Let the coordinate change map which conjugates F k | Λ − k ( B ) and RF k is denoted by ψ k +1 v ≡ H − k ◦ Λ − k : Dom( RF k ) → Λ − k ( B )where H k is the horizontal-like diffeomorphism and Λ k is dilation with each appropriateconstants s k < − k . Denote F k ◦ ψ k +1 v by ψ k +1 c . The word of length n in theCartesian product, W n ≡ { v, c } n is denoted by w n or simply w . Express the compositionsof ψ jv and ψ jc for k ≤ j ≤ n as followsΨ nk, w = ψ kw ◦ ψ k +1 w ◦ · · · ◦ ψ nw n − k where each w i is v or c and the word w = ( w w . . . w n − k ) in W n − k . The map Ψ nk, w is from B ( R n F ) to B ( R k F ). Denote the region Ψ nk, w ( B ( R n F )) by B nk, w . In particular, denote B n , w by B n w for simplicity. We see that(2.1) diam( B n w ) ≤ Cσ n where w is any word of length n in W n for some C > F is ainfinitely renormalizable H´enon-like map, then it has invariant Cantor set O F = ∞ \ n =1 [ w ∈ W n B n w and F acts on O F as a dyadic adding machine. The counterpart of the critical value ofunimodal renormalizable map is called the tip { τ F } ≡ \ n ≥ B n v where v = v n for every n ∈ N . The word w ∈ W ∞ for each w ∈ O is called the address of w . Similarly, the word with finite length w n ∈ W n corresponding the region, B n w n is calledthe address of box. Moreover, since each box, B n w n contains a unique periodic point withminimal period, 2 n , the address of periodic point is also defined as that of B n w n . The firstsuccessive finite concatenation of the given address, w is called the subaddress of w . ByDistortion Lemma and the average Jacobian with invariant measure, we see the followinglemma. emma 2.1. For any piece B n w at any point w = ( x, y, z ) ∈ B n w , the Jacobian determinantof F n is (2.2) Jac F n ( w ) = b n F (1 + O ( ρ n )) where b is the average Jacobian of F for some < ρ < . Then there exists the asymptotic expression of Jac R n F for the map F ∈ I (¯ ε ) with b F and the universal function. Theorem 2.2 ([dCLM] and [Nam]) . For the map F ∈ I (¯ ε ) with small enough positivenumber ¯ ε , the Jacobian determinant of n th renormalization of F is as follows (2.3) Jac R n F = b n F a ( x ) (1 + O ( ρ n )) where b F is the average Jacobian of F and a ( x ) is the universal positive function for n ∈ N and for some ρ ∈ (0 , . Denote the tip, τ F n to be τ n for n ∈ N . The definitions of tip and Ψ nk, v imply that Ψ nk, v ( τ n ) = τ k for k < n . Then after composing appropriate translations, tips on each level moves to theorigin as the fixed point Ψ nk ( w ) = Ψ nk, v ( w + τ n ) − τ k for k < n . Notations with the subscript, v is strongly related to the tip. For instance, B nk, v contains the tip, τ k for every n > k and Ψ nk, v is the map from the tip, τ n to the tip τ k forevery n > k . Thus in order to emphasize the tip on every deep level, we sometimes use thenotation B nk, tip or Ψ nk, tip instead of B nk, v or Ψ nk, v . Moreover, if we need to distinguish threedimensional notions from two dimensional one, then we use the subscript, 2 d . For example, d Ψ nk , d B nk, v , d t n, k , d S nk ( w ) and so on. Ψ nk . The map Ψ nk is separated nonlinear part and dilation part after reshuffling(2.4) Ψ nk ( w ) = t n, k u n, k d n, k α n, k σ n, k σ n, k x + S nk ( w ) yz + R nk ( y ) where α n, k = σ n − k ) (1 + O ( ρ k )) and σ n, k = ( − σ ) n − k (1 + O ( ρ k )). The non-linear map x + S nk ( w ) has following asymptotic with the universal diffeomorphism v ∗ ( x ). Lemma 2.3.
Let x + S nk ( w ) be the first coordinate map of three dimensional coordinatechange map in (2.4) for infinitely renormalizable H´enon-like map. Then (2.5) | [ x + S n ( x, y, z )] − [ v ∗ ( x ) + a F, y + a F, yz + a F, z ] | = O ( ρ n ) where constants | a F, | , | a F, | and | a F, | are O (¯ ε ) for ρ ∈ (0 , . Moreover, for each fixed y and z , the above asymptotic has C convergence with the variable x . The constants t n, k , u n, k and d n, k converges to some numbers — say t ∗ , k , u ∗ , k and d ∗ , k respectively — super exponentially fast as n → ∞ . Moreover, estimation of the aboveconstants is following(2.6) | t n, k | , | u n, k | , | d n, k | ≤ C ¯ ε k or k < n and for some constant C >
0. Lemma 5.1 in [Nam2] contains the detailedcalculation for these constants. Moreover, Lemma 5.2 in [Nam2] implies that(2.7) k R nk k C ≤ Cσ n for some C > n . Recall the following definitions for later useΛ − n ( w ) = σ n · w, ψ n +1 v ( w ) = H − n ( σ n w ) , ψ n +1 c ( w ) = F n ◦ H − n ( σ n w ) ψ n +1 v ( B ( R n +1 F )) = B n +1 v , ψ n +1 c ( B ( R n +1 F )) = B n +1 c for each n ∈ N . Let H´enon-like map satisfying ε ( w ) = ε ( x, y ), thatis, ∂ z ε ≡ toy model H´enon-like map . Denote the toy model map by F mod . Then theprojected map π xy ◦ F mod = F d from B to R is exactly two dimensional H´enon-like map.If F mod is renormalizable, then we have π xy ◦ RF mod = RF d . Proposition 2.4.
Let F mod = ( f ( x ) − ε d ( x, y ) , x, δ ( w )) be a toy model diffeormorphismin I (¯ ε ) . Then n th renormalized map R n F mod is also a toy model map, that is, π xy ◦ R n F mod = R n F d for every n ∈ N . Moreover, ε d,n ( x, y ) = ( b ) n a ( x ) y (1 + O ( ρ n )) where b is the averageJacobian of two dimensional map, F d = π xy ◦ F mod . Let b mod be the average Jacobian of F mod ∈ I (¯ ε ). Define another number, b as the ratio b mod /b . Then by the above Proposition ∂ z δ n ≍ b n for every n ∈ N , which is anotheruniversal number. Let the following map be a perturbation of toy model map, F mod ( w ) =( f ( x ) − ε d ( x, y ) , x, δ ( w ))(2.8) F ( w ) = ( f ( x ) − ε ( w ) , x, δ ( w ))where ε ( w ) = ε d ( x, y ) + e ε ( w ). Thus ∂ z ε ( w ) = ∂ z e ε ( w ). If k e ε k is sufficiently small, then F iscalled a small perturbation of F mod . Let us consider the block matrix form of DF .(2.9) DF = D e F d ∂ z ε ∂ x δ ∂ y δ ∂ z δ = A BC D ! , DF mod = DF d ∂ x δ ∂ y δ ∂ z δ = A C D ! where D e F d = (cid:18) f ′ ( x ) − ∂ x ε ( w ) − ∂ y ε ( w )1 0 (cid:19) and DF d = (cid:18) f ′ ( x ) − ∂ x ε d ( x, y ) − ∂ y ε d ( xy )1 0 (cid:19) respectively. Observe that if B ≡ , then F is F mod . Define m ( A ) as k A − k − and it is calledthe minimum expansion (or strongest contraction) rate of A . Lemma 2.5 (Lemma 7.4 in [Nam1]) . Let F be a small perturbation of F mod defined in (2.8) .Let A , A , B , C and D be components of the block matrix defined in (2.9) . Suppose that k D k ≤ ρ · m ( A ) for some ρ ∈ (0 , . Suppose also that k B kk C k ≤ ρ · m ( A ) · m ( D ) where ρ < κγ for sufficiently small γ > . Then there exist the continuous invariant plane fieldover the given invariant compact set, Γ . he tangent bundle T Γ B has the splitting with subbundles E ⊕ E such that(1) T Γ B = E ⊕ E .(2) Both E and E are invariant under DF .(3) k DF n | E ( x ) kk DF − n | E ( F − n ( x )) k ≤ Cµ n for some C > < µ < n ≥ T Γ B has dominated splitting over the compact invariant set Γ. Moreover,dominated splitting implies that invariant sections are continuous by Theorem 1.2 in [New].Then the maps, w E i ( w ) for i = 1 ,
3. Single invariant surfaces
The uniform boundedness of the ratio k D kk A − k < in DF means thatsup w ∈ B k D w k m ( A w ) ≤ w ∈ B . It implies thedominated splitting of tangent bundle over a given invariant compact set, Γ. If dominatedsplitting over a given compact set Γ satisfies thatsup w ∈ B k D w k m ( A w ) r ≤ r ∈ N , then we say that F has r-dominated splitting over Γ. Moreover, if k D k for DF mod is sufficiently smaller than b for all w ∈ Γ, then contracting or expanding rates, m ( A ) and k D k are separated by a uniform constant over the whole Γ. It is called pseudo hyperbolicity . Dominated splittingover the given invariant compact set, Γ with smooth cut off function implies the pseudo(un)stable manifolds at each point in Γ tangent to an invariant subbundle. However, if thedominated splitting satisfies certain conditions, then the whole compact set is contained ina single invariant submanifold of the ambient space (Theorem 3.1 below).
Definition 3.1. A C r submanifold Q which contains Γ is locally invariant under f if thereexists a neighborhood U of Γ in Q such that f ( U ) ⊂ Q .The necessary and sufficient condition for the existence of these submanifolds, see [CP] or[BC]. Theorem 3.1 ([BC]) . Let Γ be an invariant compact set with a dominated splitting T Γ M = E ⊕ E such that E is uniformly contracted. Then Γ is contained in a locally invariantsubmanifold tangent to E if and only if the strong stable leaves for the bundle E intersectthe set Γ at only one point. Moreover, the existence of invariant submanifold is robust under C perturbation by [BC].Infinitely renormalizable toy model H´enon-like map with b ≪ b satisfies the sufficient con-dition for the existence of locally invariant single surfaces by Lemma A.2. By C robustness, he ambient space of toy model maps and its sufficiently small perturbation can be reducedto a single invariant surface. Remark 3.1.
Theorem 3.1 is extended to the existence of C r invariant submanifold with r − dominated splitting. Moreover, the given invariant compact set can be extended to themaximal one. Lemma 3.2.
Let F mod be a toy model map in I (¯ ε ) . Suppose that b ≪ b where b is theaverage Jacobian of π xy ◦ F mod . Then Per F mod has the dominated splitting in Lemma 2.5.Moreover, there exists a locally invariant C single surface Q which contains Per F mod and Q meets transversally and uniquely strong stable manifold, W ss ( w ) at each w ∈ Per F mod .Proof. One of the eigenvalues of DF mod at each point is asymptotically b with the eigenvector(0 0 1) by straightforward calculation. Thus dominated splitting exists with the condition b ≪ b over any invariant compact set, in particular, Per F mod . Each cone of the vector(0 0 1) at all points is disjoint from the invariant plane field, say E pu - tangent subbundlewith pseudo unstable direction. Thus any invariant surface, Q tangent to E pu over Per F mod meets transversally the strong stable manifold. Let us show the uniqueness of intersectionpoint. Suppose that w and w ′ are intersection points between Q and W ss ( w ). If w ′ = w ,then w ′ / ∈ Per mod by Lemma A.2. Take a small neighborhood U of w ′ in the invariant surface Q . Then U converges to the neighborhood of F n ( w ) in Q as n → ∞ by Inclination Lemma.Thus Q cannot be a submanifold of the ambient space because it accumulates itself. Itcontradicts to Theorem 3.1. Hence, w is the unique intersection point. (cid:3) Recall that three dimensional H´enon-like map in I (¯ ε ) is sectionally dissipative at each pe-riodic points. Thus the invariant plane field over Per F mod contains the unstable direction ofeach periodic point. Then Q contains the set A ≡ O ∪ [ n ≥ W u (Orb( q n ))where each q n is a periodic point whose period is 2 n for n ∈ N . A is called the globalattracting set . Per as the graph of C r map. Let F mod be theH´enon-like toy model map in I (¯ ε ). Let b be the average Jacobian of F d ≡ π xy ◦ F mod andassume that b ≪ b . The set of lines perpendicular to xy − plane(3.1) [ ( x, y ) ∈ π xy ( B ) { ( x, y, z ) | z ∈ I z } is invariant under F mod . Thus the invariant section, w E ss ( w ) is constant. The above set,(3.1) contains the strong stable manifold over Γ. The angle between each tangent spaces E ssw and E puw is (uniformly) positive. Thus the maximal angle between E pu and T R is less than π . Remark 3.2. If T Γ B = E ss ⊕ E pu is r − dominated splitting, then Q which is invariantsingle surface tangent to E pu is a C r surface. Moreover, since the strong stable manifolds ateach point is the set of perpendicular lines to xy − plane, Q is the graph of C r function froma region in I x × I y to I z . et F mod ∈ I (¯ ε ) with b ≪ b . Then by above Lemma 3.2, we may assume invariant surfacestangent to the invariant plane field has the neighborhood, say also Q , of the tip, τ F mod in thegiven invariant single surface which satisfies the following properties.(1) Q is contractible.(2) Q contains τ F mod in its interior and is locally invariant under F N for big enough N ∈ N .(3) Topological closure of Q is the graph of C r map from a neighborhood of τ (cid:0) π xy ◦ F mod (cid:1) in xy − plane to I z .By C robustness of the existence of single invariant surfaces, let F be a sufficiently smallperturbation of F mod such that there exist invariant surfaces each of which is the graph of C r map from a region in the xy − plane to I z . Proposition 3.3.
Let F ∈ I (¯ ε ) . Suppose that there exists an invariant surface under F , say Q which is the graph of C r function, ξ on π xy ( B n tip ) such that k Dξ k ≤ C for some C > .Then Q n ≡ (cid:0) Ψ n tip (cid:1) − ( Q ) is the graph of a C r function ξ n on π xy (cid:0) B ( R n F ) (cid:1) such that ξ n ( x, y ) = c y (1 + O ( σ n )) for some constant c .Proof. The n th renormalization of F , R n F is (cid:0) Ψ n tip (cid:1) − ◦ F n ◦ Ψ n tip . Thus Q n ≡ (cid:0) Ψ n tip (cid:1) − ( Q )is an invariant surface under R n F . Let us choose a point w ′ = ( x ′ , y ′ , z ′ ) ∈ Q ∩ B n where B n tip ≡ Ψ n tip ( B ( R n F )) and z ′ = ξ ( x ′ , y ′ ). Thusgraph( ξ ) = ( x ′ , y ′ , ξ ( x ′ , y ′ )) = ( x ′ , y ′ , z ′ ) . Moreover, let (cid:0) Ψ n tip (cid:1) − ( x ′ , y ′ , z ′ ) = ( x, y, z ) ∈ Q n . Thus by the equation (2.4), each coordi-nates of Ψ n ≡ Ψ n tip ( w − τ n ) − τ F as follows x ′ = α n, ( x + S n ( w )) + σ n, t n, · y + σ n, u n, ( z + R n ( y ))(3.2) y ′ = σ n, · y (3.3) z ′ = σ n, d n, · y + σ n, ( z + R n ( y ))(3.4)where w ′ = ( x ′ , y ′ , z ′ ). Firstly, let us show that Q n is the graph of a well defined function ξ n from π xy ( B ( R n F )) to π z ( B ( R n F )), that is, z = ξ n ( x, y ). By the equations (3.3) and (3.4),we see that(3.5) σ n, · z = z ′ − σ n, d n, · y − σ n, R n ( y )= ξ ( x ′ , y ′ ) − σ n, d n, · y − σ n, R n ( y )= ξ (cid:0) α n, ( x + S n ( w )) + σ n, t n, · y + σ n, u n, ( z + R n ( y )) , σ n, · y (cid:1) − σ n, d n, · y − σ n, R n ( y ) . Define a function as below G n ( x, y, z ) = ξ (cid:0) α n, ( x + S n ( w )) + σ n, t n, · y + σ n, u n, ( z + R n ( y )) , σ n, · y (cid:1) − σ n, d n, · y − σ n, R n ( y ) − σ n, · z. hen the partial derivative of G n over z is as follows ∂ z G n ( x, y, z ) = ∂ x ξ ◦ (cid:0) α n, ( x + S n ( w )) + σ n, t n, · y + σ n, u n, ( z + R n ( y )) , σ n, · y (cid:1) · (cid:2) α n, · ∂ z S n ( w ) + σ n, u n, (cid:3) − σ n, . Recall that α n, = σ n (1 + O ( ρ n )), σ n, = ( − σ ) n (1 + O ( ρ n )), k ∂ z S n k = O (cid:0) ¯ ε (cid:1) and | u n, | = O (cid:0) ¯ ε (cid:1) . Then k ∂ z G n k ≥ (cid:2) − k ∂ x ξ k (cid:2) σ n C ¯ ε + σ n C ¯ ε (cid:3) + σ n (cid:3) (1 + O ( ρ n ))for some positive C and C . Since k Dξ k ≤ C for some C > k ∂ z G n k is away from zerouniformly for small enough ¯ ε >
0. By implicit function theorem, z = ξ n ( x, y ) is a C r functionlocally on a neighborhood of at every point ( x, y ) ∈ π xy ( B ( R n F )). Furthermore, since Q n iscontractible, ξ n ( x, y ) is defined globally by C r continuation of the coordinate charts.By the equations (3.3) and (3.4) with chain rule, we obtain the following equations ∂ x ξ · ∂x ′ ∂x = σ n, · ∂ x ξ n ∂ x ξ · ∂x ′ ∂y + ∂ y ξ · σ n, = σ n, d n, + σ n, · ∂ y ξ n + σ n, · ( R n ) ′ ( y ) . Each partial derivatives of ξ n as follows by the equation (3.2),(3.6) ∂ξ n ∂x = 1 σ n, · ∂ x ξ · (cid:20) α n, (cid:0) ∂ x S n ( w ) (cid:1) + σ n, u n, · ∂ξ n ∂x (cid:21) ∂ξ n ∂y = 1 σ n, · ∂ x ξ · (cid:20) α n, ∂ y S n ( w ) + σ n, t n, + σ n, u n, (cid:16) ∂ξ n ∂y + ( R n ) ′ ( y ) (cid:17)(cid:21) + ∂ y ξ − d n, − ( R n ) ′ ( y ) . Recall the facts that σ n, ≍ ( − σ ) n , α n, ≍ σ n for each n ∈ N Thus (cid:13)(cid:13)(cid:13) ∂ξ n ∂x (cid:13)(cid:13)(cid:13) ≤ k ∂ x ξ k C σ n ≤ C σ n for some C >
0. Recall also that k ∂ y S n k ≤ C ¯ ε for some C > t n, , u n, and d n, converge to the numbers t ∗ , , u ∗ , , and d ∗ , respectively superexponentially fast.In the above equation (3.6), each partial derivatives ∂ x ξ and ∂ y ξ converges to the origin as n → ∞ because all points in the domain of ξ are in B n ≡ Ψ n ( B ( R n F )) and diam( B n ) ≤ Cσ n . Thus both derivatives ∂ x ξ ( x, y ) and ∂ y ξ ( x, y ) converges to ∂ x ξ ( τ F ) and ∂ y ξ ( τ F ) as n → ∞ respectively. However, the quadratic or higher order terms of ∂ξ n ∂y converges to zeroexponentially fast by the equation (2.7), that is, k R nk k C ≤ Cσ n . Hence, we obtain that ξ n ( x, y ) = c y (1 + O ( σ n ))where c = ∂ x ξ ( τ F ) · t ∗ , + ∂ y ξ ( τ F ) − d ∗ , − u ∗ , . (cid:3) . Universality of conjugated two dimensionalH´enon-like map Let F ∈ I (¯ ε ) be a sufficiently small perturbation of the given model map F mod ∈ I (¯ ε ).Let Q n and Q k be invariant surfaces under R n F and R k F respectively for k < n . Then byLemma 3.3, Ψ nk is the coordinate change map between R k F n − k and R n F from level n to k such that Ψ nk ( Q n ) ⊂ Q k . Let us define C r two dimensional H´enon-like map d F n, ξ on level n as follows d F n, ξ ≡ π ξ n xy ◦ R n F | Q n ◦ ( π ξ n xy ) − (4.1)where the map ( π ξ n xy ) − : ( x, y ) ( x, y, ξ n ( x, y )) is a C r diffeomorphism on the domain oftwo dimensional map, π xy ( B ). In particular, the map F d, ξ is defined as follows(4.2) F d, ξ ( x, y ) = ( f ( x ) − ε ( x, y, ξ ) , x )where graph( ξ ) is a C r invariant surface under the three dimensional map F : ( x, y, z ) ( f ( x ) − ε ( x, y, z ) , x, δ ( x, y, z )). Let us assume that 2 ≤ r < ∞ . ByLemma 3.3, the invariant surfaces, Q n and Q k are the graphs of C r maps ξ n ( x, y ) and ξ k ( x, y )respectively. The map d Ψ nk, ξ, tip is defined as the map satisfying the following commutativediagram ( Q n , τ n ) π ξ n xy, n (cid:15) (cid:15) Ψ nk, v , tip / / ( Q k , τ k ) π ξ k xy, k (cid:15) (cid:15) ( d B n , τ d, n ) d Ψ nk, ξ, tip / / ( d B k , τ d, k )where Q n and Q k are invariant C r surfaces with 2 ≤ r < ∞ of R n F and R k F respectivelyand π ξ n xy, n and π ξ k xy, k are the inverses of graph maps, ( x, y ) ( x, y, ξ n ) and ( x, y ) ( x, y, ξ k )respectively.Using translations T k : w w − τ k and T n : w w − τ n , we can let the tip move to theorigin as the fixed point of new coordinate change map, Ψ nk ≡ T k ◦ Ψ nk, tip ◦ T − n . Thus dueto the above commutative diagram, corresponding tips in d B j for j = k, n is changed tothe origin. Let π xy ◦ T j be T d, j for j = k, n . This origin is also the fixed point of the map d Ψ nk, ξ := T d, k ◦ d Ψ nk, ξ, tip ◦ T − d, n where T d, j = π xy, j ◦ T j with j = k, n . By straightforwardcalculation, we obtain the expression of d Ψ nk, ξ as follows d Ψ nk, ξ = π ξ k xy, k ◦ Ψ nk ( x, y, ξ n )= π ξ k xy, k ◦ α n, k σ n, k t n, k σ n, k u n, k σ n, k σ n, k d n, k σ n, k x + S nk, ξ yξ n + R nk ( y ) (cid:0) α n, k ( x + S nk, ξ ) + σ n, k t n, k y + σ n, k u n, k ( ξ n + R nk ( y )) , σ n, k y (cid:1) (4.3)where S nk, ξ = S nk ( x, y, ξ n ( x, y )). ThenJac d Ψ nk, ξ = det (cid:18) α n, k (1 + ∂ x S nk, ξ + ∂ z S nk, ξ · ∂ x ξ n ) + σ n, k u n, k ∂ x ξ n • σ n, k (cid:19) = σ n, k (cid:0) α n, k (1 + ∂ x S nk, ξ + ∂ z S nk, ξ · ∂ x ξ n ) + σ n, k u n, k ∂ x ξ n (cid:1) . (4.4)If F ∈ I (¯ ε ) has invariant surfaces as the graph of C r maps defined on I x × I y at every level,then d Ψ k +1 k, ξ is the conjugation between ( d F k, ξ ) and d F k +1 , ξ for each k ∈ N . Then twodimensional map F d, ξ is called formally infinitely renormalizable map with C r conjugation.Moreover, the map defined on the equation (4.3) with n = k + 1, d Ψ k +1 k, ξ is the inverse of thehorizontal map ( x, y ) ( f k ( x ) − ε k ( x, y, ξ k ) , y ) ◦ ( σ k x, σ k y )by Proposition 4.1 below. Proposition 4.1.
Let the coordinate change map between ( d F k, ξ ) and d F k +1 , ξ be d Ψ k +1 k, ξ which is the conjugation defined on (4.3) . Then d Ψ k +1 k, ξ = H − k, ξ ◦ Λ − k for every k ∈ N where H k, ξ ( x, y ) = ( f k ( x ) − ε k ( x, y, ξ k ) , y ) and Λ − k ( x, y ) = ( σ k x, σ k y ) .Proof. Recall the definitions of the horizontal-like diffeomorphism H k and its inverse, H − k as follows H k ( w ) = ( f k ( x ) − ε k ( w ) , y, z − δ k ( y, f − k ( y ) , H − k ( w ) = ( φ − k ( w ) , y, z + δ k ( y, f − k ( y ) , . Observe that H k ◦ H − k = id and f k ◦ φ − k ( w ) − ε k ◦ H − k ( w ) = x for all points w ∈ Λ − k ( B ).Then if we choose the set σ k · graph( ξ k +1 ) ⊂ Λ − k ( B ), then the similar identical equationholds. By the definition of d Ψ nk, ξ , the following equation holds d Ψ k +1 k, ξ ( x, y ) = π ξ k xy ◦ Ψ k +1 k ◦ ( π ξ k +1 xy ) − ( x, y )= π ξ k xy ◦ Ψ k +1 k ( x, y, ξ k +1 )= π ξ k xy ◦ H − k ◦ Λ − k ( x, y, ξ k +1 )= π ξ k xy ◦ H − k ( σ k x, σ k y, σ k ξ k +1 )( ∗ ) = π ξ k xy (cid:0) φ − k ( σ k x, σ k y, σ k ξ k +1 ) , σ k y, ξ k ( φ − k , σ k y ) (cid:1) = ( φ − k ( σ k x, σ k y, σ k ξ k +1 ) , σ k y ) . In the above equation, ( ∗ ) is involved with the fact that H − k ◦ Λ − k ( graph( ξ k +1 )) ⊂ graph( ξ k ).Let us calculate H k, ξ ◦ d Ψ k +1 k, ξ ( x, y ). The second coordinate function of it is just σ k y . Thefirst coordinate function is as follows f k ◦ φ − k ( σ k x, σ k y, σ k ξ k +1 ) − ε k (cid:0) φ − k ( σ k x, σ k y, σ k ξ k +1 ) , σ k y, ξ k ( φ − k , σ k y ) (cid:1) f k ◦ φ − k ( σ k x, σ k y, σ k ξ k +1 ) − ε k ◦ H − k ( σ k x, σ k y, σ k ξ k +1 )= σ k x. Then H k, ξ ◦ d Ψ k +1 k, ξ ( x, y ) = ( σ k x, σ k y ). However, since H k, ξ ◦ (cid:0) H − k, ξ ( x, y ) ◦ Λ − k ( x, y ) (cid:1) =( σ k x, σ k y ), by the uniqueness of inverse map d Ψ k +1 k, ξ = H − k, ξ ◦ Λ − k . (cid:3) Lemma 4.1 enable us to define the renormalization of two dimensional C r H´enon-like mapsas an extension of the renormalization of analytic two dimensional H´enon-like maps.
Definition 4.1.
Let F : ( x, y ) ( f ( x ) − ε ( x, y ) , x ) be a C r H´enon-like map with r ≥ F is renormalizable, then RF , the renormalization of F is defined as follows RF = (Λ ◦ H ) ◦ F ◦ ( H − ◦ Λ − )where H ( x, y ) = ( f ( x ) − ε ( x, y ) , y ) and the linear scaling map Λ( x, y ) = ( sx, sy ) for theappropriate number s < − F is renormalizable n times, then the above definition can be applied to R k F for 1 ≤ k ≤ n successively. The two dimensional map d F n, ξ with the C r function ξ n is the same as R n F d, ξ by Lemma 4.1 and the above definition. Thus the map d F n, ξ is realized to be R n F d, ξ andcalled the n th renormalization of F d, ξ . Recall that O F is the sameas O F | Q which is the critical Cantor set restricted to the invariant surface Q . By the C r conjugation π ξxy between F | Q and F d, ξ , the ergodic invariant measure on O F d, ξ is defined asthe push forward measure µ on O F by the map π ξxy , that is, ( π ξxy ) ∗ ( µ ) ≡ µ d, ξ . In particular,it is defined as µ d, ξ (cid:0) π ξxy ( O F ∩ B n w ) (cid:1) = µ d, ξ (cid:0) π ξxy ( O F ) ∩ π ξxy ( B n w ) (cid:1) = 12 n . Since O F | Q is independent of any particular surface, so is π ξxy ( O F ). Then we express thismeasure to be µ d because the measure, µ d, ξ is also independent of ξ . Let us define the average Jacobian of F d, ξ b d = exp Z O F d log Jac F d, ξ dµ d . This average Jacobian is independent of the surface map ξ because every invariant surfacescontains the same critical Cantor set, O F d . Lemma 4.2.
Let F be in I (¯ ε ) which is a sufficiently small perturbation of toy model mapwith b ≫ b . Suppose that invariant C r surfaces Q n with ≤ r < ∞ under R n F contains Per R n F . Suppose also that Q n = graph ( ξ n ) where ξ n is C r map from I x × I y to I z . Let R n F d, ξ be π ξ n xy ◦ F n | Q n ◦ ( π ξ n xy ) − for each n ≥ . Then Jac R n F d, ξ = b n , d a ( x )(1 + O ( ρ n )) where b , d is the average Jacobian of F d, ξ and a ( x ) is the universal function of x for somepositive ρ < . roof. Lemma 2.1 could be applied for C r H´enon-like map for r ≥
2. Thus we obtainJac F n d, ξ = b n , d (1 + O ( ρ n )) . Moreover, the chain rule implies thatJac R n F d, ξ = b n , d Jac d Ψ n , ξ, tip ( x, y )Jac d Ψ n , ξ, tip ( R n F d, ξ ( x, y )) (1 + O ( ρ n )) . After letting the tip on every level move to the origin by appropriate linear map, the equation(4.4) implies thatJac d Ψ n , ξ = σ n, (cid:16) α n, · ∂ x (cid:0) x + S n ( x, y, ξ n ) (cid:1) + σ n, u n, · ∂ x ξ n (cid:17) . (4.5)Then in order to have the universal expression of Jacobian determinant, we need the asymp-totic of following maps ∂ x (cid:0) x + S n ( x, y, ξ n ) (cid:1) and σ n, α n, ∂ x ξ n By Lemma 2.3, x + S n ( x, y, ξ n ) = v ∗ ( x ) + a F, y + a F, y · ξ n + a F, ( ξ n ) + O ( ρ n ) . The above asymptotic has C convergence with the variable, x . Then ∂ x (cid:0) x + S n ( x, y, ξ n ) (cid:1) = v ′∗ ( x ) + a F, y · ∂ x ξ n + 2 a F, · ξ n · ∂ x ξ n + O ( ρ n ) . where v ∗ ( x ) is the universal function for some ρ ∈ (0 , k ∂ x ξ n k ≤ C σ n . Then ∂ x (cid:0) x + S n ( x, y, ξ n ) (cid:1) = v ′∗ ( x ) + O ( ρ n ) . (4.6)By the equation (3.6) in Proposition 3.3, σ n, α n, ∂ξ n ∂x = ∂ x ξ (¯ x, ¯ y ) · (cid:20) ∂ x S n ( x, y, ξ n ) + σ n, α n, u n, ∂ξ n ∂x (cid:21) Thus we obtain that σ n, α n, ∂ξ n ∂x = ∂ x ξ (¯ x, ¯ y )1 − u n, ∂ x ξ (¯ x, ¯ y ) · (cid:2) ∂ x S n ( x, y, ξ n ) (cid:3) where (¯ x, ¯ y ) ∈ Ψ n , v ( B ( R n F d, ξ )) for all big enough n . Thus (¯ x, ¯ y ) converges to the origin as n → ∞ exponentially fast by the equation (2.1).diam( d Ψ n , ξ ( B )) ≤ diam(Ψ n ( B )) ≤ Cσ n for some C >
0. Recall that the map, ∂ x ξ (¯ x, ¯ y ) converges to ∂ x ξ (0 ,
0) exponentially fast and u n, converges to u ∗ , super exponentially fast. Then σ n, α n, ∂ξ n ∂x = ∂ x ξ (0 , − u ∗ , ∂ x ξ (0 , v ′∗ ( x ) + O ( ρ n ) . (4.7)Let ( x ′ , y ′ ) = R n F d, ξ ( x, y ). ThenJac d Ψ n , ξ ( x, y )Jac d Ψ n , ξ ( x ′ , y ′ ) = 1 + ∂ x ( S n , ξ ( x, y )) + σ n, α n, u n, ∂ x ξ n ( x, y )1 + ∂ x ( S n , ξ ( x ′ , y ′ )) + σ n, α n, u n, ∂ x ξ n ( x ′ , y ′ )(4.8) here S n ( x, y, ξ n ) = S n , ξ ( x, y ). The translation does not affect Jacobian determinant andeach translation from tip to the origin converges to the map w τ ∞ exponentially fastwhere τ ∞ is the tip of two dimensional degenerate map, F ∗ ( x, y ) = ( f ∗ ( x ) , x ). Then by thesimilar calculation used in Universality Theorem in [dCLM], the equation (4.8) converges tothe following universal function exponentially fast.lim n →∞ Jac d Ψ n , ξ, tip ( x, y )Jac d Ψ n , ξ, tip ( x ′ , y ′ ) = v ′∗ ( x − π x ( τ ∞ )) + u ∗ , ∂ x ξ ( π xy ( τ F ))1 − u ∗ , ∂ x ξ ( π xy ( τ F )) v ′∗ ( x − π x ( τ ∞ )) v ′∗ ( f ∗ ( x ) − π y ( τ ∞ )) + u ∗ , ∂ x ξ ( π xy ( τ F ))1 − u ∗ , ∂ x ξ ( π xy ( τ F )) v ′∗ ( f ∗ ( x ) − π y ( τ ∞ ))= v ′∗ ( x − π x ( τ ∞ )) v ′∗ ( f ∗ ( x ) − π y ( τ ∞ )) ≡ a ( x ) . (cid:3) Theorem 4.3 (Universality of C r H´enon-like maps with C r conjugation for 2 ≤ r < ∞ ) . Let H´enon-like map F d, ξ be the C r map defined on (4.2) for ≤ r < ∞ . Suppose that F d, ξ is infinitely renormalizable. Then R n F d, ξ ( x, y ) = ( f n ( x ) − ( b d ) n a ( x ) y (1 + O ( ρ n )) , x )(4.9) where b d is the average Jacobian of F d, ξ and a ( x ) is the universal function for some <ρ < .Proof. By the smooth conjugation of two dimensional map and F n | Q n , we see that R n F d, ξ ( x, y ) = ( f n ( x ) − ε n ( x, y, ξ n ) , x )Denote ε n ( x, y, ξ n ) by ε n, ξ n ( x, y ). Then the Jacobian of R n F d, ξ is ∂ y ε n, ξ n ( x, y ). By Lemma4.2, ∂ y ε n, ξ n ( x, y ) = ( b d ) n a ( x )(1 + O ( ρ n )). Then ε n, ξ n ( x, y ) = ( b d ) n a ( x ) y (1 + O ( ρ n )) + U n ( x ) . The map U n ( x ) which depends only on the variable x can be incorporated to f n ( x ). (cid:3) Recall that the conjugation between R n F d, ξ, tip and (cid:0) R k F d, ξ (cid:1) n − k is d Ψ nk, ξ . Recall also that σ n, k = ( − σ ) n − k (1 + O ( ρ k )) and α n, k = σ n − k ) (1 + O ( ρ k )). Theorem 4.4.
Let R k F ∈ I (¯ ε k ) be the map which has invariant surfaces Q k ≡ graph ( ξ k ) tangent to E pu over the critical Cantor set. Then the coordinate change map, d Ψ nk, ξ isexpressed as follows (4.10) d Ψ nk, ξ ( x, y ) = (cid:0) α n, k ( x + d S nk ( x, y )) + σ n, k · d t n, k · y, σ n, k y (cid:1) where x + d S nk ( x, y ) has the asymptotic x + d S nk ( x, y ) = v ∗ ( x ) + a F, k y + O ( ρ n − k ) for | a F, k | = O ( ε k ) and ρ ∈ (0 , . roof. By Lemma 4.1, the coordinate change map, d Ψ nk, ξ is the composition of the inverseof horizontal diffeomorphisms with linear scaling maps as follows H − k, ξ ◦ Λ − k ◦ H − k +1 , ξ ◦ Λ − k +1 ◦ · · · ◦ H − n, ξ ◦ Λ − n . Then after reshuffling non-linear and linear parts separately by direct calculations and lettingthe tip move to the origin by appropriate translations on each levels, the coordinate changemap is of the form in (4.10). In order to estimate d S nk ( x, y ), the recursive formulas of thefirst and the second partial derivatives of d S nk ( x, y ) are required. However, the calculation inSection 7.2 in [dCLM] can be used because analyticity does not affect any recursive formulasof derivatives and furthermore it just requires C r map for r ≥
2. Hence, recursive formulaswith same estimations are applied to d S nk ( x, y ). Thus we have the following estimation x + d S nk ( x, y ) = v ∗ ( x ) + a F, k y + O ( ρ n − k )where | a F, k | = O ( ε k ). Alternatively, let us choose the equation (4.3) d Ψ nk, ξ ( x, y ) = (cid:0) α n, k ( x + S nk, ξ ( x, y )) + σ n, k t n, k y + σ n, k u n, k ( ξ n + R nk ( y )) , σ n, k y (cid:1) where S nk, ξ ( x, y ) = S nk ( x, y, ξ n ( x, y )). By Proposition 3.3, the map ξ n ( x, y ) = c y + η ( y ) + O ( ρ n )where the map η ( y ) is quadratic or higher order terms with k η k C ≤ C σ n − k for some C > | u n, k | ≤ C ¯ ε k and k R nk k C ≤ C σ n − k for some positive C and C . Recall that the constants, α n, k = σ n − k ) (1 + O ( ρ n )) and σ n, k = ( − σ ) n − k (1 + O ( ρ n )).Hence, we appropriately define each terms of d Ψ nk, ξ d S nk ( x, y ) = S nk, ξ ( x, y ) + σ n, k α n, k u n, k [ ξ n ( x, y ) − c y + R nk ( y )] d t n, k = t n, k + u n, k c which are as desired. (cid:3) Let d t k +1 , k be d t k for simplicity. Similarly, denote α k +1 , k and σ k +1 , k to be α k and σ k respectively. The following corollary and the proof is the same as those of analytic maps in[dCLM]. For the sake of completeness, the proof is written below. Corollary 4.5.
Let F d, ξ be the infinitely renormalizable C r H´enon-like map with singleinvariant surfaces tangent to E pu over the critical Cantor set. Let d S nk be the coordinatechange map between R k F d, ξ and R n F d, ξ defined in Theorem 4.4. Then t k ≍ − ( b d ) k for every k ∈ N .Proof. Compare the derivative of Λ k ◦ H k, ξ at the tip and the derivative of (cid:0) d Ψ k +1 k, ξ (cid:1) − atthe origin as follows (cid:18) − d t k (cid:19) = (cid:18) α k σ k (cid:19) (cid:18) • − s k · ∂ y ε n, ξ n ( τ k )0 1 (cid:19) Thus d t k = α k · s k · ∂ y ε n, ξ n ( τ k ) where s k ≍ −
1. Since by Lemma 4.2, − ∂ y ε n, ξ n ( τ k ) ≍ − Jac R n F d, ξ ≍ − ( b d ) k . hen d t k ≍ − ( b d ) k for each k ∈ N . (cid:3) Q n .Lemma 4.6. Let F d, ξ be a C r infinitely renormalizable two dimensional H´enon-like mapfor ≤ r < ∞ . Then F d, ξ has no continuous invariant line field over the critical Cantorset. Especially, every invariant line fields are discontinuous at the tip.Proof. Universality Theorem 4.3 and the estimation of scaling map, Ψ nk in Theorem 4.4imply the universal expression of H´enon-like maps and of horizontal map similar to those ofanalytic ones. Then the proof discontinuity of invariant line field is essentially the same asthe proof of Theorem 9.7 in [dCLM]. (cid:3) Theorem 4.7.
Let F ∈ I (¯ ε ) be a sufficiently small perturbation of toy model map with b ≪ b . Let Q be an invariant surface under F which is tangent to the continuous invariantfield, say E , over O F . Then any invariant line field in E over O F is discontinuous at thetip.Proof. The proof is the same as that of Theorem 7.8 in [Nam1] with the above Lemma 4.6. (cid:3)
The geometric properties of critical Cantor set — non existence of continuous invariant linefield and unbounded geometry of critical Cantor set — are showed in the invariant surface.These negative results on the invariant surfaces are also valid on three dimensional analyticH´enon-like maps in no time.
5. Density of conjugated maps in C r H´enon-like maps
The renormalization for analytic H´enon-like map is extended to C r H´enon-like maps byinvariant C r single surfaces of analytic three dimensional map. We would show that the set of C r H´enon-like maps from invariant surfaces is open and dense in C r infinitely renormalizableH´enon-like maps in the parameter space of average Jacobian for any given 2 ≤ r < ∞ (Theorem 5.5). Lemma 5.1.
Let F mod ∈ I (¯ ε ) be the infinitely renormalizable toy model three dimensionalH´enon-like map. Assume that b ≪ b and there exist invariant C r single surfaces which aretangent to E pu over the critical Cantor set, O F mod and these surfaces is the graph of C r mapfrom I x × I y to I z . Let a sufficiently small perturbation of F mod with parameter t as follows (5.1) F t ( x, y, z ) = ( f ( x ) − ε ( x, y ) + tz, x, δ ( x, y, z )) for small enough | t | . Then F t has also invariant C r single surfaces tangent to E pu over itscritical Cantor set.Proof. The existence of invariant cone fields of DF mod and a small perturbation of DF mod by Lemma 7.3 and Lemma 7.4 in [Nam1]. Existence of single invariant surfaces for F mod isdue to Section 3. (cid:3) enote an invariant single surface of F t by graph ( ξ t ) where ξ is the C r map from I x × I y to I z . Thus the C r H´enon-like map from invariant surface, π xy ◦ F t | graph ( ξ t ) ≡ F d,t is defines asfollows(5.2) F d,t ( x, y ) = ( f ( x ) − ε ( x, y ) + tξ t ( x, y ) , x ) . Let F d be a C r H´enon-like map. The unimodal part f of the following map F d ( x, y ) = ( f ( x ) − ε ( r ) ( x, y ) , x )can be approximated arbitrary closely by analytic maps in C r topology. Then we mayassume that f is analytic and ε ( r ) ( x, y ) is C r . Moreover, two variable C r map can bealso approximated by analytic maps, for instance, multivariate Bernstein polynomials in C r topology. See [ ? ]. Any analytic H´enon-like maps in I (¯ ε ) can be approximated by maps in(5.2). Lemma 5.2.
The set of two dimensional C r H´enon-like map in (5.2) is a dense subset oftwo dimensional C r H´enon-like map in I (¯ ε ) for ≤ r < ∞ . Lemma 5.3.
The critical Cantor set, O F d of two dimensional C r H´enon-like map movescontinuously as F d in infinitely H´enon-like maps.Proof. By construction of the critical Cantor set, for a given word w n ∈ W n , the uniqueperiodic point w n with period 2 n of the region B n w n is C r by Implicit Function Theorem.Each point w ∈ O F is the limit of w n as n → ∞ for the given word w ∈ W ∞ which contains w n as a finite subaddress of w for every n ∈ N . Since two dimensional box, B n w n ( F d ) is π xy (cid:0) B n w n ( F ) (cid:1) of three dimensional map F , the uniform convergence of three dimensionalboxes as n → ∞ implies that of two dimensional ones. Then the critical Cantor set movescontinuously as F d . (cid:3) Recall the maps in (5.1) and (5.2) for | t | < r where r is sufficiently small such that(1) For every | t | < r , there exist single invariant surfaces tangent to E pu over the criticalCantor set as the graph from I x × I y to I z .(2) Jac F d, t is positive on ( − r, r ) × B . Corollary 5.4.
The average Jacobian b d,t ≡ b ( F d,t ) for | t | < r moves continuously on t forsufficiently small r > .Proof. The average Jacobian of F d,t is defined explicitly as follows b d,t = exp Z O t log(Jac F d,t ) dµ t = exp Z O t log (cid:18) ∂ε∂y + t ∂ξ t ∂y (cid:19) dµ t where µ t is the unique F d,t -invariant probability measure on each critical Cantor set O t ≡O F d,t . By Lemma 5.3, O t moves continuously. Then the integral is also continuous with t . (cid:3) Remark 5.1.
If the H´enon-like map F t in I (¯ ε ) is analytic and it is extendible holomorphi-cally, then the critical Cantor set moves holomorphically with t by Lemma 5.6 in [dCLM].Define that a C r H´enon-like map, F d is embedded in analytic three dimensional H´enon-likemap in I (¯ ε ) only if F d is conjugated by a C r map to F | Q where Q is a C r invariant surfacetangent to E pu over the critical Cantor set. heorem 5.5. Let F d,b be an element of parametrized C r H´enon-like maps for b ∈ [0 , in I (¯ ε ) where b is the average Jacobian of F d,b for ≤ r < ∞ . Then for some ¯ b > , the set ofparameter values, an interval [0 , ¯ b ] on which the map F d,b is embedded in three dimensionalanalytic H´enon-like maps in I (¯ ε ) contains a dense open subset.Proof. The density of the set of conjugated map from invariant surfaces is due to Lemma5.2. The openness is involves with Lemma 5.1 and Corollary 5.4. (cid:3)
Notes.
The definition of renormalizability of C r H´enon-like map is just extension of thatof analytic H´enon-like maps. However, hyperbolicity of renormalization operator for C r H´enon-like maps at the fixed point is not proved yet. In previous sections, using singleinvariant surfaces in three dimensional analytic H´enon-like maps, we construct C r conjuga-tion between maps in single invariant surfaces and two dimensional maps. It defines infiniterenormalization of C r H´enon-like maps in this class. Moreover, direct calculations of asymp-totics in [dCLM] to this article, the smoothness of invariant surfaces seems to be sufficientfor r = 2. However, the hyperbolicity of period doubling operator of one dimensional mapsrequires C ǫ maps with arbitrary small but positive number ǫ in [Dav] and moreover, H´enonrenormalization contains that of one dimensional maps as degenerate maps. On the otherhand, since invariant surfaces are constructed by invariant cone fields, these surfaces cannotbe C ∞ or analytic. Existence of any single invariant C ∞ or non-flat analytic surfaces tangentto E pu over the critical Cantor set is not known yet.
6. Unbounded geometry on the Cantor set
Let the subset of critical Cantor set on each pieces be O w ≡ B n w ∩O where w ∈ W n = { v, c } n is the word of length n . We may assume that every box region is (path) connected and simplyconnected. Suppose that each topological region, B n w compactly contains O w and moreover B n w is disjoint from O \ O w for every word w . Assume also that every B n w is forward invariantunder F n for all word w and every n ∈ N . Bounded geometry is defined for given box regionswhich satisfy the followingdist min ( B n +1 w v , B n +1 w c ) ≍ diam( B n +1 w ν ) for ν ∈ { v, c } diam( B n w ) ≍ diam( B n +1 w ν ) for ν ∈ { v, c } for all w ∈ W n and for all n ≥
0. The proof of unbounded geometry of critical Cantor setrequires to compare the diameter of boxes and the minimal distance of two adjacent boxes.In order to compare these quantities, we would use the maps, Ψ nk , R k F and Ψ k with the twopoints w = ( x , y , z ) and w = ( x , y , z ) in O R n F . Let us each successive image of w j under Ψ nk , R k F and Ψ k be ˙ w j , ¨ w j and ... w j for j = 1 , w j ✤ Ψ nk / / ˙ w j ✤ R k F / / ¨ w j ✤ Ψ k / / ... w j Let the coordinates of the point, ˙ w j be ( ˙ x j , ˙ y j , ˙ z j ). The points ¨ w j and ... w j also have the similarcoordinate expressions. Let S and S be the (path) connected set on R . If π x ( S ) ∩ π x ( S )contains at least two points, then this intersection is called the x − axis overlap or horizontal verlap of S and S . Moreover, we say S overlaps S on the x − axis or horizontally.Let F d be an infinitely renormalizable two dimensional H´enon-like map and b be the averageJacobian of F d . Then unbounded geometry of the critical Cantor set depends on Universalitytheorem and the asymptotic of the tilt, − t k ≍ b k but it does not depend on the analyticityof the map. The infinitely renormalizable C r H´enon-like maps defined by invariant surfaceshas Universality by Theorem 4.3 and the asymptotic of the tilt − d t k ≍ b k by Corollary 4.5.Then unbounded geometry of the critical Cantor set in [dCLM] and [HLM] is applicable to C r H´enon-like map defined by invariant surfaces.Observe that dist min ( S , S ) ≤ dist( w , w ) for all w ∈ S and w ∈ S and diam( S ) ≥ dist( w, w ′ ) for all w, w ′ ∈ S . Lemma 6.1.
Let F d be an infinitely renormalizable C r H´enon-like maps defined by invariantsurfaces which is tangent to E pu over O F . Suppose that two dimensional box d B n − k v v ( R k F d ) overlaps d B n − k v c ( R k F d ) on the x − axis where v = v n − k − . Then for all sufficiently large k and n with k < n , we have the following estimate dist min ( d B n w v , d B n w c ) ≤ C b k σ k σ n − k diam( d B n w v ) ≥ C σ n − k ) σ k where w = v k cv n − k − ∈ W n for some positive constants C and C .Proof. The proof is the same as the analytic case because unbounded geometry dependsonly on the universality theorem and asymptotic of the tilt − d t k ≍ b k . Then we can adaptthe proof for analytic maps in [HLM]. For the sake of completeness, we describe the proofbelow. Choose two points w = ( x , y ) and w = ( x , y ) in d B v ( R n F d ) ∩ O R n F d and d B c ( R n F d ) ∩ O R n F d respectively in order to estimate the minimal distance between twoboxes.The expression of d Ψ nk, ξ in Theorem 4.4 and overlapping assumption implies the coordinatesof the points, ( ˙ x j , ˙ y j ), (¨ x j , ¨ y j ) and (... x j , ... y j ) for j = 1 , x − ˙ x = 0 and ˙ y − ˙ y = σ n, k ( y − y )The special form of H´enon-like map, R k F d and coordinate change map, d Ψ nk, ξ imply that(6.1) ... y − ... y = σ k, (¨ y − ¨ y ) = σ k, ( ˙ x − ˙ x ) = 0By mean value theorem and the fact that (¨ x j , ¨ y j ) = R k F d ( ˙ x j , ˙ y j ) for j = 1 , x − ¨ x = f k ( ˙ x ) − ε k ( ˙ x , ˙ y ) − (cid:2) f k ( ˙ x ) − ε k ( ˙ x , ˙ y ) (cid:3) = − ε k ( ˙ x , ˙ y ) + ε k ( ˙ x , ˙ y )= − ∂ y ε k ( η ) · ( ˙ y − ˙ y )= − ∂ y ε k ( η ) · σ n, k ( y − y )where η is some point in the line segment between ( ˙ x , ˙ y ) and ( ˙ x , ˙ y ). Thus by Theorem4.4 and the equation (6.1), we obtain that... x − ... x = π x ◦ d Ψ k , ξ (¨ x , ¨ y ) − π x ◦ d Ψ k , ξ (¨ x , ¨ y )= α k, (cid:2) (¨ x + d S k (¨ x , ¨ y )) − (¨ x + d S k (¨ x , ¨ y )) (cid:3) + σ k, (cid:2) d t k, · (¨ y − ¨ y ) (cid:3) α k, (cid:2) v ′∗ (¯ x ) + O (¯ ε + ρ k ) (cid:3) (¨ x − ¨ x ) . (6.2)Then by the fact that ∂ y ε k ≍ b k where b is the average Jacobian of F d , we can estimatethe minimal distancedist min ( d B n w v , d B n w c ) ≤ (cid:12)(cid:12) ... x − ... x (cid:12)(cid:12) + (cid:12)(cid:12) ... y − ... y (cid:12)(cid:12) ≤ σ k (cid:12)(cid:12) ¨ x − ¨ x (cid:12)(cid:12) · v ′∗ (¯ x )(1 + O ( ρ k )) ≤ C b k σ k σ n − k where v ∗ ( x ) is the positive universal function for some C >
0. Take any two differentpoints, ( x , y ) and ( x , y ) in the box d B v ( R n F d ) ∩ O R n F d to estimate the diameter of d B n w v . Thus the special forms of R k F d , d Ψ nk, ξ and the equation (6.2) implies thatdiam( d B n w v ) ≥ (cid:12)(cid:12) ... y − ... y (cid:12)(cid:12) = σ k, · (¨ y − ¨ y )= (cid:12)(cid:12) σ k, · ( ˙ x − ˙ x ) (cid:12)(cid:12) = (cid:12)(cid:12) σ k, (cid:2) π x ◦ d Ψ nk, ξ ( ˙ x , ˙ y ) − π x ◦ d Ψ nk, ξ ( ˙ x , ˙ y ) (cid:3)(cid:12)(cid:12) = (cid:12)(cid:12) σ k, α n, k (cid:2) v ′∗ ( e x )( x − x ) + O (¯ ε k + ρ n − k ) (cid:3)(cid:12)(cid:12) ≥ C σ n − k ) σ k where v ∗ ( x ) is the positive universal function for some C > (cid:3) Unbounded geometry on the critical Cantor set holds if we choose n > k such that b k ≍ σ n − k for every sufficiently large k ∈ N . This is true on the parameter space of average Jacobian, b almost everywhere with respect to Lebesgue measure. Theorem 6.2 ([HLM]) . The given any < A < A , < σ < and any p ≥ , the set ofparameters b ∈ [0 , for which there are infinitely many < k < n satisfying A < b p k σ n − k < A is a dense G δ set with full Lebesgue measure. Recall that toy model map has universal numbers — the average Jacobian, b mod , the averageJacobian of two dimensional map, π xy ◦ F mod , b , mod and the ratio of these two numbers, b , mod ≡ b mod /b , mod . If b , mod ≪ b , mod , then each of these numbers can be generalizedto a sufficiently small perturbation of toy model map. In particular, the number b , mod isgeneralized to the average Jacobian of F d, ξ , say b , by Theorem 4.3. Another number b isjust defined as the ratio, b F /b . Then unbounded geometry of Cantor attractor of F | Q oninvariant surface is extended to those of same Cantor set for three dimensional map, F . Theorem 6.3.
Let F be three dimensional H´enon-like map in I (¯ ε ) which is a small pertur-bation of toy model map with b ≪ b . Then for each sufficiently small fixed positive number b , the parametrized H´enon-like map F b for b ∈ [ b ◦ , b • ] where b is the average Jacobian oftwo dimensional C r H´enon-like map, F d, ξ for b ≪ b ◦ < b • . Then there exists G δ subset S with full Lebesgue measure of [ b ◦ , b • ] such that the critical Cantor set, O F b has unboundedgeometry. roof. The box on the invariant surface Q , Q B n w is defined as the image of the box, d B n w of two dimensional H´enon-like map under the graph map ( x, y ) ( x, y, ξ ) for every n ∈ N and every word w ∈ W n . By Proposition 3.3, the minimal distance between two boxes onthe surface and that between two boxes on xy − plane with the same word are comparablewith each other for all words. Furthermore, there exist three dimensional boxes, B n w suchthat Q ∩ B n w ⊃ Q B n w for every word w because Q is an invariant surface which compactlycontains the critical Cantor set. Then by Lemma 6.1, we havedist min ( d B n w v , d B n w c ) ≍ dist min ( Q B n w v , Q B n w c )dist min ( B n w v , B n w c ) ≤ dist min ( Q B n w v , Q B n w c ) ≤ C b k σ k σ n − k for the word w = v n − k − cv k and moreover,diam( d B n w v ) ≍ diam( Q B n w v )diam( B n w v ) ≥ diam( Q B n w v ) ≥ C σ n − k ) σ k for the word w = v n − k − cv k and for positive constants C and C independent of w and n .One box overlaps its adjacent box on the x − axis in three dimension if and only if so doesin two dimension because there exists an invariant surface as the graph from the plane to z − axis. Then b k ≍ σ n − k for all sufficiently large k in the G δ subset which has full measure in the parameter space[ b ◦ , b • ] by Theorem 6.2. Hence, dist min ( B n w v , B n w c ) ≤ Cσ k diam( B n w v ) for some C > (cid:3)
Appendix APeriodic points and critical Cantor set
Let us take a word, w = ( w w w . . . w n . . . ) as an address. The word of the first n concate-nations, w n = ( w w w . . . w n ) is defined as the subaddress of the word w . Lemma A.1.
Let F be the H´enon-like map in I (¯ ε ) with sufficiently small positive ¯ ε . Thenthe set of accumulation points of Per F is the critical Cantor set O F .Proof. The region B n w n ≡ Ψ n , w n ( B ( R n F )) contains the periodic point, Ψ n , w n ( β ( R n F )) withperiod 2 n . By construction of the critical Cantor set, every point O F , say w is as follows { w } = \ n ≥ B n w n for the corresponding words, w n are the subaddresses of w ∈ W ∞ ≡ { v, c } ∞ for all n ∈ N .Since diam( B n w n ) ≤ Cσ n for all word w n and for all n ∈ N , every points in O F is containedin the set of accumulation points of Per F . For the reverse inclusion, recall the following facts— For any H´enon-like map F ∈ I (¯ ε ), the region B v ∪ B c contains all periodic points of F . The number of periodic points with any given single period, 2 n is always finite.— The region B N w N compactly contains B n w n where n > N and the word w N is a subad-dress of the word w n .Take any point, say w , in the set of accumulation point of Per F . We may assume that thereexists a sequence of periodic points, { q n k } which converge to w as k → ∞ where the periodof each q n k is 2 n k and n k is increasing and n k → ∞ as k → ∞ . Observe that the periodicpoint q n k is Ψ n k , w nk ( β ( R n k F )) for some address w n k . We claim that there exists a periodicpoint, q n k of which region B n k w nk contains w . If not, then Orb F (cid:0) B n k w nk (cid:1) is disjoint from w .However, every periodic points of which period is greater than q n k are in Orb F (cid:0) B n k w nk (cid:1) . Itcontradicts the convergence of periodic points to w . Then we may assume that the region B n k w nk contains w and the sequence Q ≡ { q n m | m > k } . Denote the region B n k w nk by B k for each k . Since every points q n m ∈ Q are a periodic points under R n k F in B ( R n k ), eachregion, B m for m > k is compactly contained in B k and moreover, B m converges to w as m → ∞ . Each region B m has its own address and the address converges to a word w ∈ W ∞ as m → ∞ . This construction implies that the sequence of B m converges to a point withthe address w in the critical Cantor set. Hence, the accumulation point, w is contained in O F . (cid:3) Lemma A.2.
Let F be the three dimensional H´enon-like map in I (¯ ε ) for small enough ¯ ε > . Then W s ( w ) ∩ Per F = { w } for each w ∈ Per F .Proof. The fact that F ∈ I (¯ ε ) implies the existence of the critical Cantor set. Note that anygiven periodic points of F has period, 2 k for some k ∈ N . For any two periodic points, p and q , we may assume that these points are fixed points under F k for large enough k ∈ N .If both p and q are in any same stable manifold, then dist( F n ( p ) , F n ( q )) → n → ∞ .However, dist( F km ( p ) , F km ( q )) is fixed for every m ∈ N . Thus p is the same as q .Any point w in the critical Cantor set has its address of which length is infinity and thesequence of boxes containing w with the address which is the first finite concatenations ofthe address of w . Thus each point in the critical Cantor set is the limit of box domain,that is, { w } = \ N ≥ B N w N where w N is the subaddress of w for all N ∈ N . Since B N w N areforward invariant under F N +1 , for any given periodic point, say q both the box domain B N w N and q are invariant under F N +1 for all big enough N . Moreover, due to the fact thatdiam( B N w N ) = Cσ n for some C >
0, we may assume that B N w N is disjoint from { q } . Thendist( F N m ( q ) , F N m ( w )) ≥ c for all m ≥ c >
0. Then W s ( w ) for each w ∈ O F does not contain any other point in Per F . Similarly, W s ( β ) for each β ∈ Per F doesnot contain any other point in Per F .There exist two disjoint neighborhoods B n w n and B n w ′ n of w ∈ O F and w ′ ∈ O F respectivelyfor all sufficiently large n . Both B n w n and B n w ′ n are forward invariant under F n +1 . We mayassume that B n w n and B n w ′ n are disjoint and the minimal distance, dist min ( B n w n , B n w ′ n ) ≥ ε > n . Suppose that both w and w ′ are contained in the same stable manifold, W s ( w ) or W s ( w ′ ). However, dist W ( w, w ′ ) ≥ ε for all n ∈ N . It contradicts the uniformcontraction along strong stable manifold. Hence, W s ( w ) ∩ Per F = { w } for each w ∈ Per F . (cid:3) eferences [BC] Christian Bonatti and Sylvain Crovisier. Center manifolds for partially hyperbolic set without strongunstable connections, 2014.[CEK] Pierre Coullet, Jean-Pierre Eckmann, and Hans Koch. Period doubling bifurcations for families ofmaps on R n . Journal of Statistical Physics,
Journal of Statistical Physics, C ǫ mappings. Communications in Mathematical Physics, C r unimodal mappings. Annals of Mathematics,
Nonlinearity,
Renormalization of three dimensional H´enon-like map.
PhD thesis, Stony BrookUniversity, December 2011.[Nam1] Young Woo Nam, Renormalization of three dimensional H´enon-map I : Universality and reductionof ambient space, preprint (2014), available at http://arxiv.org/abs/1408.4289 , August 2014.[Nam2] Young Woo Nam, Invariant space under H´enon renormalization : Intrinsic geometry of Cantorattractor, preprint (2014), available at http://arxiv.org/abs/1408.4619 , August 2014.[New] Sheldon Newhouse. Cone fields, domination and hyperbolicity. In
Modern dynamical systems andapplication, pages 419-432. Cambridge University Press, 2004.