Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Stable Sets

It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\fR}^2$. A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a computer-assisted proof of existence of a"universal"area-preserving map $F_*$ -- a map with orbits of all binary periods $2^k, k \in \fN$. In this paper, we consider {\it infinitely renormalizable} maps -- maps on the renormalization stable manifold in some neighborhood of $F_*$ -- and study their dynamics. For all such infinitely renormalizable maps in a neighborhood of the fixed point $F_*$ we prove the existence of a"stable"invariant set $\cC^\infty_F$ such that the maximal Lyapunov exponent of $F \arrowvert_{\cC^\infty_F}$ is zero, and whose Hausdorff dimension satisfies $${\rm dim}_H(\cC_F^{\infty}) \le 0.5324.$$ We also show that there exists a submanifold, $\bW_\omega$, of finite codimension in the renormalization local stable manifold, such that for all $F\in\bW_\omega$ the set $\cC^\infty_F$ is"weakly rigid": the dynamics of any two maps in this submanifold, restricted to the stable set $\cC^\infty_F$, is conjugated by a bi-Lipschitz transformation that preserves the Hausdorff dimension.

Abstract. It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of R 2 . A renormalization approach has been used in (Eckmann et al 1982) and (Eckmann et al 1984) in a computer-assisted proof of existence of a "universal" areapreserving map F * -a map with orbits of all binary periods 2 k , k ∈ N. In this paper, we consider infinitely renormalizable maps -maps on the renormalization stable manifold in some neighborhood of F * -and study their dynamics.
For all such infinitely renormalizable maps in a neighborhood of the fixed point F * we prove the existence of a "stable" invariant Cantor set C ∞ F such that the Lyapunov exponents of F | C ∞ F are zero, and whose Hausdorff dimension satisfies dim H (C ∞ F ) < 0.5324. We also show that there exists a submanifold, Wω, of finite codimension in the renormalization local stable manifold, such that for all F ∈ Wω the set C ∞ F is "weakly rigid": the dynamics of any two maps in this submanifold, restricted to the stable set C ∞ F , is conjugated by a bi-Lipschitz transformation that preserves the Hausdorff dimension.
To prove universality one usually introduces a renormalization operator on a functional space, and demonstrates that this operator has a hyperbolic fixed point. The renormalization approach to universality has been very successful in onedimensional dynamics, and has led to explanation of universality in unimodal maps (Epstein 1986, Epstein 1989, Lyubich 1999, critical circle maps (de Faria 1992, de Faria 1999, Yampolsky 2002, Yampolsky 2003 and holomorphic maps with a Siegel disk (McMullen 1998, Yampolsky 2007, Gaidashev and Yampolsky 2007. There is, however, at present no deep understanding of universality in conservative systems, other than in the "trivial" case of the universality for systems "near integrability" (Koch 2002, Koch 2004, Gaidashev 2005, Kocić 2005, Khanin et al 2007. It is worth noting that universality in conservative systems seems to be completely different from that in one-dimensional and dissipative maps. As it has been shown in (Collet et al 1980, de Carvalho et al 2005, Lyubich and Martens 2008, the case of very dissipative systems is largely reducible to the one-dimensional Feigenbaum-Coullet-Tresser universality. For families of area-preserving maps a universal infinite period-doubling cascade was observed by several authors in the early 80's (Derrida and Pomeau 1980, Helleman 1980, Benettin et al 1980, Bountis 1981, Collet et al 1981. The existence of a hyperbolic fixed point for the period-doubling renormalization operator has been proved with computer-assistance in (Eckmann et al 1984).
In (Gaidashev and Johnson 2009) we used the method of covering relations (see, e.g. (Zgliczyński 1997, Zgliczyński and Gidea 2004, Kokubu et al 2007, Zgliczyński 2009, CAPD 2009)) in rigorous computations to construct hyperbolic sets for all maps in some neighborhood of the fixed point of the renormalization operator. The Hausdorff dimension of these hyperbolic sets has been estimated with the help of the Duarte Distortion Theorem (see, e.g. (Duarte 2000)) which enables one to use the distortion of a Cantor set to find bounds on its dimension.
In this paper, we prove that infinitely renormalizable maps in a neighborhood of existence of the hyperbolic sets also admit a "stable" set. This set is a bounded invariant set, such that the maximal Lyapunov exponent at any point of this set is zero. Together with our result from (Gaidashev and Johnson 2009), this demonstrates that for all reversible area-preserving infinitely renormalizable maps in some neighborhood of the renormalization fixed point, there are coexisting hyperbolic and stable sets.
We also address the issues of rigidity of the stable set and invariance of its Hausdorff dimension. Similar issues have been investigated in (de Carvalho et al 2005) for attractors of very dissipative two-dimensional maps, where it has been shown that the regularity of conjugacy of attractors for two infinitely renormalizable maps F andF has a definite upper bound where Jac(F) is the "average" Jacobian of the map F . The authors of ( de Carvalho et al 2005) put forward two questions: 1) whether the Hausdorff dimension of the attractor of an infinitely renormalizable map depends only on its average Jacobian, and 2) how regular is the conjugacy when Jac(F ) = Jac(F ). In this regard, we obtain a partial result along similar lines in the "extreme" case of area-preserving maps (constant Jacobian equal to one): we prove that there exists a subset of locally infinitely renormalizable maps such the actions of any two maps from this subset on their stable sets are conjugate by a bi-Lipschitz map which preserves the Hausdorff dimension. We can not make a definite conclusion about whether this subset is equal to the whole set of locally infinitely renormalizable maps, or strictly smaller, because a sharp bound on the convergence rate of renormalizations of infinitely renormalizable maps is not known to date.
Finally, we provide an upper bound on the Hausdorff dimension of the stable set for all infinitely renormalizable maps.

Renormalization for area-preserving reversible maps
An "area-preserving map" will mean an exact symplectic diffeomorphism of a subset of R 2 onto its image.
Recall, that an area-preserving map can be uniquely specified by its generating function S: Furthermore, we will assume that F is reversible, that is For such maps it follows from (2) that .
It is this "little" s that will be referred to below as "the generating function". If the equation −s(y, x) = u has a unique differentiable solution y = y(x, u), then the derivative of such a map F is given by the following formula: The period-doubling phenomenon can be illustrated with the area-preserving Hénon family (cf. (Bountis 1981)) : Figure 1. The geometry of the period doubling. p k is the further elliptic point that has bifurcated from the hyperbolic point p ′ k .
Maps H a have a fixed point ((−1 + √ 1 + a)/a, (−1 + √ 1 + a)/a) which is stable for −1 < a < 3. When a 1 = 3 this fixed point becomes unstable, at the same time an orbit of period two is born with H a (x ± , x ∓ ) = (x ∓ , x ± ), x ± = (1 ± √ a − 3)/a. This orbit, in turn, becomes unstable at a 2 = 4, giving birth to a period 4 stable orbit. Generally, there exists a sequence of parameter values a k , at which the orbit of period 2 k−1 turns unstable, while at the same time a stable orbit of period 2 k is born. The parameter values a k accumulate on some a ∞ . The crucial observation is that the accumulation rate is universal for a large class of families, not necessarily Hénon. Furthermore, the 2 k periodic orbits scale asymptotically with two scaling parameters (7) λ = −0.249 . . . , µ = 0.061 . . .
To explain how orbits scale with λ and µ we will follow (Bountis 1981). Consider an interval (a k , a k+1 ) of parameter values in a "typical" family F a . For any value α ∈ (a k , a k+1 ) the map F α possesses a stable periodic orbit of period 2 k . We fix some α k within the interval (a k , a k+1 ) in some consistent way; for instance, by requiring that the restriction of F 2 k α k to a neighborhood of a stable periodic point in the 2 k -periodic orbit is conjugate, via a diffeomorphism H k , to a rotation with some fixed rotation number r. Let p ′ k be some unstable periodic point in the 2 k−1periodic orbit, and let p k be the further of the two stable 2 k -periodic points that bifurcated from p ′ k . Denote with d k = |p ′ k − p k |, the distance between p k and p ′ k . The new elliptic point p k is surrounded by invariant ellipses; let c k be the distance between p k and p ′ k in the direction of the minor semi-axis of an invariant ellipse surrounding p k , see Figure 1. Then, where ρ k is the ratio of the smaller and larger eigenvalues of DH k (p k ). This universality can be explained rigorously if one shows that the renormalization operator where Λ F is some F -dependent coordinate transformation, has a fixed point, and the derivative of this operator is hyperbolic at this fixed point. It has been argued in (Collet et al 1981) that Λ F is a diagonal linear transformation. Furthermore, such Λ F has been used in (Eckmann et al 1982) and (Eckmann et al 1984) in a computer assisted proof of existence of a reversible renormalization fixed point F * and hyperbolicity of the operator R.
We will now derive an equation for the generating function of the renormalized .
If the solution of (10) is unique, then z(x, y) = z(y, x), and it follows from (9) that the generating function of the renormalized F is given by (11)s(x, y) = µ −1 s(z(x, y), λy).
Definition 2.2. The Banach space of functions s(x, y) = ∞ i,j=0 c ij x i y j , analytic on a bi-disk |x − 0.5| < ρ, |y − 0.5| < ρ, for which the norm |c ij |ρ i+j is finite, will be referred to as A(ρ).
As we have already mentioned, the following has been proved with the help of a computer in (Eckmann et al 1982) and (Eckmann et al 1984): Theorem 1. There exist a polynomial s a ∈ A s (ρ) and a ball B r (s a ) ⊂ A s (ρ), r = 6.0 × 10 −7 , ρ = 1.6, such that the operator R EKW is well-defined and analytic on B r (s a ).
Furthermore, its derivative DR EKW | Br(sa) is a compact linear operator, and has exactly two eigenvalues δ 1 and δ 2 of modulus larger than 1, while Finally, there is an s * ∈ B r (s a ) such that The scalings λ * and µ * corresponding to the fixed point s * satisfy Remark 2.3. The bound (15) is not sharp. In fact, a (lower) bound on the largest eigenvalue of DR EKW (s * ), restricted to the tangent space of the stable manifold, is not known.
The interval enclosures of λ * and µ * will be denoted The corresponding interval enclosure for the linear map The bound on the fixed point generating function s * will be called s * : while the bound on the renormalization fixed point F * will be referred to as F * : the third iterate of this bound will be referred to as G * . It follows from Theorem 1, that there exists a codimension 2 local stable manifold W s loc (s * ) ⊂ B r (s a ). Definition 2.4. A reversible map F of the form (4) such that s ∈ W s loc (s * ) is called infinitely renormalizable. The set of all reversible infinitely renormalizable maps is denoted by W.
Definition 2.5. The set of reversible maps F of the form (4) with s ∈ B ̺ (s * ) will be referred to as F * (̺). Denote,

Hyperbolic sets for maps in F *
In this Section we will recall some of our results from the satellite paper (Gaidashev and Johnson 2009).
We will start by introducing several classical definitions which will be helpful in understanding our Theorem 2 below. An invariant set C is called hyperbolic for the map F if there is a Riemannian metric on a neighborhood U of C, and β < 1 < δ, such that for any p ∈ C and n ∈ N the tangent space T F n (p) U admits a decomposition in two equivariant subspaces: , on which the sequence of differentials is hyperbolic: Definition 3.3. Let X be a metric space, A ⊂ X and d ∈ [0, ∞). Suppose that B = {B i } is some cover of A whose elements are open sets. We will denote The d-dimensional Hausdorff content of A is defined as In (Gaidashev and Johnson 2009) we have demonstrated that all maps in a neighborhood of the fixed point admit a hyperbolic set in their domain of analyticity.
Theorem 2. The following holds for all F ∈ F * . i) There exist connected open sets D ⊂ C 2 and D 3 ⊂ C 2 such that the maps F and G ≡ F • F • F are analytic on D and D 3 , respectively.
ii) The map F possesses a hyperbolic fixed point p 0 = p 0 (F ) ∈ D, such that 1) P x p 0 ∈ (0.57761843, 0.57761989), and P u p 0 = 0, where P x,u are projections on the x and u coordinates; 2) DF (p 0 ) has two negative eigenvalues.
iii) The map G admits a locally maximal invariant hyperbolic set C G : where ∆ = ∆ 0 ∪ ∆ 1 and ∆ 0 ⊂ D 3 , ∆ 1 ⊂ D 3 are compact sets, diffeomorphic to rectangles, with non-empty interior, that constitute a Markov partition for G| CG . Furthermore, the Hausdorff dimension of C G satisfies: where ε ≈ 0.00013 e −7499 is strictly positive.
iv) The local stable manifold W s loc (p 0 ) ∩ ∆ 0 is a graph over the x-axis with the angle of the slope bounded away from 0 and π/2. Remark 3.4. The bounds on the rectangles ∆ 0 and ∆ 1 of the Markov partition for C G are given in Table 1.
One can construct a convergent sequence of approximations of the hyperbolic sets C G in a straightforward way. Define recursively: The following Lemma has been proved in (Gaidashev and Johnson 2009). Table 1. The rectangles that approximate the Markov partition for the horseshoe of G. The rectangles are spanned by vectors e s 0 = (0.788578889012330, −0.614933602760558), e u 0 = T (e s 0 ) and e s 1 = (0.750925931392967773, 0.660386436536671957), e u 1 = T (e s 1 ), respectively. The length of the sides of the rectangles ∆ ′ 0 and ∆ ′ 1 is 2 · stable/unstable scale · |e u,s 0,1 |.
In this paper we will complement Theorem 2, and show that the third iterate of F also supports a stable set in its domain of analyticity.

Statement of results
Recall, that a map H : X → Y between two metric spaces X and Y is called bi-Lipschitz, if there is a constant C ≥ 1, such that for any two points p and q in X i.e. if dist X (p, q) and dist Y (H(p), H(q)) are commensurate. We denote commensurability of two quantities by "≍". A classical result from analysis states that such maps preserve the Hausdorff dimension.
Consider the dyadic group, where lim ← − stands for the inverse limit. An element w of the dyadic group can be represented as a formal power series w → ∞ k=0 w k+1 2 k . The odometer, or the adding machine, p : {0, 1} ∞ → {0, 1} ∞ is the operation of adding 1 in this group.
We are now ready to state our main theorem.
Main Theorem 1. There exists ̺ > 0 such that any F ∈ W(̺) admits a "stable" Cantor set C ∞ F ⊂ D, that is the set on which the maximal Lyapunov exponent is equal to zero, with the following properties.
1) The set C ∞ F is a Hausdorff limit of invariant hyperbolic sets of vanishing hyperbolicity. 2 3) The restriction of the dynamics F | C ∞ F is topologically conjugate to the adding machine.
where H is a bi-Lipschitz map.
Remark 4.1. We have obtained the following rigorous computer bound which is clearly dependent on our choice of coordinates for F 's. At the same time, ω ≤ µ * /|λ * | < 0.246 (independently of the choice of a coordinate system in which maps F are considered), while ν, the renormalization convergence rate on W(̺), is less than 0.85. Therefore, it seems likely that the submanifold W ω (̺) = W(̺).

Some notation and definitions
We will use the following notation for the sup norm of a function h and a transformation H defined on some set S ⊂ R 2 or C 2 : where P x and P u are projections on the corresponding components. We will also use the notation | · | for the l 2 norm for vectors in R 2 .
With DG * : p → DG * (p) we denote an interval matrix valued function such that where D 3 is the domain of G * , and the bound on the operator norm of DG for We will also use the following abbreviations for maps, transformations and scalings

A stable invariant set as a Hausdorff limit of hyperbolic sets
According to Theorem 2 the fixed point F * possesses a hyperbolic set for its third iterate. By the stability property of such sets, there exists a neighborhood B r ′ (s * ) such that all maps F of the form (4) The following holds on Λ k Also, define the following sequence of sets is a conjugacy of G and G * on C k * : In the rest of this Section we will be studying the sequence of sets C k G . We will demonstrate that the limit set C ∞ G exists, is stable, in the sense that the maximal Lyapunov exponent on C ∞ G is zero, bounded, closed and invariant under G. We will also show that there exists an ω > 0 such that for any F ∈ W ω (̺) the sets C ∞ G are weakly rigid: there exists a bi-Lipschitz (see (27)) conjugacy H G between C ∞ * and C ∞ G .
The stable set C ∞ G is an analogue of the attracting set for dissipative Hénonlike maps constructed (de Carvalho et al 2005). The (more standard) approach of ( de Carvalho et al 2005) is based on the so called presentation functions; it also demonstrates that the attracting set is Cantor and that the restriction of the dynamics to it is homeomorphic to the adding machine. We outline a similar procedure in Section 7.
The approach of this Section is more technical than that method of presentation functions; one of the our goals in pursuing it was to demonstrate that the stable set is a Hausdorff limit of hyperbolic Cantor sets with vanishing hyperbolicity. On the other hand the more compact method of presentation functions shows that the stable set is indeed Cantor, and that the stable dynamics is that of an odometer (an adding machine).
We will first demonstrate boundedness of sets C k G . Given a set S ⊂ D 3 on which an iterate G i for all G ∈ G * is defined, we use the notation G * i (S) as a shorthand for ∪ G∈G * G i (S). Notations Λ * n (S) and T n (S) are used in a similar sense.
Let {j l } m l=1 , be the index set such that α j l = 0: (39) i = 2 j1 + 2 j2 + . . . + 2 jm , j m ≤ k − 1, m ≤ k. Consider G i on a subset of D 3 where this iterate is defined: For convenience, we will denote T n,m,G = Λ −1 m,G •Λ n,G •G n , T n,0,G = Λ n,G •G n , T n = Λ n * •G * , T n ≡ Λ * n •G * , and also use the following notation for compositions of these maps: where j 0 ≡ 0. In this notation, the iterate G i can be written as We apply the formula (44) to write the action of G i on H k,G (Λ k * (C * )): The set E 1 was chosen so that Λ * (∆) ⊂ E 1 , where ∆ is as in Theorem 2, and Λ * (E 1 ) ⊂ E 1 . Therefore, . Now, to demonstrate the invariance of the set E, we verify that Figure 3. Invariance of the set E under the action of T n . (Progs 2009) for programs used in this verification).
These inclusions imply (see Figure 3) that for any sequence The set E is depicted in Figure 2.
The next technical property of the set E will be required in the proof of weak rigidity in Section 8. This "separation" property has been verified directly on the computer.
Lemma 6.2. The projections of sets E 2 and E 4 on the horizontal axis are separated from 0 and from rescalings of themselves: dist(P x (E 2 ∪ E 4 ), 0) is strictly positive, and |λ − | sup p∈E2∪E4 |P x p| < inf p∈E2∪E4 |P x p|.
Remark 6.3. We have computed (see (Progs 2009)) the upper and lower bounds on the norms of DT 1 v and D(G * • Λ * ) to be as follows: We will also denote A = max{A 1 , A 3 }.
The next Lemma, albeit straightforward, will be important in our proofs of convergence of sets C k G and existence of a bi-Lipschitz conjugacy between the limit sets.
Proof. By the strong structural stability property of the hyperbolic sets (see e.g. Theorem 18.1.3 and 18.2.1 in (Katok and Hasselblatt 1995) and, in fact, if ̺ is sufficiently small then there exists a constant C ′ such that for all F ∈ W(̺) where the "constant" C ′′ (̺) decreases to zero with the size of the local manifold W(̺), therefore, Finally, and similarly for P u (H G k − Id). The claim follows.
In several following propositions and theorems we will have to use a number of "constants" c i (̺) all of which have the property lim ̺→0 c i (̺) = 0.
Proposition 6.5. There exists ̺ > 0 such that for all F ∈ W(̺) the sets V k G and C k G converge in the Hausdorff metric, specifically: 0))) ≤ const θ k , the limit set is closed, and satisfies ((0, 0)).

Proof. Clearly,
Let the binary expansion of i < 2 k be as in (39). Recall, that according to Lemma 6.1 (43).
Let s k ≡ Λ k,G (s) ∈ V k,0 G and p k+1 ≡ Λ k+1,G (p) ∈ V k+1,0 G be any two points in the corresponding sets. Since According to Lemma 6.1 the sequences Suppose, out of m differences j n − j n−1 , n = 1, . . . , m, q are larger than 1 and m − q equal to 1. Then, where . The more often DG * in DT 1 has to be evaluated on E 1 , that is, the more often the bound A 1 (see Remark 6.3) appears in the product in (56), the worse the resultant bound. Recall that m ≤ k and j m ≤ k − 1. Therefore, if m ≤ k 2 , then all differences j n − j n−1 may be larger than 1 (q = m), and If m > k 2 then there are at most q = k − m differences j n − j n−1 that are larger than 1: and since A 2 3 /|λ − |A 1 < 1 we get in this case Any point in V k,i G can be represented as G i (s k ) for some s k ∈ V k,0 G , and any point in V k+1,i G can be represented as G i (p k+1 ) for some p k+1 ∈ V k+1,0 G , therefore A similar computation holds for inverse iterates This demonstrates that the Hausdorff distance between components V k,i G and components V k+1,i G , V k+1,2 k +i G decreases with k at a geometric rate. An identical argument for sets C k G (rather than V k G ) shows that these sets converge in the Hausdorff metric at the same rate θ. We define the set C ∞ G as the set of all limit points of sequences {p k }, p k ∈ C k G . Such set is clearly closed. Finally, to show (55), we again notice that if s k,i ∈ V k,i , then there exists a point We will now show that the set C ∞ G is invariant for G. Lemma 6.6. For any F ∈ W(̺) the sets C k G are invariant under G. The same is true about the set C ∞ G . Proof. This follows from a simple computation: . By Proposition 6.5, a point p ∞ ∈ C ∞ G is a limit point of some sequence {p k }, We will now address the convergence properties of transformations i H k,G .
Proposition 6.7. There exists ̺ > 0 such that for all F ∈ W(̺) the following holds.
1) The transformations G i • Λ k,G • Λ −k * • G −i * are defined and analytic on V k,i * for all k ∈ N and 0 ≤ i < 2 k , and satisfy

2) For any
). By Lemma 6.1 the iterate G i , 1 ≤ i < 2 k , is well-defined and analytic on Λ * k (U k * ) for all G ∈ G * .
Proving (57) is similar to (56) and arguments that follow it: where the function C(̺) is as in Lemma 6.4. We will now demonstrate (58) in two steps.
Step (1). Write The image of V k,i * under the inverse of this map is contained in E 2 ∪ E 4 for all 1 ≤ q ≤ m: The map G jm is defined and analytic on E 1 ∪ E 3 and maps it into E 2 ∪ E 4 , and therefore G jm and maps it into E 2 ∪ E 4 . Because of (63) we also have for any n > j q : where c 4 (̺) is another constant decreasing to zero together with ̺, and c 5 (̺) is the maximum of c 3 (̺) and c 4 (̺).
As the result of the above discussion, we have where h jm is some function analytic on J −1 G * ,m,i (V k,i * ) and satisfying (66) |h jm | J −1 G * ,m,i (V k,i * ) ≤ c 6 (̺)ν jm .
Step (2). At the next step, to obtain the bound (58) we will use an inductive argument.
Suppose that for q ≤ m This is certainly satisfied for q = m (see (65) and (66)). We prove that a representation similar to (67) holds for q − 1 with a similar bound on h jq−1 . First, Again, consider the map in the brackets: The first norm in (69) has been estimated in (64). To provide a bound on the second norm we will use the fact that and that if j q − 1 − j q−1 = 0 then ≤ c 6 (̺)ν jq−1 + ac 6 (̺) Therefore, the last equality being the definition of c 1 (̺).
2) To demonstrate (59) we notice that It follows from a computation similar to (32) that The above proposition implies, that if p is in the limit set C ∞ * , then there exist integers i and K, dependent on p, and a sequence of points p k,i ∈ C k,i * , k ≥ K, that converge to p: lim k→∞ p k,i = p. We have from (38) Bounds (57)-(59) imply that the limit exists.
We will finally demonstrate that the limit set C ∞ G is stable. Recall, the definition of the upper Lyapunov exponent of (p, v) ∈ (D ∩ R 2 ) × R 2 with respect to G: where is some norm in R 2 . The maximal Lyapunov exponent of p ∈ (D ∩ R 2 ) with respect to G is defined as Lemma 6.8. For any F ∈ W(̺) and p ∈ C k G the maximal Lyapunov exponent χ(p; G) satisfies Proof. Let i = q2 k + n, n = 2 j1 + 2 j2 + . . . + 2 jm < 2 k and p ∈ H k,G (Λ k * (C * )). Denote where we have used the representation (40). According to Lemma 6.1 Finally, Clearly, the above result implies stability of the limit set: Corollary 6.9. For any F ∈ W(̺) and p ∈ C ∞ G the maximal Lyapunov exponent χ(p; G) is equal to zero.

The stable set as a Cantor set
In this Section we will sketch the construction of the stable set using the method of presentation functions. The construction of this Section is almost identical to that of ( de Carvalho et al 2005), and we will therefore omit many details. In fact, we will attempt to use the notation similar to that of ( de Carvalho et al 2005).
Given F ∈ W(̺) define the two presentation functions Lemma 7.1. For every F ∈ W(̺) there exists a simply connected closed set B F such that and (74) max{ Dψ F 0 BF , Dψ F 1 BF } ≤ ϑ, ϑ = 0.272. Proof. First, we verify the following on the computer: . We also check that the sets ψ F 0 (B) ⊂B and ψ F 1 (B) ⊂B are disjoint. Second, we verify that the boundary of the ellipseB ⊂B, intersects each of ψ F 0 (B), ψ F 1 (B) along a single arc. Therefore, the set Figure 7) is simply connected, and satisfies the claim.
One can view {0, 1} n as an additive group of residues mod 2 n via an identification w → n−1 k=0 w k+1 2 k .
Let p : {0, 1} n → {0, 1} n , be the operation of adding 1 in this group. The following Lemma has been proved in (de Carvalho et al 2005), and it's proof holds in our case of area-preserving maps word by word: Lemma 7.2.
1) The above families of pieces are nested: Since the setB from Lemma 7.1 contains (0, 0), so does each piece B n 0 n . It follows from part 3) of Lemma 7.2 that the set w∈{0,1} n B n w contains iterates G i ((0, 0)) up to order 2 n . Therefore, the Cantor set C ∞ F is the closure of the orbit of zero, and is equal to ). Recall the definition (28) of the dyadic group. Lemma 7.2 implies the following: B n w1w2...wn .

"Weak" rigidity
In this Section we will demonstrate that the map H G is bi-Lipschitz for a subset of infinitely renormalizable maps.
Commensurability, together with convergence property (70) implies that the limit H G is a bi-Lipschitz transformation.
Define the following points: where the the last four lines are understood as definitions of points p, s ∈ C G k and p * , s * ∈ C * . For any j < k and F ∈ W ω (̺) there exists c ′ 7 (̺) such that |λ Gj | ≤ |λ * |+c ′ 7 (̺)ω j , therefore where c 7 (̺) and c ′′ 7 (̺) are some constants. This, together with (53) implies the following bound for any p * in C * and p = H G k (p * ) and all j < k Next, suppose that q is the smallest integer such that j q =ĵ q and j l =ĵ l , l < q. For definitiveness, supposeĵ q > j q . We expand, as before, and similarly for Gî * . Our immediate goal will be to show that |p k,i − s k,î | and |p * k,i − s * k,î | are commensurate. To this end we will show that the distances between the images of points p,s and p * ,s * under the consecutive application of the three maps T [i] q−1 1 ,G , (. . .) and {. . .} in (77) stay commensurate. We will perform this in three steps.
Remark 8.2. We would like to emphasize that the commensurability property (88) holds only for i =î, and therefore does not imply that the hyperbolic sets C k G and C k * are bi-Lipschitz conjugate. In case i =î a positive lower bound (79) does not exist, which would invalidate the arguments that follow.

Some concluding remarks
We have demonstrated that the Hausdorff dimension of the stable set for the maps F in the subset W ω (̺) of the infinitely renormalizable maps is independent of F , and that the stable dynamics for two infinitely renormalizable maps in W ω (̺) is bi-Lipschitz-conjugate. This is quite weaker than the corresponding result about the invariance of the Hausdorff dimension of the Feigenbaum attractor for all infinitely renormalizable unimodal maps (see (Pa luba 1989, Rand 1988, McMullen 1996, de Melo and Pinto 1999). On the other hand, it does demonstrate that one should expect at least some kind of rigidity of invariant sets for infinitely renormalizable maps in conservative dynamics -rigidity which was absent in dissipative maps (see (de Carvalho et al 2005)).
Our proof of the bi-Lipschitz property of the conjugacy between stable sets C ∞ G and C ∞ G balances two phenomena that, in a sense, work against each other: convergence of renormalizations of maps G ∈ W(̺) versus the fact that the rates of contraction of distances in different directions by maps Λ F and Λ F • F are essentially different. A careful look at the proof shows that the bi-Lipschitz property is achieved if the convergence rate ν is sufficiently small to "counteract" the relative size of contractions. However, this is not the case with the upper bound (15) on ν at hand. Although this upper bound is by no means sharp, it does indicate that one might need to choose a submanifold W ω (̺) of W(̺) on which the convergence rate is smaller.
Another obvious issue for investigation is whether the bi-Lipschitz conjugacy of the stable sets extends to their neighborhood as a C 1+ǫ map. Again, this is the case for the conjugacies between attractors of the unimodal maps (see (Rand 1988, McMullen 1996, de Melo and Pinto 1999), and it is not for very dissipative maps where, as we have already mentioned, the regularity of the conjugacy of attractors for two maps F andF has a definite upper bound (1).