Topological and geometric hyperbolicity criteria for polynomial automorphisms of C 2

. We prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of C 2 . Along the way we also show that a sufﬁcient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.

The result is not obvious since the conjugacy φ cannot detect that a point in N pJ 1 q is postcritical. Here and throughout the paper we use indices 0 and 1 to label the dynamical objects (Julia set, etc.) respectively associated to f 0 and f 1 .
Proof. Notice first that the conjugacy φ sends periodic points to periodic points. The topological dynamics around a periodic point determines its type (attracting, repelling, neutral) so it follows that all periodic points of f 1 are hyperbolic. In particular f 1 has no parabolic points.
Since the Julia set is the accumulation set of periodic orbits we infer that φpJ 0 q " J 1 . Now a rational map without parabolic points is hyperbolic if and only if its critical set is disjoint from the Julia set. This property holds for f 1 by the topological conjugacy, so the result follows. (1) It is not enough in the proposition to assume that φ is a conjugating homeomorphism J 0 Ñ J 1 . Indeed it is well known that z 2`1 4 is topologically conjugate on its Julia set to any quadratic polynomial in the main cardioid (e.g. z 2 ). As we will comment below, a similar phenomenon holds for complex Hénon maps (see [RT]).
(2) If we suppose a priori that degpf 0 q " degpf 1 q " d we can relax the assumption on φ by assuming only that φ is any injective continuous map defined in a neighborhood N 0 of J 0 and such that φ˝f 0 " f 1˝φ wherever these compositions makes sense. Indeed by the invariance of domain theorem, φpN 0 q is an open subset of the plane. We only have to shows that it contains Jpf 1 q. Indeed f 0 has only finitely many non-repelling periodic points, so J 0 contains p n " d n repelling periodic points of period d for large n. Thus by the topological conjugacy, φpJ 0 q contains p n repelling periodic points of f 1 , which are equidistributed to the equilibrium measure µ f1 whose support is Jpf 1 q. Therefore φpJ 0 q " J 1 and we are done.
1.3. Anosov diffeomorphisms. The problem of topological invariance of hyperbolicity in real dynamics has been popularized in particular by A.
Katok. The answer is already quite subtle for the simplest case of Anosov diffeomorphisms of the 2-torus. Indeed there exist examples of C 2 diffeomorphisms f of the 2-torus which are not hyperbolic but still globally topologically conjugate to a linear Anosov map. This can be done by either carefully deforming a linear Anosov map until some saddle fixed point becomes neutral by preserving the geometry of the stable and unstable foliations (see [K]), or by deforming the foliations until reaching a cubic heteroclinic tangency (see [E, BDV]).
If we now impose the conjugacy to be Hölder then there are different regimes depending on the precise Hölder regularity. First, it can be arranged that in the previous examples the conjugacy and its inverse are Hölder continuous [Go], and thus hyperbolicity is not invariant under Hölder conjugacy. On the other hand if the conjugacy is sufficiently close to being bi-Lipschitz -namely, the product of the Hölder exponents of φ and φ´1 is larger than 1/2-then f is Anosov [F] (see [Go]).
1.4. A conjecture. The most natural way to address Question 1.1 would be to find a topological criterion ensuring hyperbolicity for a complex Hénon map, in the spirit of the one-dimensional condition J XPCpf q " H. Strictly speaking, a Hénon map admits no critical points; nevertheless there are ways to give a reasonable meaning to this condition -which are more differentialgeometric than topological, though. For instance, in the dissipative regime, the condition that there are no critical points on J naturally corresponds to the existence of a dominated splitting and, provided dissipation is strong enough, a good analogue of the one-dimensional situation was achieved in [LP].
A variant is to translate the condition J X PCpf q " H into a regularity property of the geometry of the forward and backward Julia sets J`{´near J. In this respect it was shown in [BS8] that if in some neighborhood of J, J`and J´are the supports of two Riemann surface laminations which are transverse along J, then f is hyperbolic. In §2 below we reprove and generalize this result in several ways.
Back to our initial problem, even if it is unclear how to design a purely topological criterion for hyperbolicity, one may ask whether hyperbolicity is invariant under topological conjugacy.
Here is a precise analogue of Proposition 1.2 for complex Hénon maps: Conjecture 1.4. Let f 0 and f 1 be two polynomial automorphisms of C 2 with non-trivial dynamics, and assume that f 0 is hyperbolic. Suppose that there exists respective neighborhoods N 0 and N 1 of J 0 " J › 0 and J › 1 and a homeomorphism φ : N 0 Ñ N 1 such that φ˝f 0 " f 1˝φ wherever these compositions makes sense. Then f 1 is hyperbolic.
Here are some comments on this conjecture: (1) It was shown in [D2] (see also [GP]) that for a complex Hénon map hyperbolicity on J › implies hyperbolicity. This explains why we can restrict to a neighborhood of J › 0 instead of J 0 , and opens the way to an analysis of hyperbolicity based on periodic points.
(2) If we add the hypothesis that f 0 and f 1 have the same dynamical degree, then by using the equidistribution of periodic orbits from [BLS2] and arguing as in Remark 1.3 we can relax the assumption that φpN pJ › 0 qq contains J › 1 .
(3) As observed in Remark 1.3, the conjecture is false if the conjugating homeomorphism φ is only defined on J 0 " J › 0 (see [RT]). (4) The conjecture is true if φ is obtained by deformation in the following sense: it was shown in [BD] that if there is a weakly stable holomorphic family pf λ q connecting f 0 and f 1 , then f 1 is hyperbolic 1 .
1.5. Quasi-hyperbolicity. The methods in this paper are closely related to the notion of quasihyperbolicity. If p is a saddle point and r ą 0, we let W s{u r ppq denote the connected component of W s{u ppq X Bpp, rq containing p. Following [BS8] map f is said quasi-hyperbolic if there exists positive constants r and B such that for every saddle periodic point p: (i) W s{u r ppq is closed in Bpp, rq and (ii) the area of W s{u r ppq is bounded by B. If φ : N 0 Ñ N 1 is a topological conjugacy as in Conjecture 1.4 then φ preserves stable and unstable manifolds, so if (i) holds for f 0 it will also hold for f 1 (after possibly shrinking r). It was shown in [BGS] that if f is quasi-hyperbolic then there exist stable and unstable manifolds W s{u pxq through each point x P J › . Furthermore f is uniformly hyperbolic (on J › and thus J) if and only if there is no tangency between W s and W u . Thus if we know that f is already quasihyperbolic, then the additional condition of hyperbolicity is a topological invariant in the sense of the conjecture. At this stage, however, it remains an open question whether quasi-hyperbolicity is a topological property.
1.6. Results and outline. In §2 we establish several sufficient conditions for hyperbolicity based on the geometry of local stable and unstable manifolds of saddle periodic points. A first sufficient condition for hyperbolicity, which essentially follows from [BS8], is that these local stable and unstable manifolds have uniform size and the angle between them is uniformly bounded from below. We give a self-contained proof of this result (see Theorem 2.9). We further show that the transversality assumption is superfluous (Theorems 2.13 and 2.14), and that, as it might be expected, in the dissipative case it is enough to control the geometry of unstable manifolds (Theorem 2.19).
In §3 we prove Conjecture 1.4 in the case where f 1 is dissipative (Theorem 3.6). In the conservative case the conjecture holds provided φ is Hölder continuous (Theorem 3.8). The general case remains open 2 .
2. Geometric criteria for hyperbolicity 2.1. Size of a submanifold at a point and u/s regularity. Endow C 2 with the Euclidean metric. A bidisk of size r is the image of Dp0, rq 2 under some affine isometry. A curve V in C 2 is a graph over an affine line L if the orthogonal projection onto L is injective when restricted to V . Then there is a well-defined notion of slope of a holomorphic curve with respect to L.
Definition 2.1. A curve V through p is said to have size r at p if there exists a neighborhood of p in V that is a graph of slope at most 1 over a disk of radius r in the tangent space T p V .
If ∆ be a disk of size r at p, fixing orthonormal coordinates px, yq so that p " 0 and T p V " ty " 0u, we get that the connected component of ∆ through p in the bidisk Dp0, rq 2 is a graph ty " ϕpxqu over the first coordinate with |ϕ 1 | ď 1 and ϕ 1 p0q " 0. In particular if ∆ is immersed and has size r at p, then it is a submanifold in Bp0, r{ ? 2q (because a bidisk of size r contains a ball of radius r{ ? 2). We now recall a few concepts from [BD]. A point x P J › is said u-regular (resp. s-regular) if there exists r ą 0 and a sequence of saddle points pp n q converging to x such that W u pp n q (resp. W s pp n q) is of size r at p n . In this case it can be shown that the sequence of disks W u r pp n q (resp. W s r pp n q) converges in the C 1 topology to a (smooth) holomorphic disk of size r at x which we denote by W u r pxq (resp. W s r pxq) (see [BD,Prop. 4.2]). This notation is meant to emphasize that at this stage W s{u r pxq need not be an stable/unstable manifold in the usual sense. We use the notation W u{s loc pxq for an unspecified neighborhood of x in W u r pxq. We say that x is regular if it is u-and s-regular and W u loc pxq and W s loc pxq do not coincide, and transverse regular if they are transverse. In particular we have the implications u-and s-regular ð regular ð transverse regular.
It is easy to see that if x is a saddle point, then x is regular (for instance because it generates homoclinic intersections, hence it belongs to a horseshoe) and W s loc pxq and W u loc pxq coincide with the classical local stable and unstable manifolds of x.
We define a local stable set 9 W s loc, pxq " ty : distpf n pxq, f n pyqq ă @n ě 0, and lim nÑ8 distpf n pxq, f n pyqq " 0u Lemma 2.2. Suppose that there is a complex disk ∆ such that x P ∆ Ă 9 W s loc, pxq. If x is s-regular, then ∆ coincides with W s pxq locally at x. The analogous result holds for 's' replaced by 'u'. Lemma 2.3. If x is Pesin regular, and x is u-and s-regular, then W s{u loc pxq agree locally at x with the Pesin manifolds W s{u Pesin . Further, W s loc pxq ‰ W u loc pxq, so x is regular in the sense defined above, and in fact it is transverse regular.

More generally we have:
2 Notice that the Jacobian is not invariant under topological conjugacy: the Hénon map pz, wq Þ Ñ pz 2`c`a w, zq is conjugate to a horseshoe for any Jacobian a, when |c| " |a|.
Lemma 2.4. If x is a Pesin regular point which is regular, then W s loc pxq and W u loc pxq coincide with the classical Pesin local stable and unstable manifolds of x.
Proof. Denote temporarily the Pesin local unstable manifolds by W u Pesin pxq. If pp n q is a sequence of saddle points converging to x, then W u pp n q " W u loc pp n q must coincide with or be disjoint from W u Pesin pxq, and converge to W u pxq in the C 1 topology. Since both W u Pesin pxq and W u pxq contain x, by the Hurwitz theorem we conclude that W u Pesin pxq locally coincides with W u pxq.
We say that x P J › uniformly u-regular if the uniform size property for local unstable manifolds holds for any sequence pp n q converging to x. If required we can specify the size r in the terminology. Uniform s-regularity is defined similarly. We say that x is uniformly (resp. transverse) regular if it is uniformly u-and s-regular, and (resp. transverse) regular.
The following result will play an important role in this paper (of course it admits an identical s-regular version).
Proposition 2.5. The following assertions are equivalent: (1) Every point in J › is uniformly u-regular.
(2) There exists a uniform r ą 0 such that for every saddle periodic point p, W u ppq has size r at p. (3) There exists a uniform r ą 0 and a dense set D of saddle periodic points such that for every p P D, W u ppq has size r at p. (4) There exists a lamination W u by Riemann surfaces in a neighborhood of J › which extends the family of local unstable manifolds of saddle points. Proof.
By the persistence of proper intersections, the same holds for r ∆ and r ∆ 1 , whenever r ∆ and r ∆ 1 are holomorphic disks which are respectively 1{100 close to ∆ and ∆ 1 . Now if ∆ is a disk of size r at x, by the Schwarz Lemma, ∆ X Bp0, r{10q remains 1{100 close to T x ∆. Taking the contraposite we see that if ∆ and ∆ 1 are disks of size r respectively at x and x 1 , with distpx, x 1 q ă 1{1000, then their tangent spaces must be 1{4-close to each other, in particular they are graphs over a disk of radius r{4 relative to the same orthogonal projection. Now by (3), for every x P J › there is a holomorphic disk W u r pxq of size r through x and these disks are either disjoint or locally coincide because local unstable manifolds of saddle points are disjoint. By the previous discussion, the disks W u r{4 pyq are disjoint graphs over some direction for y close to x, so they form a lamination by the Lambda Lemma of [MSS]. Thus we get the desired lamination structure in the r{5-neighborhood of J › .
Remark 2.6. Under the assumptions of Proposition 2.5, there exists a neighborhood N of J › and a lamination W u of N by Riemann surfaces which extends the family of local unstable manifolds of saddle points. Beware however that it does not a priori imply that J´X N is laminated nor that it coincides with SupppW u q: indeed, J´is the closure of global unstable manifolds, which could recur to N in a complicated fashion (this point is a main issue in [D2]).

Existence of invariant laminations and hyperbolicity.
Recall that a complex Hénon map f is said to be hyperbolic if J is a hyperbolic set. As was noted above, by [D2] (see also [GP]) it is actually enough to check hyperbolicity on J › : this opens the way to hyperbolicity criteria based on periodic points.
Theorem 2.7 ( [D2]). If J › is a hyperbolic set for f , then f is hyperbolic.
A geometric criterion for hyperbolicity based on the existence and transversality of unstable laminations was established in [BS8,Thm 8.3]. By incorporating the result of Theorem 2.7 it reads as follows.
Theorem 2.8 ( [BS8]). Let f be a complex Hénon map. Assume that there exists a neighborhood of J › and Riemann surface laminations L˘of J˘such that L`and L´intersect transversally at all points of J › . Then f is hyperbolic.
It is convenient to formulate this result in the language of uniform regularity. The following is an essentially equivalent statement (see however Remark 2.6).
Theorem 2.9. Let f be a complex Hénon map. If every point in J › is uniformly regular and transverse then f is hyperbolic.
Let us give a self-contained proof of this theorem, which basically follows the approach of [BS8,Thm 8.3]. First, recall from Proposition 2.5 that if every point in J › is uniformly uregular, then exists r ą 0 and a lamination W u in the r-neighborhood of J › , extending the unstable manifolds of saddle points. Recall also the dynamical Green function G`, defined by G`pxq " lim nÑ8 d´n log`}f n pxq}. It is a non-negative continuous psh function in C 2 , with the property that tG`" 0u " K`.
Proposition 2.10. Let f be a complex Hénon map. Assume that every point in J › is uniformly u-regular and that for every x P J › , G`| W u loc pxq ı 0. Then f is uniformly expanding in the direction of T W u along J › .
This condition on G`will be used several times in the sequel; it means that G`does not vanish identicaly on any neighborhood of x in W s pxq.
Proof. Let r be such that for any saddle point p, W u ppq has size 5r at p. Then by Proposition 2.5, W u defines a lamination in the r-neighborhood of J › such that for every saddle point p, W u ppq coincides with the local unstable manifold of p.
We have to show that f is uniformly expanding along W u | J › , that is, there exists C ą 0 and λ ą 1 such that for every x P J › , every k ě 1 and e P T x W u pxq,ˇˇDf k x peqˇˇě Cλ k |e| (where |¨| denotes the Riemannian metric induced by the standard Hermitian structure of C 2 ). By continuity it is enough to prove this property on the (dense) set S of saddle periodic points. For this, we will construct a metric |¨| 1 on T W u | S which is equivalent to the ambient one (with uniform constants) and such that for every p P S, and e P T p W u ppq, |Df p peq| 1 ě λ |e| 1 .
For every saddle point p, the global unstable manifold is biholomorphic to C, so its uniformisation ψ u p : C Ñ W u ppq is unique up to a multiplicative factor at the source. In particular f is affine is these parameterizations, and there is a well-defined notion of a round disk in W u ppq, which is f -invariant. For e P T p W u ppq and η ą 0 we define |e| η in the style of the Kobayashi metric: For every x P J › (not necessarily a saddle), G`| W u loc pxq is not identically 0 near x so we infer that for every r ą 0, sup G`| W u r pxq ą 0. From the continuity of the Green function, the compactness of J › , and the lamination structure we infer the existence of constants r ą 0 and η 1 1 ą η 1 ą 0 such that for every x P J › , η 1 ď sup G`| W u r pxq ď η 1 1 . Recall that for near any x P J › , up to a unitary change of coordinates, W u is a union of graphs over a disk of size r and slope bounded by 1 (relative to some projection π). Thus if p P S is close to x, and ψ u p is as above, it follows that π˝ψ u p | Dp0,ρη 1 q is a univalent holomorphic function. Set η 2 " η 1 {2. The Koebe distortion theorem together with the uniform continuity of the Green function imply that for ρ ă ρ η2 , π˝ψ u p pDp0, ρqq is approximately a round disk (with uniform distortion bounds). From this uniformity, we infer that there exists λ ą 1 such that for every p P S and e P T p W u ppq, |e| η2{d ě λ |e| η2 . Set η 3 " η 2 {d. The invariance relation of the Green function G`implies that |df p peq| dη3 " |e| η3 . From this we get that for every p P S and e P T p W u ppq |df p peq| η2 ě λ |e| η2 . Finally, again from the uniform continuity of the Green function and bounded distortion, we get that |¨| η2 is (uniformly) equivalent to |¨| on T W u | S so the proof is complete.
Remark 2.11. By a standard procedure, up to reducing λ it is possible to construct a continuous metric Then one easily checks that |¨| 2 is well-defined, continuous, and satisfiešˇD The next result implies that if f is uniformly regular, then the second assumption of Proposition 2.10 holds.
Proposition 2.12 (see [BD,Prop. 4.7]). Let f be a complex Hénon map. If x P J › is regular then G`| W u loc pxq ı 0. Proof. If pp n q is a sequence of distinct saddle points converging to x, then W s r pp n q is a sequence of disjoint submanifolds converging to W s r pxq. Since by assumption W u r pxq and W s r pxq are distinct, then for large n W s r pp n q must possess transverse intersection points with W u r pxq close to x: if W u r pxq and W s r pxq are transverse this is clear, and if they are tangent this follows from [BLS1, Lemma 6.4]). Then the inclination lemma implies that pf n | W u loc pxq q is not a normal family of holomorphic mappings, therefore G`is not harmonic on W u loc pxq, thus not identically zero, and we are done.
Proof of Theorems 2.8 and 2.9. Under the assumptions of Theorem 2.9, it follows directly from Propositions 2.10 and 2.12 that f is uniformly expanding along T W u | J › and contracting along T W s | J › , that is, J › is a hyperbolic set. Then we conclude from Theorem 2.7 that f is hyperbolic.
To establish Theorem 2.8, it is enough to check that the existence of the transverse laminations L`and L´imply uniform transverse regularity. We first observe that for any saddle point p, W s ppq locally coincides with the leaf L`ppq of L`through p, and likewise in the unstable direction. Indeed since the leaves of L`are contained in J`, for every disk ∆ contained in such a leaf, pf n | ∆ q ně0 is a normal family. Now if Ll oc ppq ‰ W s loc ppq then either they are transverse and it follows from the inclination lemma that pf n | Ll oc ppq q ně0 is not normal. Otherwise by [BLS1,Lemma 6.4] for any x P J › close to p, Ll oc pxq is transverse to W s loc ppq and similarly pf n | Ll oc pxq q ně0 is not a normal family. In both cases we reach a contradiction. It then follows from Proposition 2.5 that every point in J › is uniformly regular and transverse and we conclude as before.
It turns out that the transversality assumption in Theorem 2.8 is unnecessary, that is, uniform regularity rules out the possibility of tangencies.
Theorem 2.13. Let f be a complex Hénon map. If every point in J › is uniformly regular then f is uniformly hyperbolic.
Proof. By Proposition 2.5 there exist laminations W u and W s in a neighborhood of J › extending the family of local stable and unstable manifolds of periodic points, and by Proposition 2.10 and 2.12 we get that f is uniformly expanding along W u and f´1 is uniformly expanding along W s . To prove the theorem we thus have to show that these laminations are transverse at all points of J › . Let T be the tangency locus, that is the set of points x P J › such that W s pxq and W u pxq are tangent at x. This is a closed invariant set. Assume by way of contradiction that it is non empty. Then it supports an ergodic invariant measure ν. Let be the Lyapunov exponents of ν. Since f is uniformly expanding/contracting along W u{s we infer that χ´ă 0 ă χ`. By Oseledets' theorem, there exists an associated invariant measurable decomposition T x C 2 " E´pxq ' E`pxq defined ν-a.e. such that the growth rate of vectors in E˘pxq is governed by χ˘. By Pesin's theory (see e.g. [FHY]) for ν-a.e. x there are local stable and unstable manifolds W s Pesin pxq and W u Pesin pxq respectively tangent to the characteristic directions associated to the negative and positive exponent. But by Lemma 2.4, W s{u Pesin pxq locally coincides with W s{u pxq, so we infer that E`pxq " E´pxq a.e. which contradicts the Oseledets theorem. This contradiction finishes the proof.
If f is not volume preserving we can further relax the previous criterion.
Theorem 2.14. Let f be a complex Hénon map with |Jacpf q| ‰ 1. If every point in J › is uniformly u-and s-regular then f is uniformly hyperbolic.
Proof. The difference with Theorem 2.13 is that T can now contain local leaves so Proposition 2.10 does not apply. Let r is the uniform size of local s/u manifolds along J › . Without loss of generality assume that |Jacpf q| ă 1. Note that if W u loc pxq " W s loc pxq then W u r pxq " W s r pxq. Denote by T 1 the set of points x P J › such that W u r pxq " W s r pxq. Then T 1 is also closed and invariant. Indeed if W u r pxq " W s r pxq then clearly W u loc pf pxqq " W s loc pf pxqq, hence W u r pf pxqq " W s r pf pxqq. Thus f pT 1 q Ă T 1 , and the closedness of T 1 follows directly from the continuity of x Þ Ñ W u{s r pxq. Assume by way of contradiction that T 1 is non-empty. Then it supports an ergodic invariant measure ν. Since f is dissipative its Lyapunov exponents satisfy χ´ă 0 ď χ`. For every x P T 1 , W u r pxq " W s r pxq is contained in J`X J´so it is a Fatou disk under forward and backward iteration. The following lemma relates these disks to the Oseledets decomposition.
Lemma 2.15. Let f be a complex Hénon map and ν be an ergodic invariant measure whose Lyapunov exponents satisfy χ´ă 0 ď χ`, and T x C 2 " E´pxq ' E`pxq be the associated measurable decomposition. If ν-a.e. point is u-regular then for ν-a.e. x, W u loc pxq is tangent to E`pxq at x.
Assuming this result for the moment, let us conclude the proof. The contradiction hypothesis implies that for ν-a.e. x, W u loc pxq " W s loc pxq. By Pesin's theory a ν-generic point x admits a local strong stable manifold W s Pesin pxq, which is tangent to E´pxq, and by Lemma 2.4 it coincides with W s loc pxq. On the other hand, by Lemma 2.15, W u loc pxq is a.s. transverse to E´pxq. This contradiction shows that T 1 is empty. Therefore every point in J › is regular and applying Theorem 2.13 finishes the proof.
Proof of Lemma 2.15. Since W u loc pxq is contained in J´,´f´n| W u loc pxq¯i s a normal family, so it follows from the Cauchy estimates that }df´n x pe u pxqq} is bounded, where e u pxq is any tangent vector to W u pxq at x. On the other hand the Oseledets theorem asserts that almost surely, if epxq is any non-zero vector such that epxq R E`pxq, }df´n x pepxqq} grows exponentially at rate |χ´|. Hence e u pxq P E`pxq and we are done.
2.3. Unstable lamination, dominated splitting and hyperbolicity. It is natural to expect that in the dissipative setting, uniform u-regularity is enough to characterize hyperbolicity. Indeed, uniform u-regularity should provide uniform expansion along some field of directions, which, together with volume contraction yields uniform hyperbolicity. The basic technical tool needed to implement this idea is that of dominated splitting. Recall that a dominated splitting on some invariant set Λ is a splitting of the form T C 2 | Λ " E s ' E c for which there exists C ą 0 and λ ă 1 such that Then this splitting is automatically continuous, and if |Jacpf q| ď 1 the direction E s is contracting. The existence of a dominated splitting for f along J is a way to formalize the "absence of critical points" on J.
Our first result can be viewed as a version of [LP] in a (greatly) simplified setting.
Proposition 2.16. Let f be a complex Hénon map with |Jacpf q| ď 1. If every point in J › is uniformly u-regular and if f admits a dominated splitting on J › , then f is hyperbolic.
Proof. Dominated splitting implies the existence of a strong stable lamination W s in a neighborhood of J › , hence points of J › are uniformly s-regular. Then if |Jacpf q| ă 1, the result follows directly from Theorem 2.14. In the general case we just have to repeat the proof of Theorem 2.14, the only difference being that dissipativity was used there to show that ν has a negative exponent while here this follows from the dominated splitting assumption.
The idea of dominated splitting shows that hyperbolicity already holds under the assumptions of Proposition 2.10: Proposition 2.17. Let f be a complex Hénon map with |Jacpf q| ď 1. If every point in J › is uniformly u-regular and for every x P J › , G`| W u loc pxq ı 0 then f is hyperbolic. Applying Proposition 2.12 yields the following corollary, which generalizes (and gives a new approach to) Theorem 2.13.
Corollary 2.18. Let f be a complex Hénon map with |Jacpf q| ď 1. If every point in J › is regular and uniformly u-regular then f is hyperbolic.
Proof of Proposition 2.17. By the cone criterion for dominated splitting (see [S,Prop. 2.2]) it is enough to prove that for every x P J › there exists a cone C x about T x W u pxq in T x C 2 such that the field of cones pC x q xPJ › is strictly contracted by the dynamics. Then the result follows from Proposition 2.16. By Proposition 2.10 and Remark 2.11 there is a continuous Riemannian metric on T W u | J › which is immediately expanded by the dynamics. Let pe x q xPJ › be a field of tangent vectors to W u of unit norm relative to this metric, and f x be orthogonal to e x in T x C 2 (relative to the ambient Riemannian structure) and such that detpe x , f x q " 1. For small ε, define a continuous field of cones C ε where |λ x | ě λ 0 ą 1 and J is the Jacobian, so |J| ď 1. Since the frame pe x , f x q is continuous, ap¨q is bounded. Then one checks easily that if ε is so small that λ 0´ε }a} ą 1, then Hence the field of cones pC x q xPJ › is strictly contracted by the tangent dynamics and we are done (note that a similar argument appears in [D1]).
The next result shows that uniform expansion can indeed be deduced from the geometric property of uniform u-regularity. Assume that every x P J › is uniformly regular of size 4r. Recalling the construction of global unstable manifolds from local ones, for x P J › we define It follows from this definition that f´1pW u pxqq " f´1pW u r pxqqYW u pf´1pxqq, hence f´1pW u pxqq contains W u pf´1pxqq and it is not a priori clear that the W u pxq define an invariant family of curves. However, if W u pxq is biholomorphic to C for every x P J › then f´1pW u pxqq " W u pf´1pxqq, for otherwise W u pf´1pxqq would strictly contain f´1pW u pxqq, and it would be a complex submanifold of C 2 biholomorphic to the Riemann sphere, which is contradictory. The following theorem confirms the expectation that the parabolicity of leaves in J´is associated with expansion (compare e.g. [LM,§4]).
Theorem 2.19. Let f be a dissipative complex Hénon map. If every point x P J › is uniformly u-regular and in addition W u pxq is biholomorphic to C, then f is hyperbolic.
Remark that the definition of W u pxq in (1) a priori depends on r. The theorem shows that if these manifolds are biholomorphic to C, this is actually not the case.
Proof. For every x P J › , fix a uniformization ψ u x : CÑW u pxq such that ψ u x p0q " x, which is normalized by |pψ u q 1 p0q| " 1. For η ą 0, define R η pxq to be the maximal radius of a round disk in C such that G`˝ψ u x | Dp0,Rηpxqq ď η (this is similar but not identical to the definition of ρ η in Proposition 2.10). Since W u pxq is an entire curve contained in J´, G`| W u pxq is unbounded so R η pxq is finite. We claim that for every η ą 0, there exists C η ą 0 such that (2) for every x P J › , C´1 η ď R η pxq ď C η .
Indeed, fix x P J › and let us show that R η is locally uniformly bounded from above and below in a neighborhood of x. Then by compactness these bounds will be uniform on J › . Viewed in the unstable parameterizations f is affine so it maps circles to circles. Let ∆ u px, Rq " ψ u x pDp0, Rqq. We first claim that there exists k ě 0 such that f´kp∆ u px, R η pxqq is contained in W u r pf´kpxqq. Indeed by definition of W u pxq, for every x 1 P B∆ u px, 4R η pxqq, there exists k ě 0 such that f´kpx 1 q P W u r pf´kpxqq. As in the proof of Proposition 2.10, the Koebe distortion theorem implies that there is a coordinate π : W u 4r pf´kpxqq Ñ C such that that for s ď r, πp∆ u pf´kpxq, sqq is approximately a disk of radius s. Now f´k p∆ u px, R η pxqqq is a round disk in the affine coordinate, and it possesses a boundary point in W u r{2 pf´kpxqq, so it follows that it is completely contained in W u r pf´kpxqq. Therefore, replacing x by f´kpxq we can assume that ψ u x pDp0, R η qq Ă W u r pxq. By uniform u-regularity, for y close to x, W u r pyq is a graph of slope at most 1 over a disk of size r relative to the projection π. Thus from Koebe distortion again, we infer that for y close to x, the distance induced by the normalized affine structure along the W u r pyq is equivalent to the ambient distance. In particular there exists a constant K depending only on r such that for y close to x and η as above, Finally by the Hölder continuity of G`distpy, tG`" ηuq is bounded from below by Cη θ , and if distpy, xq ď r it is bounded from above by Cr. This completes the proof of (2). Then, from the invariance relation of G`we have f p∆ u px, R η pxqqq " ∆ u pf pxq, R dη pf pxqqq Ą ∆ u pf pxq, R η pf pxqqq, hence for every n ě 0 we infer that f n p∆ u px, R η pxqqq Ą ∆ u pf n pxq, R η pf n pxqqq. In particular for every x P J › and every n ě 1 we have that that f n p∆ u px, C η qq Ą ∆ u pf n pxq, C´1 η q. Again since f is affine in the unstable parameterizations we deduce that for every t ą 0, f n p∆ u px, tC η qq Ą ∆ u pf n pxq, tC´1 η q.
Finally taking the derivative at t " 0 we conclude that }Df n x | TxW u } ě pC η q´2. This bound in turns implies the existence of a dominated splitting along J › . This follows from the criterion of Bochi-Gourmelon [BG, Thm A] (see also Yoccoz [Y]). Indeed since f has constant Jacobian, for x P J › the singular values of Df n x are σǹ and σń " J n {σǹ , where J " |Jacpf q| ă 1, and σǹ ě pC η q´2. Therefore σǹ σń " pσǹ q 2 J n ě 1 C 4 η J n so [BG] applies and we get a dominated splitting on J › . Applying proposition 2.16 concludes the proof.
Remark 2.20. If J´is globally laminated (outside a finite set of periodic points, say) one might expect that the additional assumption that W u pxq » C for every x in Theorem 2.19 would follow from the density of unstable manifolds of saddle points. Unfortunately there are examples of minimal Riemann surface laminations containing both parabolic and hyperbolic leaves (see [Gh,Thm 6.6]) 2.4. Concluding remarks.
2.4.1. Uniform s-regularity on J › does not imply hyperbolicity. Indeed there are examples of Hénon mappings with parabolic points and a dominated splitting on J › (see [RT, LP]). It would be interesting to know whether uniform s-regularity on J › implies the existence of a dominated splitting.
2.4.2. The only property of J › that was used in the various hyperbolicity criteria in this section is that J › is a closed invariant set in which saddle periodic points are dense. So in all these results we could with an arbitrary closed invariant set Λ, in which saddle points are dense. The notion of uniform u-regularity has to be replaced by uniform u-regularity along Λ, meaning that the uniform size of unstable manifolds holds only for sequences of saddle points in Λ, and likewise for s-regularity. Then there are statements analogous to Theorems 2.9, 2.13, 2.14 and 2.19, in which uniform regularity is replaced by uniform regularity along Λ, and the conclusion is that Λ is a hyperbolic set.

A topological criterion for hyperbolicity
In this section we work in the setting of Conjecture 1.4: We assume that f 0 and f 1 are two complex Hénon maps such that f 0 is hyperbolic, and that there exist respective neighborhoods N 0 and N 1 of J 0 " J › 0 and J › 1 and a conjugating homeomorphism φ : N 0 Ñ N 1 . Our purpose is to show that f 1 is hyperbolic on J › 1 .
Proposition 3.1. Let f 0 and f 1 be as in Conjecture 1.4. Then φpJ 0 q " J › 1 . If f 1 is dissipative then all periodic points of f 1 on J › 1 are saddles. If f 1 is conservative the same holds provided φ is Hölder continuous.
Proof. The first assertion is a direct consequence of the equidistribution of periodic orbits. Indeed the topological conjugacy shows that f 0 and f 1 have the same entropy, hence the same dynamical degree. Since periodic orbits equidistribute towards the maximal entropy measure, we get that φ › µ 0 " µ 1 . Since Supppµ 1 q " J › 1 , we infer that φpJ 0 q " J › 1 . (On the other hand it is unclear at this stage whether J › 1 " J 1 .) Any periodic point on J › 1 admits a neighborhood in which it is topologically conjugate to a saddle. Let p P J › 1 be some periodic point which we may suppose fixed. Assume that f 1 is dissipative. Then if p is not a saddle it is semi-attracting. By the hedgehog theory of [FLRT, LRT] there exists in some neighborhood of p a non-trivial totally invariant set H made of points which do not converge to p under backward nor forward iteration: indeed there is a subsequence q n such that f qn Ñ id on H. This is not compatible with the local conjugacy to a saddle fixed point, therefore we conclude that p is a saddle.
If f is conservative and p is not a saddle, then it is neutral. Since φ is Hölder, then points in φpW s loc pφ´1ppqqq converge to the origin exponentially fast. On the other hand for every ε ą 0 there exists a norm on C 2 for which p1´εq }v} ď }df p pvq} ď p1`εq }v} .
Indeed if df p is diagonalizable this is clear since the eigenvalues have modulus 1, and otherwise we can make df p triangular with the off-diagonal term as small as we wish, and take an adapted norm. Then if x is close to p and f n 1 pxq Ñ p we infer that }f n pxq´f n ppq} ě p1´2εq n }x´p} which is contradictory if ε is small enough. Thus again we conclude that all periodic points on J › 1 are saddles.
For a saddle point p, we now denote by W s r ppq the component of W s ppq X Bpp, rq containing p. We fix r 0 such that for every x P J 0 , W s r0 pxq (resp. W u r0 pxq) is a properly embedded holomorphic disk with the property that there exist uniform C ą 0 and 0 ă λ ă 1 such that for every x 1 P W s r0 pxq (resp. W u r0 pxq), distpf n pxq, f n px 1 qq ď Cλ |n| when n Ñ 8 (resp n Ñ´8). We also assume for further reference that f has product structure in the 2r 0 -neighborhood of J › 0 . Proposition 3.2. Let f 0 and f 1 be as in Conjecture 1.4. There exists r 1 ą 0 such that for any saddle periodic point p for f 1 , W s r1 ppq (resp. W u r1 ppq) is a submanifold of Bpp, r 1 q.
Proof. Without loss of generality we treat the case of stable manifolds and assume that p is fixed. If r 0 is as above there exists r 1 ą 0 such that for every y P J › 1 , Bpy, r 1 q Ť φpBpφ´1pyq, r 0 qq. We claim that for every saddle fixed point p for f 1 , The right inclusion is obvious since belonging to W s ppq is characterized by the topological property that f n pyq Ñ p. For the left inclusion, just observe that φ´1`W s r1 ppq˘is a connected subset of W s pφ´1ppqq X Bpφ´1ppq, r 0 q containing p, hence it is contained in W s r0 pφ´1ppqq. To show that it is properly embedded, we first observe that there exists r " rppq such that W s r ppq is properly embedded in Bpp, rq. By the invariance of domain theorem φ´1pW s r ppqq is a neighborhood of φ´1ppq in W s pφ´1ppqq. Thus it follows that there exists n " nppq such that f n 0 pW s r0 pφ´1ppqqq Ă φ´1pW s r ppqq. Then from (3) we get that f n 1 pW s r1 ppqq Ă W s r ppq, so W s r1 ppq Ă f´n 1 W s r1 ppq. From this we conclude that W s r1 ppq is properly embedded in Bpp, r 1 q, as desired.
Remark 3.3. At this stage we know that stable manifolds are properly embedded in a ball of uniform size, but since in the last argument the quantities n and r are a priori not uniform in p, we have no uniformity for the geometry of W s r1 ppq. Obtaining such a uniformity will be the purpose of the forthcoming arguments.
Lemma 3.4 (Tubular neighborhood lemma). If ∆ is a subvariety in Bp0, 2rq of size r at 0 then there exists η " ηprq such that if ∆ 1 is a subvariety in Bp0, 2rq such that d H p∆, ∆ 1 q ă η in Bp0, 2rq, then ∆ 1 is a branched cover over ∆ in Bp0, r{2q. (Here d H denotes the Hausdorff distance.) Proof. After a unitary change of coordinates, ∆ is a graph y " ψpxq of slope at most 1 in the bidisk Dp0, rq 2 . Since ψ 1 p0q " 0 by the Schwarz lemma we have |ψ 1 pxq| ď |x| {r so actually |ψpxq| ď r{2. It follows that if η ă r{4 and d H p∆, ∆ 1 q ă η in Dp0, rq 2 , ∆ 1 is horizontal in this bidisk. Thus it is a branched cover over the first coordinate, hence over ∆. The "tube argument" alluded to in the title consists in applying the previous lemma to construct invariant laminations in the setting of the conjecture. Here is a sample statement. Proof. Let p be as in the statement of the proposition, and assume that W s ppq has size r at p. Reducing r if necessary, we may assume that φ´1pW s r ppqq is contained in a flow box of the stable lamination of f 0 . Without loss of generality we may also assume that r ă r 1 {4, where r 1 is as in Proposition 3.2. We will show that there exists a neighborhood V of p such that if q P V is another periodic point, then W s r pqq is a graph over W s ppq in Bp0, r{2q. This implies that p is uniformly s-regular. Uniform u-regularity is proven in the same way, and the transversality property is obvious since p is a saddle. We know that for any saddle point q P J › 1 , the stable manifold W s r1 pqq is properly embedded in Bpq, r 1 q. In addition, by (3) it is contained in φpW s r0 pφ´1pqqqq. By the uniform continuity of φ there exists a neighborhood V of p such that for every saddle point q P V , W s r pqq is η-close to W s r ppq in Bpp, 2rq, where η is as in the tubular neighborhood lemma. Thus W s r pqq is a branched cover over W s r ppq in Bpp, r{2q, and to conclude the proof it remains to show that this cover has degree 1. By the product structure of f 0 in pJ › 0 q 2r0 we have that W s r0 pφ´1ppqq X W u r0 pφ´1ppqq " φ´1ppq ( hence W s r1 ppq X W u r1 ppq " tpu. Thus, reducing η if necessary, to compute the degree of this branched cover it is enough to count the number of intersection points, with multiplicity, between W s r pqq and W u r1 ppq. Applying the product structure again we get that W s r pqq X W u r1 ppq is a single point. Furthermore it is well-known that the order of contact between two smooth complex curves in C 2 is a topological invariant. Indeed if we consider two smooth curves C and D with an isolated intersection at 0 P C 2 and intersect them with a small sphere S about 0, then C X S winds n times about D X S where n is the intersection multiplicity. So we conclude that the intersection W s r pqq X W u r1 ppq is transverse and we are done.
3.3. Proof of Conjecture 1.4 in the dissipative case.
Theorem 3.6. Let f 0 and f 1 be two polynomial automorphisms of C 2 with non-trivial dynamics, and assume that f 0 is hyperbolic and that f 1 is dissipative. Suppose that there exists respective neighborhoods N 0 and N 1 of J 0 " J › 0 and J › 1 and a homeomorphism φ : N 0 Ñ N 1 such that φ˝f 0 " f 1˝φ where these compositions makes sense. Then f 1 is hyperbolic.
To prove the theorem, let W s 0 and W u 0 be the stable and unstable laminations in N 0 and L s 1 and L u 1 be their respective images under φ. At this stage L s 1 and L u 1 are topological laminations by topological disks in N 1 .
Define Ω to be the set of points x P J › 1 such that there exists a neighborhood V of x in J › 1 such that for every y P V , L s 1 pyq and L u 1 pyq are holomorphic and of uniform size in V . Note that they must be transverse by the topological invariance of the order of contact between smooth curves. By construction, Ω is open in J › 1 and completely invariant (i.e. f pΩq " Ω). Proposition 3.5 shows that Ω contains all saddle points.
The main step of the proof is the following: Lemma 3.7. Let f 0 and f 1 be as in Theorem 3.6. Then any invariant measure supported on J › 1 gives full mass to Ω.
Theorem 3.6 follows easily. Indeed, if non-empty, the complement of Ω in J › 1 is a closed invariant set hence if it is non-empty it supports an invariant measure ν. By the lemma, νpΩq " 1 hence the contradiction. Therefore we conclude that Ω " J › 1 , in particular all points in J › 1 are uniformly regular, and the result follows from Theorem 2.9.
Proof of Lemma 3.7. The method is to adapt the "tube argument" to Pesin stable manifolds. Under the assumptions of the theorem, let ν be any invariant measure for f 1 supported on J › 1 . Then by Oseledets' Theorem for ν-a.e. x there exist Lyapunov exponents χ 1 pxq ď χ 2 pxq satisfying χ 1 pxq`χ 2 pxq " log |Jacpf q|. In addition since ν is not concentrated on a periodic orbit we have χ 2 pxq ě 0 a.e. hence χ 1 pxq ă 0 since |Jacpf q| ă 1. By the Pesin stable manifold theorem, for ν-a.e. x, there exists a local stable manifold W s loc pxq which can characterized as the set of points y sufficiently close to x such that lim sup 1 n log distpf n 1 pyq, f n 1 pxqq ă 0. Pick any point x such that W s loc pxq exists. We will show that both L s 1 and L u 1 are laminations by Riemann surfaces near x.
Observe first that by hyperbolicity of f 0 , the local stable manifold of φ´1pxq is the set of points z near φ´1pxq such that distpf n 0 pzq, f n 0 pφ´1pxqqq Ñ 0 as n Ñ`8. Hence φ´1pW s loc pxqq Ă W s loc pφ´1pxqq. Since φ´1 is continuous and injective, by the invariance of domain theorem, φ´1pW s loc pxqq is neighborhood of φ´1pxq in W s loc pφ´1pxqq. Thus W s loc pxq coincides with L s 1 pxq in a neighborhood of x.
Let r be so small that W s loc pxq has size r at x and W s loc pxq " L s 1 pxq in Bpx, 2rq. Then after a unitary change of coordinates as in Lemma 3.4, W s loc pxq is a graph of the form y " ψpxq over Dp0, rq. Denote this graph by ∆ s r . For small η ą 0, we define Tub η " Tub η p∆ s r q " tpx, yq, |x| ă r, |y´ψpxq| ă ηu . We say that a submanifold M of Tub η (which extends to some neighborhood of Tub η ) is horizontal if M X BTub η Ă px, yq P Tub η , |x| " r ( and similarly it is vertical if M X BTub η X t|x| " ru " H. As already observed, if M is horizontal it is a branched covering over the first coordinate, and similarly if it is vertical the restriction of px, yq Þ Ñ y´ψpxq to M X Tub η is a branched covering over Dp0, ηq. Exactly as in Proposition 3.5, if q P J › is a saddle point sufficiently close to x, W s loc pqq is horizontal in Tub η . Now by the transversality of W s 0 and W u 0 , there exists a neighborhood N of φ´1pxq such that for any z P N , the distance between W u 0 pzq and φ´1pB∆ s r q is bounded from below by a uniform positive constant. By continuity of φ, for any y " φpzq P φpN q and reducing η if necessary we get that distpB∆ s r , L u 1 pyqq ą 2η. By Proposition 3.2, if q is a saddle point close to x, W u r1 pqq " L u 1 pqq X Bpq, r 1 q is a submanifold in Bpq, r 1 q for a uniform r 1 (which we may assume to be large with respect to r and η). So we conclude that it is a submanifold in a neighborhood of Tub η , which must be vertical in Tub η . Thus we have shown that if q is a saddle periodic point sufficiently close to x, the local stable and unstable manifolds of q are respectively horizontal and vertical in Tub η , with a single transverse intersection point (for transversality again we use the topological invariance of the order of contact). Hence both have covering degree 1 respectively over the horizontal and vertical directions in Tub η , i.e. they are graphs. Then by the Schwarz Lemma they have uniformly bounded geometry. So we conclude that x belongs to Ω, and the proof is complete.
3.4. The conservative case. In the conservative case we can only prove Conjecture 1.4 in the case of a Hölder conjugacy.
Theorem 3.8. Let f 0 and f 1 be two polynomial automorphisms of C 2 with non-trivial dynamics, and assume that f 0 is hyperbolic and f 1 is conservative.
Suppose that there exists respective neighborhoods N 0 and N 1 of J 0 " J › 0 and J › 1 and a Hölder continuous homeomorphism φ : N 0 Ñ N 1 such that φ˝f 0 " f 1˝φ where these compositions makes sense. Then f 1 is hyperbolic.
The proof is identical to that of Theorem 3.8, the only difference is that in Lemma 3.7 we need a different argument to show that any ergodic invariant measure ν for f 0 admits a negative Lyapunov exponent (this issue already appeared in the proof of Proposition 3.1). So Theorem 3.8 follows from: Proof. This follows from standard Pesin-theoretic considerations. Let ν be an invariant measure for f 1 . Without loss of generality we can assume that ν is ergodic so it admits two Lyapunov exponents χ 1 ď χ 2 with χ 1`χ2 " 0. Assume by way of contradiction that χ 1 " χ 2 " 0. The Oseledets-Pesin reduction theorem (see [KM,Thm. S.2.10], note that it does not require ν to be hyperbolic) asserts that for every ε ą 0 there exists a measurable cocycle C ε with values in GL 2 pCq such that for ν-a.e. x, the matrix A ε pxq :" C ε pf 1 pxqq´1¨pDf 1 q x¨Cε pxq satisfies e´ε ď }A ε pxq} ď e ε and e´ε ď › › pA ε pxqq´1 › › ď e ε . Then, the Pesin theorem on existence of regular neighborhoods (see [KM,Thm. S.3.1]) implies that there is a measurable function q such that for ν-a.e. x, f behaves likes pDf 1 q x on Bpx, qpxqq and furthermore e´ε ă qpf 1 pxqq{qpxq ă e ε . More precisely there is a change of coordinates Ψ x defined on Bpx, qpxqq such that Ψ f1pxq˝f˝Ψx is ε C 1 -close to its differential at x, which equals A ε pxq. Now by the Hölder conjugacy to f 0 , for every x there exists y close to x such that distpf n 1 pyq, f n 1 pxqq decreases like e´α n for some α ą 0. If we pick ε small as compared to α, then for generic x we have that f n 1 pyq P Bpf n pxq, qpf n pxqq for every large n. It follows that for large k distpf n`k 1 pyq, f n`k 1 pxqq ě Ce´2 εk distpf n 1 pyq, f n 1 pxqq, which is contradictory, and the proof is complete.