J I Zhuchao

NON-UNIFORM HYPERBOLICITY IN POLYNOMIAL SKEW PRODUCTS

Keywords: May 27, 2021. 2010 Mathematics Subject Classification. 37F10, 37F15, 37F50, 32H50 Non-uniform hyperbolicity, Topological Collet-Eckmann condition, polynomial skew product, Fatou component, non-wandering domain theorem

Let f : C 2 → C 2 be a polynomial skew product which leaves invariant an attracting vertical line L. Assume moreover f restricted to L is non-uniformly hyperbolic, in the sense that f restricted to L satisfies one of the following conditions: 1. f |L satisfies Topological Collet-Eckmann and Weak Regularity conditions. 2. The Lyapunov exponent at every critical value point lying in the Julia set of f |L exist and is positive, and there is no parabolic cycle. Under one of the above conditions we show that the Fatou set in the basin of L coincides with the union of the basins of attracting cycles, and the Julia set in the basin of L has Lebesgue measure zero. As an easy consequence there are no wandering Fatou components in the basin of L.

1. Introduction 1.1. Background. In one-dimensional complex dynamics, i.e. in the theory of dynamics of rational maps on Riemann sphere P 1 , the classical Fatou-Julia dichotomy partitions the Riemann sphere into the Fatou set and the Julia set. Let f be a rational map on P 1 , the Fatou set F (f ) is defined as the largest open subset of P 1 in which the sequence of iterates (f n ) n≥0 is normal. Its complement is the Julia set J(f ). A Fatou component is a connected component of F (f ). A Fatou component is called wandering if it is not pre-periodic. One can show that the Fatou set is either empty or an open and dense subset. The dynamics on the Fatou set is completely understood, due to the work of Fatou, Julia, Siegel and Herman, supplemented with Sullivan's non-wandering domain theorem [START_REF] Sullivan | Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia problem on wandering domains[END_REF]: the orbit of any point in the Fatou set eventually lands in an attracting basin, a parabolic basin, a Siegel disk or a Herman ring. See Milnor [START_REF] Milnor | Dynamics in One Complex Variable[END_REF] for a self-contained proof.

If in addition f satisfies some non-uniformly hyperbolic conditions, the measurable dynamics of f can also be understood. There are various hyperbolic conditions, such as uniform hyperbolicity, sub-hyperbolicity, the Collet-Eckmann condition (CE for short), the Topological Collet-Eckmann condition (TCE for short), the condition that Lyapunov exponent at all critical values exist and is positive, and f has no parabolic cycles (Positive Lyapunov for short), the Weak regularity condition (WR for short), etc. We say that f satisfies TCE if there is an "Exponential shrinking of components" on the Julia set, see the precise definition in Definition 2.6.

Theorem. (Przytycki,Smirnov [38,Theorem 4.3]) Let f be a TCE rational map on P 1 with degree at least 2 and such that J(f ) = P 1 . Then the Fatou set F (f ) is equal to the union of a finite number of attracting basins, and the Julia set J(f ) has Hausdorff dimension strictly smaller than 2 (hence it has area zero).

In higher dimensional complex dynamics, one of the major problems is to study the dynamics of holomorphic endomorphisms of P k , k ≥ 2. The Fatou and Julia sets can be defined similarly. Unlike the one-dimensional case, little is known about the dynamics on the Fatou set in higher dimension. It is known that Sullivan's non-wandering domain theorem does not hold in general. Indeed Astorg, Buff, Dujardin, Peters and Raissy [START_REF] Astorg | A two-dimensional polynomial mapping with a wandering Fatou component[END_REF] constructed a holomorphic endomorphism h : P 2 -→ P 2 induced by a polynomial skew product, possessing a wandering Fatou component.

A polynomial skew product f is a polynomial map from C 2 to C 2 , of the following form:

f (t, z) = (g(t), h(t, z)),
where g is a one variable polynomial and h is a two variables polynomial. We assume that g and h have degree at least 2. In the rest of the paper a polynomial map or a rational map is asked to have degree at least 2. See Jonsson [START_REF] Jonsson | Dynamics of polynomial skew products on C 2[END_REF] for a systematic study of such polynomial skew products, see also Dujardin [START_REF] Dujardin | A non-laminar dynamical Green current[END_REF], Astorg and Bianchi [START_REF] Astorg | Bifurcations in families of polynomial skew products[END_REF], Boc-Thaler, Fornaess and Peters [START_REF] Boc-Thaler | Fatou components with punctured limit sets[END_REF] for related studies. As the definition suggests, the polynomial skew product leaves invariant a foliation by vertical lines, hence one-dimensional tools can be used. Our first purpose is to study the dynamics of a polynomial skew product on its Fatou set.

We assume h has the expression h(t, z) = i+j≤n a i,j t i z j .

If in addition we assume the polynomial skew product f satisfies deg g = deg h = n, and a 0,n = 0, then f extends to P 2 holomorphically. In this case the polynomial skew product is called regular. The regular polynomial skew products form a sub-class of holomorphic endomorphisms on P 2 .

To investigate the Fatou set of f , let π 1 be the projection to the t-coordinate, i.e.

π 1 : C 2 → C, π 1 (t, z) = t.
We first notice that π 1 (F (f )) ⊂ F (g), and passing to some iterate of f , we may assume that the points in F (g) will eventually land into an immediate basin or a Siegel disk (no Herman rings for polynomials), thus we only need to study the following semi-local case:

(1.1) f = (g, h) : ∆ × C → ∆ × C,
where g(0) = 0 which means the line L : {t = 0} is invariant and ∆ is an immediate attracting or a parabolic basin or a Siegel disk of g. The map f is called attracting, parabolic or elliptic respectively when g ′ (0) is attracting, parabolic, elliptic. The examples of wandering domains constructed in [START_REF] Astorg | A two-dimensional polynomial mapping with a wandering Fatou component[END_REF] are parabolic polynomial skew products. At this stage it remains an interesting problem to investigate the existence of wandering domains for attracting polynomial skew products, one part of our main theorem answer this question in the negative way under the non-uniformly hyperbolic condition.

In the geometrically attracting case, by Koenigs' Theorem, (1.1) is locally conjugated to (1.2) f (t, z) = (λt, h(t, z)),

where λ = g ′ (0). Beware that h is no longer a polynomial in t.

In the super-attracting case by Böttcher's Theorem, (1.1) is locally conjugated to f (t, z) = (t m , h(t, z)), m ≥ 2.

In both attracting or super-attracting cases

h(t, z) = a 0 (t) + a 1 (t)z + • • • + a d (t)z d
is a polynomial in z with coefficients a i (t) holomorphic in t in a neighborhood of 0. We furthermore assume that a d (0) = 0, which means the degree of h(t, z) in z is constant for t ∈ ∆. This condition is needed in the proof of the main theorem, and it is satisfied for regular polynomial skew products. In the rest of the paper, an attracting polynomial skew product is assumed to have the normal form (1.2), and ∆ denotes a small disk centered at 0.

1.2. Main theorem and outline of the proof. In this paper we show that under the nonuniformly hyperbolic hypothesis, we can exclude the existence of wandering domain, and give a classification of the dynamics on the Fatou set, and show that the Julia set has Lebesgue measure zero.

Theorem. (Main Theorem) Let f be an attracting polynomial skew product, let p = f | L be the restriction of f on the invariant fiber L. Assume that p satisfies one of the following conditions: 1. p satisfies TCE and WR. 2. p satisfies Positive Lyapunov. Then the Fatou set of f coincides with the union of the basins of attracting cycles, and the Julia set of f has Lebesgue measure zero.

We let ∞ be the point at infinity of L. Since ∞ can be seen as an attracting fixed point, the Fatou set of f is never empty. The definitions of TCE condition, WR condition and Positive Lyapunov condition are given in section 2. The basins of attracting cycles are clearly non-wandering, as a consequence there are no wandering domains in the basin of L. In the rest of the paper we shall prove the main theorem for f geometrically attracting. In the super-attracting case the proof is completely similar, and left to the reader. Note that in that case the non existence wandering Fatou components was established by Lilov [START_REF] Lilov | Fatou theory in two dimensions[END_REF] (see also [START_REF] Zhuchao | Non-wandering fatou components for strongly attracting polynomial skew products[END_REF]).

The proof of the main theorem is divided into several steps. In section 2 we recall some preliminaries, and introduce some one-dimensional techniques such as the Koebe distortion lemma, Przytycki's lemma and Denker-Przytycki-Urbanski's lemma (DPU lemma for short). The one-dimensional non-uniformly hyperbolic theory is also introduced. Then proof of the main theorem goes as follows.

Step 1: We start with some definitions. Let f : ∆ × C → ∆ × C be an attracting polynomial skew product, the critical set of f is defined by Crit := (t, z) ∈ ∆ × C | ∂h ∂z (t, z) = 0 . We let the radius of ∆ be small enough so that each connected component of Crit intersect L at exactly one point. We define Crit' := {the union of the connected components of Crit that intersect J(p)} . Definition 1.1. Let x ∈ ∆ × C be a point in the immediate basin of L, we say that x slowly approach Crit' if for every α > 0, dist v (f n (x), Crit') ≥ e -αn for every sufficiently large n.

Here dist v denote the vertical distance which means that dist v (x, y) = |π 2 (x)π 2 (y)|, where π 2 is the projection to the z-coordinate, i.e.

π 2 : C 2 → C, π 2 (t, z) = z. Let also D v (x, r) denote the vertical disk, D v (x, r) = {y ∈ ∆ × C : π 1 (y) = π 1 (x), dist v (x, y) < r}.
This notion of slow approach was introduced by Levin, Przytycki and Shen in one-dimensional complex dynamics in [START_REF] Levin | The Lyapunov exponent of holomorphic maps[END_REF]. Next we show that Theorem 1.2. Lebesgue a.e. x ∈ ∆ × C slowly approach Crit'. This is proved in sections 3 and 4. In section 3 the existence of a stable manifold at each critical value in J(p) and the properties of renormalization maps associated with a critical value variety are studied. In section 4 we use the techniques developed in section 3 to prove Theorem 1.2.

Step 1 is where we need the TCE and WR conditions or the Positive Lyapunov condition, in the remaining steps the TCE condition alone is sufficient for the proof.

Step 2: We define vertical Lyapunov exponent at one point as follows: Definition 1.3. Let x ∈ ∆ × C be a point in the immediate basin of L, the lower vertical Lyapunov exponent is defined by

χ -(x) := lim inf n→∞ 1 n log |Df n | x (v)|.
Where v = (0, 1) is the unit vertical tangent vector.

It is well-known that the one-dimensional attracting basins of p extend to two-dimensional attracting basins, for example see [START_REF] Lilov | Fatou theory in two dimensions[END_REF] or [START_REF] Zhuchao | Non-wandering fatou components for strongly attracting polynomial skew products[END_REF]Section 3]. These two-dimensional attracting basins correspond to non-wandering Fatou components. We let W s (J(p)) denote the stable set of J(p),

W s (J(p)) := x ∈ ∆ × C : lim n→∞ dist(f n (x), J(p)) = 0 .
It is easy to see that assuming p satisfies TCE, W s (J(p)) is the union of the wandering Fatou components together with the Julia set J(f ). We show that Theorem 1.4. If x ∈ W s (J(p)) slowly approach Crit', then χ -(x) ≥ log µ Exp , where µ Exp > 1 is the constant appearing in the definition of the TCE condition. This is proved in section 5. Then by Theorem 1.2 Lebesgue a.e. point x ∈ W s (J(p)) satisfies χ -(x) ≥ log µ Exp . This already implies the non-existence of wandering Fatou component thanks to the fact that points in Fatou set can not have positive Lyapunov exponent. Thus W s (J(p)) coincide with the Julia set J(f ).

Step 3: Finally by using an adaption of a so called "telescope argument" in [27, Theorem 1.5], we show that Theorem 1.5. The Julia set J(f ) has Lebesgue measure zero. This is proved in section 6, and the proof is complete.

In Appendix A we study the relations between TCE condition, CE condition, WR condition and Positive Lyapunov condition. In Appendix B we exhibit some families of polynomial maps satisfying the conditions in the main theorem. 1.3. Previous results. The first result of non-wandering domain theorem for polynomial skew products goes back to Lilov [START_REF] Lilov | Fatou theory in two dimensions[END_REF]. In his PhD thesis Lilov proved that super-attracting polynomial skew products do not have wandering Fatou components. He actually showed a stronger result, namely that there can not have vertical wandering Fatou disks.

In the geometrically attracting case, there are many works trying to understand the dynamics in the atrracting basin of the invariant line. Peters and Vivas showed in [START_REF] Peters | Polynomial skew-products with wandering Fatou-disks[END_REF] that there is an attracting polynomial skew product with a wandering vertical Fatou disk. This result does not answer the existence question of wandering Fatou components, but showed that the question is considerably more complicated than in the super-attracting case. On the other hand, by using a different strategy from Lilov's, Peters and Smit in [START_REF] Peters | Fatou components of attracting skew-products[END_REF] showed that the non-wandering domain theorem holds in the attracting case, under the assumption that the dynamics on the invariant fiber is sub-hyperbolic. The author showed that the non-wandering domain theorem holds in the attracting case, under the assumption that the multiplier is sufficiently small, following Lilov's strategy, see [START_REF] Zhuchao | Non-wandering fatou components for strongly attracting polynomial skew products[END_REF].

In the parabolic case, the examples of wandering domains are constructed in [START_REF] Astorg | A two-dimensional polynomial mapping with a wandering Fatou component[END_REF], as we have mentioned. See also the recent paper [START_REF] Astorg | Wandering domains arising from Lavaurs maps with Siegel disks[END_REF] and [START_REF] Hahn | A polynomial automorphism with a wandering Fatou component[END_REF].

The elliptic case was studied by Peters and Raissy in [START_REF] Peters | Fatou components of polynomial elliptic skew products[END_REF]. See Raissy [START_REF] Raissy | Polynomial skew-products in dimension 2: Bulging and Wandering Fatou components[END_REF] for a survey of the history of the investigation of wandering domains for polynomial skew products.

To the best of our knowledge, Theorem 1.5 is the first time where the zero measure of Julia set is shown for non-hyperbolic p (it is in general not true when no conditions of p are assumed, as even in one dimension Julia set can has positive Lebesgue measure, cf. [START_REF] Buff | Quadratic Julia sets with positive area[END_REF] and [START_REF] Avila | Lebesgue measure of Feigenbaum Julia sets[END_REF]). The previous results we have mentioned only consider the dynamics on the Fatou set. However when p is uniformly hyperbolic, it is well-known that the Julia set has zero measure and the Fatou set coincide with the union of the basins of attracting cycles. In fact the stable set of W s (J(p)) is foliated by stable manifolds, and this foliation is absolutely continuous hence W s (J(p)) has Lebesgue measure zero.

Acknowledgements. I would like to thank my adviser Romain Dujardin for his advice, help and encouragement during the course of this work. I also would like to thank Jacek Graczyk for useful discussion, Theorem B.1 is due to him.

Preliminaries

In this section we introduce some one-dimensional tools used in the proof of the main theorem. Let f : P 1 → P 1 be a rational map , let Crit denote the set of critical points, and let Crit' denote the set of critical points lie in the Julia set. Let CV (f ) be the critical value set. We fix a Riemannian metric on P 1 , and D(x, ε) denotes a small disk centered at x with radius ε.

2.1. Some technical lemmas. In [35, Lemma 1], Przytycki introduced a fundamental lemma which concerns the recurrence properties of small neighborhood of Crit'. Lemma 2.1 (Przytycki). Let c ∈ Crit'. There exist a constant C > 0 such that for every ε > 0 and n > 0, if

f n (D(c, ε)) ∩ D(c, ε) = ∅, then n ≥ C log 1
ε . We define a positive valued function on P 1 as follows Definition 2.2. Let c ∈ Crit ′ , define a positive valued function φ c (x) by

φ c (x) := -log dist (x, c), if x = c ∞, if x = c.
In terms of using the function φ c (x), the Lemma 2.1 can be reformulated as: there exists a constant Q > 0, such that for every x ∈ P 1 , c ∈ Crit', n ≥ 1, we have min (φ c (x), φ c (f n (x))) ≤ Qn.

In a later paper by Denker, Przytycki, Urbanski [15, Lemma 2.3], Lemma 2.1 was generalized as the following DPU lemma. We let φ(x) := max c∈Crit' φ c (x).

Lemma 2.3 (Denker-Przytycki-Urbanski

). There exist a constant Q > 0 such that for all n ≥ 0 we have

n-1 j=0 except M terms φ(f j (x)) ≤ Qn,
where the summation over all but at most M = #Crit' indices.

Note that the original statement differs slightly. This formulation also appeared in [START_REF] Zhuchao | Non-wandering fatou components for strongly attracting polynomial skew products[END_REF]Lemma 2.5]. Lemma 2.1 will be used in section 4 and Lemma 2.3 will be used in section 6.

Next we introduce a version of the Koebe distortion lemma for multivalent maps. We refer to [START_REF] Przytycki | Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: on non-renormalizable quadratic polynomials[END_REF]Lemma 1.4] and [39, Lemma 2.1] for more details. Consider a disk D(x, δ) of radius δ centered at x, let W be a connected component of f -n (D(x, δ)), assume that f n restricted to W is D-critical, that is f n has at most D critical points counted with multiplicity. Then f n | W has distortion properties similar to univalent maps. In the following we assume δ smaller than diam P 1 /2. Lemma 2.4. For each ε > 0 and D < ∞ there are constants C 1 (ε, D) > 0 and C 2 (ε, D) > 0 such that the following holds.

Let D(x, δ) denotes the ball in P 1 centered at x with radius δ. Assume that W is a simply connected domain in P 1 and F : W → D(x, δ) is a proper holomorphic map. Let W ′ ⊂ W be a connected component of F -1 (D(x, δ/2)). Assume further that P 1 \ W contains a disk of radius ε and that F is D-critical on W . Then for every y ∈ W ′ , (2.1)

|F ′ (y)| diam (W ′ ) ≤ C 1 δ.
In addition W ′ contains a disk B of radius r around every pre-image of

F -1 (x) contained in W ′ , with (2.2) r ≥ C 2 diam(W ′ ). Assume further that W ′′ is a connected component of F -1 (B ′ ), where B ′ ⊂ D(x, δ/2) is a disk, then there exist a constant C 3 (ε, D) > 0 such that (2.3) diam W ′′ diam W ′ ≤ C 3 diam(B ′ ) δ 2 -D .
Finally if R is a measurable subset of D(x, δ/2), there exist a constant C 4 (ε, D) > 0 such that

(2.4) meas F -1 (R) ∩ W ′ meas W ′ ≤ C 4 meas(R) δ 2 2 -D .
where meas denotes the Lebesgue measure induced by the Riemannian metric on P 1 .

Note that we will use this lemma only for F being a polynomial, so the assumption that W is simply connected and P 1 \ W contains a disk are automatically satisfied.

Proof. The inequalities (2.1) and (2.2) were proved in [39, Lemma 2.1] and the inequality (2.3) was proved in [START_REF] Przytycki | Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: on non-renormalizable quadratic polynomials[END_REF]Lemma 1.4]. Here we prove inequality (2.4).

By the Riemann mapping theorem there is a surjective univalent map ψ : D(0, 1) → W . We consider the composition F • ψ : D(0, 1) → D(x, δ). By the classical Koebe distortion lemma for univalent maps, (2.4) is true for D = 0. To prove (2.4), it is sufficient to prove the following: Let G : D(0, 1) → D(0, 1) be a degree D Blaschke product on the unit disk, there exist a constant C 4 (D) > 0 such that if R is a measurable set of D(0, 1/2), then

(2.5) meas G -1 (R) ≤ C 4 meas(R) 2 -D .
We let a i ∈ D(0, 1) be the critical points of G, 1 ≤ i ≤ n. Thus we have n ≤ D. We denote A = meas(R). It is sufficient to prove (2.5) for A small. For small ε we cover R by ε-disks such that N πε 2 ≤ 2A, where N is the number of disks in the covering. For δ > 0 small there exist a uniform constant M such that dist(G(x), G(a i )) ≥ δ for every i and G(x) ∈ D(0, 3/4) imply

|G ′ (x)| ≥ M δ 1/2 . If an ε-disk D ε is disjoint from the union n i=1 D(G(a i ), A 1/2
), then by the change of variable formula we have meas G -1 (D ε ) min

x∈G -1 (Dε) |G ′ (x)| 2 ≤ Dπε 2 . From min x∈G -1 (Dε) |G ′ (x)| 2 ≥ M 2 A 1/2 we get meas G -1 (D ε ) ≤ Dπε 2 M 2 A 1/2 .
Let B 1 be the union of all ε-disks disjoint from n i=1 D(G(a i ), A 1/2 ). Then we have meas

G -1 (B 1 ) ≤ Dε∩( n i=1 D(G(a i ),A 1/2 ))=∅ meas G -1 (D ε ) ≤ N Dπε 2 M 2 A 1/2 ≤ 2DA 1/2 M 2 .
Let B 2,i be the union of all ε-disks not disjoint from D(G(a i ),

A 1/2 ), for 1 ≤ i ≤ n. Let B 2 = n i=1 B 2,i . By (2.3) for small ε (ε ≤ A 1/2 , say), there is a constant C 3 (D) > 0 such that meas G -1 (D(G(a i ), A 1/2 + ε)) ≤ C 3 A 2 -D .
Thus we have meas

G -1 (B 2 ) ≤ n i=1 meas G -1 (B 2,i ) ≤ n i=1 meas G -1 (D(G(a i ), A 1/2 + ε)) ≤ DC 3 A 2 -D .
The last inequality holds since n ≤ D. Finally we have meas

G -1 (R) ≤ meas G -1 (B 1 ) + meas G -1 (B 2 ) ≤ 2DA 1/2 M 2 + DC 3 A 2 -D . Setting C 4 := DC 3 + 2D/M 2 the proof is complete.
Lemma 2.4 will be used frequently in the rest of the paper.

One-dimensional non-uniformly hyperbolic theory.

A rational map f is uniformly hyperbolic if f expands a Riemannian metric on a neighborhood of J(f ). This is equivalent to Smale's Axiom A, and is equivalent to the condition that the closure of the post critical set P C(f ) is disjoint from J(f ). The measurable dynamics of f is well-understood: the Fatou set is the union of finitely attracting basins, the Hausdorff dimension of J(f ) is equal to the Minkowski dimension of J(f ) and is smaller than 2, and there is a unique invariant probability measure µ such that supp(µ) = J(f ) which is absolutely continuous with respect to the δ-dimensional Hausdorff measure (δ is the Hausdorff dimension of J(f )). It can be shown that µ is mixing (hence ergodic) and has positive entropy. It is widely conjectured that uniformly hyperbolic maps are dense in the parameter space of fixed degree. This is known as Fatou conjecture and is a central problem in one-dimensional complex dynamics. Many weaker notions such as sub-hyperbolicity, semi-hyperbolicity have been defined. See [START_REF] Milnor | Dynamics in One Complex Variable[END_REF]Section 19], [13] for more details.

Non-uniformly hyperbolic theory, also known as Pesin theory, is a generalization of uniformly hyperbolic theory. In Pesin theory we only require an invariant hyperbolic measure rather than the presence of invariant expanding and contracting directions. In this subsection we introduce some strong notions of non-uniform hyperbolicity in one-dimensional complex dynamics.

Definition 2.5. A rational map f satisfies CE if there exists µ CE > 1 and C > 0 such that for every point c ∈ Crit' whose forward orbit does not meet other critical points, and every n ≥ 0 we have

|(f n ) ′ (f (c))| ≥ Cµ n CE .
In addition we ask that there are no parabolic cycles.

The CE condition was first introduced by Collet and Eckmann in [START_REF] Collet | Positive Liapunov exponents and absolute continuity for maps of the interval[END_REF] for S-unimodal maps of an interval. The CE condition was introduced in complex dynamics by Przytycki in [START_REF] Przytycki | Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: on non-renormalizable quadratic polynomials[END_REF]. The TCE condition was first introduced by Przytycki and Rohde in [START_REF] Przytycki | Porosity of Collet-Eckmann Julia sets[END_REF], as a generalization of the CE condition. Definition 2.6. A rational map f satisfies TCE if there exist µ Exp > 1 and r > 0 such that for every x ∈ J(f ), every n ≥ 0 and every connected component

W of f -n (D(x, r)) we have that diam (W ) ≤ µ -n
Exp . There are various equivalent characterization of the TCE condition, see [START_REF] Przytycki | Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps[END_REF]. The following inclusions are strict: uniform hyperbolicity sub-hyperbolicity CE TCE.

It was proved by Aspenberg [START_REF] Aspenberg | The Collet-Eckmann condition for rational functions on the Riemann sphere[END_REF] that the set of non-hyperbolic CE maps has positive measure in the parameter space of rational maps of fixed degree, see also Rees [START_REF] Rees | Positive measure sets of ergodic rational maps[END_REF]. In the family of unicritical polynomials, it was shown by Graczyk-Swiatek [START_REF] Graczyk | Harmonic measure and expansion on the boundary of the connectedness locus[END_REF] and Smirnov [START_REF] Smirnov | Symbolic dynamics and Collet-Eckmann conditions[END_REF] that for a.e. c ∈ ∂M d in the sense of harmonic measure f c = z d + c satisfies the CE condition, where M d is the connectedness locus, and ∂M d is the bifurcation locus. We list some useful property of TCE maps.

Proposition 2.7. Let f be a TCE map such that J(f ) = P 1 then (1) The Fatou set F (f ) is the union of attracting basins.

(2) The Hausdorff dimension δ of J(f ) is equal to the Minkowski dimension of J(f ) and is smaller than 2.

(3) There is a unique invariant probability measure µ such that supp(µ) ⊂ J(f ) and µ is absolutely continuous with respect to the conformal measure with exponent δ. Moreover µ is exponentially mixing (hence ergodic) and has positive Lyapunov exponent.

For the proof see [START_REF] Przytycki | Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps[END_REF] and [START_REF] Przytycki | Statistical properties of topological Collet-Eckmann maps[END_REF]. For more about measurable dynamics on J(f ), we refer the reader to Przytycki and Rivera-Letelier [START_REF] Przytycki | Statistical properties of topological Collet-Eckmann maps[END_REF], Graczyk and Smirnov [START_REF] Graczyk | Non-uniform hyperbolicity in complex dynamics[END_REF], and Rivera-Letelier and Shen [START_REF] Rivera-Letelier | Statistical properties of one-dimensional maps under weak hyperbolicity assumptions[END_REF].

In our presentation of the main theorem, we also ask that p satisfies WR or Positive Lyapunov. These additional conditions are used to construct stable manifold at v ∈ CV (p) ∩ J(p) in section 3.

Definition 2.8. A rational map f satisfies WR(η, ι) if there exists η, ι > 0 and C 0 > 0 such that for all v ∈ CV (f ) whose forward orbit does not meet any critical point and for every integer n ≥ 0, it holds

n-1 j=0 d(f j (v),Crit')≤η -log |f ′ (f j (v))| < nι + C 0 .
This condition means that for every v ∈ CV (f ) ∩ J(f ): the orbit of v does not come either too close nor too often to Crit'. The following WR condition is stronger than WR(η, ι) with fixed η and ι. Definition 2.9. A rational map f satisfies WR if for all v ∈ CV (f ) whose forward orbit does not meet critical points we have

lim η→0 lim sup n→∞ 1 n n-1 j=0 d(f j (v),Crit')≤η -log |f ′ (f j (v))| = 0.
The condition Positive Lyapunov is stronger than CE. Definition 2.10. A rational map f satisfies Positive Lyapunov if for every point c ∈ Crit' whose forward orbit does not meet other critical points the following limit exists and is positive

lim n→∞ 1 n log |(f n ) ′ (f (c))| > 0.
In addition we ask that there are no parabolic cycles.

The condition WR, as well as the condition Positive Lyapunov is satisfied for many polynomial maps, see Appendix B.

Stable manifolds and renormalization maps

In this section we work under the assumption that p satisfies TCE and WR(η, ι) with small ι (ι will be determined later) or p satisfies Positive Lyapunov. We construct stable manifolds at each v ∈ CV (p) ∩ J(p) under the above assumptions, we also study the renormalization maps associated to each critical value variety.

3.1. Stable manifold of a critical value. A (local) stable manifold W s loc (v) of f is an embedded complex disk of C 2 passing through v such that there exist δ > 0 that for every

x ∈ W s loc (v) and n ≥ 0, dist(f n (x), f n (v)
) decreases exponentially fast. The construction of stable manifold is classical for hyperbolic periodic points and for uniformly hyperbolic invariant sets, cf. [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF]. In Pesin's theory we can also construct stable manifold for a.e. point with respect to a hyperbolic invariant probability measure, cf. [START_REF] Barreira | Lyapunov exponents and smooth ergodic theory[END_REF]. Since we deal with a single non-uniformly hyperbolic orbit, we construct a stable manifold at v by using Hubbard and Oberste-Vorth's graph transform associated to a sequence of crossed mappings, cf. [START_REF] John | Hénon mappings in the complex domain[END_REF]. In the first three subsections we prove the following theorem: Theorem 3.1. Let f be an attracting polynomial skew product and let p = f | L be the restriction of f to the invariant fiber. Assume p satisfies either TCE and WR, or p satisfies Positive Lyapunov. Then every v ∈ J(p) ∩ CV (p) admits a local stable manifold transverse to the invariant fiber L.

We begin with some definitions.

Let B 1 = U 1 × V 1 , B 2 = U 2 × V 2 be two bi-disks. Let Ω be a neighborhood of B 1 , let f : Ω → C 2 be a holomorphic map such that f (B 1 ) ∩ B 2 = ∅. Let π 1 (resp.
W 1 ⊂ U 1 × V ′ 1 , where V ′ 1 ⊂ V 1 is a relatively compact open subset and W 2 ⊂ U ′ 2 × V 2 , where U ′ 2 ⊂ U 2 is a relatively compact open subset, such that f : W 1 → W 2 is
bi-holomorphic, and for every x ∈ U 1 , the mapping

π 2 • f | W 1 ∩(({x}×V 1 ) : W 1 ∩ ({x} × V 1 ) → V 2
is proper of degree d, and for every y ∈ V 2 the mapping

π 1 • f -1 | W 2 ∩(U 2 ×{y}) : W 2 ∩ (U 2 × {y}) → U 1 is proper of degree d.
Beware that coordinate are switched as compared to [START_REF] John | Hénon mappings in the complex domain[END_REF]: here the horizontal direction is contracted and the vertical direction is expanded.

Let B be the bidisk D(0, 1) × D(0, 1). We define the horizontal boundary as ∂ h (B) := {|x| < 1, |y| = 1}. The vertical boundary ∂ v (B) can be defined similarly.

Definition 3.3. An analytic curve X is called horizontal (resp. vertical) in B 1 if X is defined in a neighborhood of B 1 , X ∩ B 1 = ∅ and X ∩ ∂ h (B 1 ) = ∅ (resp. X ∩ ∂ v (B 1 ) = ∅). It follows that π 1 : X → U 1 (resp. π 2 : X → V 1
) is proper of degree d, for some integer d > 0. We call this d the degree of the analytic curve.

Proposition 3.4. If f : B 1 → B 2 is a degree 1 crossed mapping and X is a degree d vertical curve in B 1 , then π 2 • f : X ∩ W 1 → V 2 is proper of degree d, or equivalent to say, f (X) is a degree d vertical curve in B 2 (defined in a neighborhood of B 2 ).
For the proof see [START_REF] John | Hénon mappings in the complex domain[END_REF]Proposition 3.4]. Definition 3.5 (Dujardin). f is called Hénon-like from B 1 to B 2 if the following three conditions are satisfied [START_REF] Aspenberg | The Collet-Eckmann condition for rational functions on the Riemann sphere[END_REF]

f restricted to Ω is injective, where Ω is a neighborhood of B 1 . (2) f (∂ h B 1 ) ∩ B 2 = ∅, (3) f (B 1 ) ∩ ∂B 2 ⊂ ∂ h B 2 .
Again, note that horizontal and vertical directions are switched as compared to [START_REF] Dujardin | Hénon-like mappings in C 2[END_REF].

Proposition 3.6. If f is Hénon-like from B 1 to B 2 , then f is a crossed mapping.
For the proof see [START_REF] Dujardin | Hénon-like mappings in C 2[END_REF]Proposition 2.3].

The following theorem summarizes our approach to construct stable manifolds.

Theorem 3.7. Let B 0 = U 0 × V 0 , B 1 = U 1 × V 1 ,.
.. be an infinite sequence of bi-disks, and f i : B i → B i+1 be of degree 1 crossed mapping, with V ′ i simply connected (with notation as in Definition 3.2) such that the modulus Mod(V i \ V ′ i ) is uniformly bounded from below. Then the set

W s (fn) = {(x, y) ∈ B 0 | f n • • • • •f 0 (x, y) ∈ B n for all n ≥ 0} is a degree 1 horizontal curve in B 0 . W s (fn)
is called the stable manifold for the sequence of crossed mappings.

For the proof see [START_REF] John | Hénon mappings in the complex domain[END_REF]Corollary 3.12].

We also need some one-dimensional preparations. The following lemma is due to Przytycki and Rivera-Letelier ([37, Lemma 3.3]). Theorem 3.8. Let f be a rational function satisfying TCE with constants µ Exp > 1 and r 0 > 0. Then the following assertions hold.

There are constants C 0 > 0 and θ 0 ∈ (0, 1) such that for every r ∈ (0, r 0 ), every integer n ≥ 1, every x ∈ J(f ) and every connected component

W of f -n (B(x, r)), we have diam (W ) ≤ C 0 µ -n Exp r θ 0 .
We now use the above property to show that p is hyperbolic away from Crit' in the following sense.

Lemma 3.9. Assume p satisfies TCE. Let η > 0 be a constant. Let {x 0 , x 1 , . . . , x N -1 } ⊂ J(p) be a segment of orbit such that for every 0 ≤ i ≤ N -1, dist(x i , Crit') > η, then there exist C 1 and α uniform constants such that

|(p N ) ′ (x 0 )| ≥ C 1 η α µ N Exp .
Proof. We use the same notations C 0 and θ 0 as in Theorem 3.8. It is sufficient to prove the lemma for η small. Let r be a constant satisfying η = C 0 r θ 0 such that r < r 0 . Let W i be the pull back of D(x N , r) by p i at x N -i for 0 ≤ i ≤ N . We show that W i ∩ Crit' = ∅ for every i.

Assume that W i ∩ Crit' = ∅ for some i. By Theorem 3.8 we have diam(W i ) ≤ C 0 µ -i Exp r θ 0 ≤ η. On the other hand since dist(x i , Crit') > η for every i, we get diam(W i ) > η, which is a contradiction. Thus W i ∩ Crit' = ∅ for every i and p N restricted to W N is univalent.

By TCE we have diam W N ≤ µ -N
Exp , by the classical Koebe distortion theorem we have

|(p N ) ′ (x 0 )| diam W N ≥ r/4 = 1/4C -1/θ 0 0 η 1/θ 0 . Taking C 1 = 1/4C -1/θ 0 0 and α = 1/θ 0 we get that |(p N ) ′ (x 0 )| ≥ C 1 η α µ N Exp .
3.2. The TCE+WR case. In this subsection, for an attracting skew product f such that p satisfies TCE and WR(η, ι) with small ι, we construct a sequence of bi-disks {B i } with low exponential size such that for every integer i, f is a degree 1 crossed mapping from B i to B i+1 with B i centered at p i (v). We use the same notations as in Lemma 3.9. We fix a constant 0 < ε 0 ≪ min |λ| -1/3 -1, µ Exp -1 . In the following we choose an integer

N such that C 1 η α µ N Exp ≥ (1 + ε 0 ) N ,
where η is given by WR(η, ι), α and C 1 are as in Lemma 3.9. We subdivide the integers into blocks of the form [iN, (i + 1)N ). We say that a block of this subdivision is of first type if

(i+1)N -1 j=iN |p ′ (p j (v))| ≥ (1 + ε 0 ) N ,
and we call this subdivision is of second type if the above inequality does not hold. By Lemma 3.9 if dist(p j (v), Crit') > η for iN ≤ j < (i + 1)N , then [iN, (i + 1)N ) is of first type.

Let m ∈ [iN, (i + 1)N ) be a positive integer, if m is in a block of first type, we define

µ m := (1 + ε 0 )   (i+1)N -1 j=iN |p ′ (p j (v))|   1/N .
When m is in a block of second type, we define

µ m := (1 + ε 0 ) 2 .
Note that in both cases we have

µ m ≥ (1 + ε 0 ) 2 ≥ 1 + ε 0 .
We define

r n := r 0 n-1 m=0 a m µ m ,
where r 0 > 0 is a constant to be determined and a m := |p ′ (p m (v))|.

Lemma 3.10. There are constants C 2 , C 3 > 0 such that the following estimates of r n hold for n ≥ 0:

(1)

r n ≤ C 2 r 0 (1 + ε 0 ) -n . (2) r n ≥ C 3 r 0 e -(α+2)nι (1 + ε 0 ) -2n
, where α is as in Lemma 3.9.

Proof. To prove the first inequality, notice that for every i ≥ 0, by the definition of µ m we have

(3.1) (i+1)N -1 j=iN µ j ≥ (1 + ε 0 ) N (i+1)N -1 j=iN a j .
Notice also that for every iN ≤ m < (i + 1)N , we have

(3.2) m j=iN a j ≤ Dp N ,
where Dp is the uniform norm of Dp on the Julia set J(p).

Combining (3.1) and (3.2), for kN ≤ n < k(N + 1) we have

r n = r 0   k-1 i=0 (i+1)N -1 j=iN a j µ j   n j=kN a j µ j ≤ r 0 (1 + ε 0 ) -kN Dp N (1 + ε 0 ) n-kN = Dp N r 0 (1 + ε 0 ) -n .
Taking C 2 = Dp N the first inequality is proved.

To prove the second inequality, notice that if the block [iN, (i + 1)N ) is of first type, then

(i+1)N -1 j=iN µ j = (1 + ε 0 ) N (i+1)N -1 j=iN a j .
Assume that {i 0 , i 2 , . . . , i l-1 } ⊂ {0, 1, . . . , k -1} are all the integers such that the block [i h N, (i h + 1)N ) is of second type, 0 ≤ h ≤ l, then we have

r n = r 0   k-1 i=0 (i+1)N -1 j=iN a j µ j   n j=kN a j µ j = r 0 (1 + ε 0 ) (l-k)N   l-1 h=0 (i h +1)N -1 j=i h N a j µ j   n j=kN a j µ j ≥ r 0 (1 + ε 0 ) (l-k)N   l-1 h=0 (i h +1)N -1 j=i h N a j (1 + ε 0 ) 2   n j=kN a j C 2 (1 + ε 0 ) 2 ≥ r 0 C 2 (1 + ε 0 ) -2n   l-1 h=0
(i h +1)N -1 j=i h N a j   n j=kN a j . (3.3)
Since the block [i h N, (i h + 1)N ) is of second type, then necessarily there is an integer j satisfies i h N ≤ j < (i h + 1)N and dist(p j (v), Crit') ≤ η. By Lemma 3.9, the product of derivative between two points

x n 1 , x n 2 such that dist(x n i , Crit') > η, n 1 < i < n 2 satisfies (3.4) n 2 -1 i=n 1 +1 a j ≥ C 1 η α .
Notice that the number of such maximal blocks

[x n 1 , x n 2 ] in [i h N, (i h + 1)N ) is equal to the cardinality # j ∈ [i h N, (i h + 1)N ) : dist(p j (v), Crit') ≤ η + 1.
There is also a constant C 4 > 0 such that dist(p j (v), Crit') ≤ η implies a j ≤ C 4 η for η small. Thus by choosing η small, for j satisfying dist(p j (v), Crit') ≤ η we have

(3.5) C 1 η α ≥ C 1 a j C 4 α ≥ Ca α j ≥ a 1+α j .
Combining (3.5) and (3.4) we get

  l-1 h=0
(i h +1)N -1 j=i h N a j   n j=kN a j ≥ n j=0 d(p j (v),Crit')≤η C 1 η α a j ≥ n j=0 a 2+α j ≥ e (-nι-C 0 )(α+2) . (3.6)
Combining (3.3) and (3.6) we get

r n ≥ r 0 C 2 (1 + ε 0 ) -2n e (-nι-C 0 )(α+2) .
Setting C 3 := e -(α+2)C 0 /C 2 the conclusion follows.

The following proposition clearly implies Theorem 3.1 in the case p satisfying TCE and WR.

We let

U i := D(p i (v), r 0 (1 + ε 0 ) -3i ), V i := D(p i (v), r i ) and let B i = U i × V i for every positive integer i.
Proposition 3.11. Assume p satisfies TCE and WR(η, ι) with small ι (to be determined in the proof ). Then there exist r 0 > 0 such that for arbitrary v ∈ J(p) ∩ CV (p) and every i, the map f : B i → B i+1 is a degree 1 crossed mapping and satisfies the condition of Theorem 3.7. As a consequence there is a stable manifold at v.

Proof. We first show that for carefully chosen ι and for r 0 sufficiently small, f restricted to a neighborhood of B n is injective for every n. The WR condition implies the following Slow Recurrence property: there exist a small α(ι) > 0, such that dist(p n (v), Crit') > e -nα for all large n, see Lemma A.2 for the proof. We let ι sufficiently small so that α << log(1 + ε 0 ). By Lemma 3.10 (1) we can let r 0 small such that r n << e -αn and also r 0 (1 + ε 0 ) -3n << e -αn , for every n .

We need the following general fact: if f : W → f (W ) is a proper holomorphic map satisfying no critical points and f (W ) is simply connected, then f is injective. Thus in our case, to show f restricted to a small neighborhood of B n is injective, it is sufficient to show f (B n ) is contained in a simply connected domain which is disjoint with the critical value set of f . Let

M = sup x∈Ω ∂f ∂z , ∂f ∂t , where Ω is some compact subset such that ∆×C\Ω is in the basin of ∞. Then f (B n ) ⊂ U n+1 × D(p n+1 (v), 2M r n ).
Let l be the maximal order of the critical points in J(p), then there is a constant C > 0 such that dist(p n+1 (v), p(Crit ′ )) ≥ Ce lαn . By choosing sufficiently small r 0 and ι we get that U n+1 × D(p n+1 (v), 2M r n ) is disjoint with the critical value set of f . Thus f restricted to a small neighborhood of B n is injective.

Next we prove f : B n → B n+1 is a degree 1 Hénon-like map for every n.

By |λ| < (1 + ε 0 ) -3 we get π 1 (f (B n )) ⊂ π 1 (B n+1 ), thus f (B n ) ∩ ∂ v B n+1 = ∅, thus it is easy to verify f (B n ) ∩ ∂B n+1 ⊂ ∂ h B n+1 . To prove f (∂ h B n ) ∩ B n+1 = ∅,
we first show that if r 0 is sufficiently small, and if we let Vn := D(p n+1 (v), (1 + ε 0 ) 1 2 r n+1 ), then Vn ⊂ p(V n ) for every n. To see this, First by the definition of r n we have

a n r n = µ n r n+1 ≥ (1 + ε 0 )r n+1
, where a n = |p ′ (p n (v))|. We choose sufficiently small r 0 such that r n ≪ dist(p n (v), Crit'). Then the Koebe distortion theorem gives us D(p n+1 (v), (1 + ε 0 )

1 2 r n+1 ) ⊂ p(V n ) as desired.
For every point x ∈ ∂ h B n , let y = π 2 (x), by the above result we have dist(p(y),

V n+1 ) ≥ (1 + ε 0 ) 1/2 -1 r n+1 , thus we get dist v (f (x), B n+1 ) ≥ (1 + ε 0 ) 1/2 -1 r n+1 -dist v (f (x), p(y)) ≥ (1 + ε 0 ) 1/2 -1 r n+1 -M r 0 (1 + ε 0 ) -3n (by mean value theorem) ≥ (1 + ε 0 ) 1/2 -1 C 3 r 0 e -(α+2)nι (1 + ε 0 ) -2n -M r 0 (1 + ε 0 ) -3n , where M = sup x∈Ω ∂f ∂z , ∂f ∂t as before. By choosing ι ≪ ε 0 we get dist v (f (x), B n+1 ) > 0. Thus f (∂ h B n ) ∩ B n+1 = ∅.
It is easy to show f : B n → B n+1 has degree 1. The reason is that the forward image of a vertical disk is again a vertical disk, and π 2 • f is of degree 1 when restricted to this vertical disk. Since f keep the degree of the curve fix, f must have degree 1.

Finally we set V ′ n := p -1 V n+1 and show that the modulus of the annulus V n -V ′ n is uniformly bounded from below. Since the modulus is invariant under univalent maps, we have that

Mod(V n \ V ′ n ) = Mod (p(V n ) \ V n+1 ) ≥ Mod Vn \ V n+1 = 1 4π log(1 + ε 0 ).
Now all the conditions in Theorem 3.7 are checked, we conclude that there is a stable manifold in the sense of Theorem 3.7. Since the dynamics contracts exponentially transverse to L, this is a stable manifold in the usual sense.

3.3. The Positive Lyapunov case. Next we assume p satisfies Positive Lyapunov instead of TCE+WR(η, ι). We can then construct the stable manifold by arguing as before. Indeed, let χ v be the following vertical Lyapunov exponent (3.7)

χ v := lim n→∞ 1 n log |(p n ) ′ (v)| > 0.
We need to construct µ n such that an estimate of r n similar to that of Lemma 3.10 holds, and also show the Slow Recurrence property: for every α > 0, dist(p n (v), Crit') > e -nα for all large n. That Positive Lyapunov implies Slow Recurrence is shown in Lemma A.3.

To show an estimate of r n similar to Lemma 3.10, we define µ n := (1 + ε 0 )e χv for every n, and let r n := r 0 n-1 i=1 a i µ i , where r 0 > 0 is a constant, a i := |p ′ (p i (v))|. Lemma 3.12. There are constants C 2 , C 3 > 0 such that the following estimates of r n hold for n ≥ 0:

(1) r n ≤ C 2 r 0 (1 + ε 0 2 ) -n . (2) r n ≥ C 3 r 0 (1 + 2ε 0 ) -n .
Proof. Since (3.7) holds, for every ε > 0 small, there exist constants C 2 > 0 and C 3 > 0 such that n-1 i=1 a i ≤ C 2 (1 + ε) n e nχv and n-1 i=1 a i ≥ C 3 (1ε) n e nχv for every n ≥ 0. Now it is sufficient to choose ε = ε 0 2+ε 0 . Remark 3.13. The above proof actually only use that the ratio of limsup and liminf of (3.7) is sufficiently close to 1, thus the condition Positive Lyapunov can be replaced by a weaker condition (ratio of upper and lower Lyapunov exponent is sufficient close to 1) to make the main theorem hold.

Finally the same argument as in Proposition 3.11 gives the existence of a local stable manifold at v, in case that p satisfies Positive Lyapunov. Thus the proof of Theorem 3.1 is complete.

3.4. Renormalization map. Next we introduce the renormalization map of a critical value variety V. We let V be a component of the critical value variety that defined in a neighborhood of B 0 , where B 0 is as in Proposition 3.11. We assume that the germ of V at (0, v) does not coincide with W s loc (v) (the converse hypothesis that V = W s loc (v) makes the main theorem even easier to prove, we will see this later). Then by the definition of the stable manifold in Theorem 3.7, for N large f N (V) ⊂ B N , thus f N (V) ∩ ∂B N = ∅. By the definition of a Hénon-like map we must have f N (V) ∩ ∂B N ⊂ ∂ h B N . Thus for N large, f N (V) is a degree d vertical curve. Note that d is constant because our Hénon-like maps have degree 1. Without loss of generality, we may assume that V is a degree d vertical curve (in B 0 ), otherwise we may replace V by some f N (V). By the definition of degree 1 crossed mapping, for every n ≥ 0,

f n (V) ∩ B n is also a degree d vertical curve (in B n ).
We assume V has the parametrization V = {γ(t) : t ∈ D(0, r 0 )} of the form γ(t) = (t l , ψ(t)), where r 0 is the radius of U 0 in Proposition 3.11, l is a positive integer and ψ is holomorphic. Since V is a vertical curve, we can further assume that ψ ′ (0) = 0, for otherwise V can not be a vertical curve in B 0 when r 0 is sufficiently small.

For n ≥ 0 we let W n := f -n (B n ) ∩ B 0 . The map

ψ n := π 2 • f n • γ : γ -1 (W n ) → V n
is well defined. Definition 3.14. For every integer n ≥ 0, let ρ n be the maximal positive real number such that ψ -1 n ( 1 2 V n ) contains a disk D(0, ρ n ), where 1 2 V n denotes a disk centered at the same point of V n but with one-half radius.

Concretely, ρ n is the typical size of the piece of V which remains in B j under iteration up to time n. We can then define the renormalization map as follows Definition 3.15. For every integer n ≥ 0, the n-th renormalization map φ n is the holomorphic map from D n to C, defined by φ n (z) = ψ n • L ρn (z) for z ∈ D n , where

D n := L -1 ρn (ψ -1 n ( 1 2 V n )) is a domain in C, and for r ∈ C, L r : C → C denotes the linear map L r (z) = rz.
By the definition of D n we know that D(0, 1) ⊂ D n . The following proposition is crucial. Proposition 3.16. The renormalization map φ n has uniformly bounded (topological) degree. Moreover there exist a constant C 0 > 0 such that diam D n ≤ C 0 , and ρ n is exponentially small, namely ρ n ≤ Cµ -n CE r n , where µ CE > 1 and C > 0 are constants.

Proof. The constant µ CE corresponds to the constant appeared in the CE condition (Definition 2.5). We will see in the proof that TCE+WR(η, ι) with ι small implies CE. By our construction, f n • γ γ -1 (W n ) is a degree d vertical curve (in B n ) for every n. Thus ψ n is of uniformly bounded degree and so is φ n .

To show diam D n ≤ C 0 , we observe that by Theorem 2.4 (2.2) there exist a uniform constant

C 0 > 0 such that C 0 ρ n ≥ diam(ψ -1 n ( 1 2 V n )), thus diam D n = 1 ρ n diam ψ -1 n 1 2 V n ≤ C 0 .
To show that ρ n is exponentially small, first notice that TCE + WR(η, ι) with ι small implies CE (see Lemma A.4), and also Positive Lyapunov implies CE. Thus

|(p n ) ′ (v)| is exponentially large, that is there exist C > 0 such that |(p n ) ′ (v)| ≥ Cµ n CE with µ CE slightly smaller than µ Exp . Thus |ψ ′ n (0)| = |ψ ′ (0)||(p n ) ′ (v)| ≥ Cµ n CE |ψ ′ (0)
|, which is exponentially large. By Theorem 2.4 (2.1) we get ρ n ≤ Cµ -n CE r n , which is exponentially small.

Slow approach to Crit'

In this section our aim is to prove Theorem 1.2. First we remark that it is not true that for every vertical fiber {t = t 0 } Lebesgue a.e. (in the sense of one-dimensional Lebesgue measure) point in this fiber slowly approach Crit', as pointed out in [START_REF] Peters | Polynomial skew-products with wandering Fatou-disks[END_REF]. Indeed in [START_REF] Peters | Polynomial skew-products with wandering Fatou-disks[END_REF] the authors construct a vertical Fatou disk which comes exponentially close to Crit'. Instead, we need to select a full measure family of vertical fibers such that Lebesgue a.e. point in the fiber slowly approach Crit'. This will be proved by studying the renormalization maps along critical varieties constructed in the previous section. Together with Przytycki's lemma we can actually track the orbits of points in Crit (Lemma 4.2 and Lemma 4.3). Thus we need the non-uniform hyperbolic conditions in section 3 to make sure that the stable manifolds at each v ∈ CV (p) ∩ J(p) exist.

Let W 0 be a forward invariant open subset of F (p) satisfying p(W 0 ) ⊂ W 0 . Such a W 0 exist since F (p) is a union of attracting basins. Let W m = p -m (W 0 ), and let K m be the complement Proof. We may assume that K 0 is sufficiently close to J(p), that is,

K m = C \ W m .
K 0 ⊂ {x : dist(x, J(p)) ≤ r} ,
where r is the constant appearing in the definition of the TCE condition. Thus by definition of the TCE condition we have

K m ⊂ x : dist(x, J(p)) ≤ µ -m Exp for m ≥ 0.
Denote N ε the number of ε-disks needed to cover J(p), by Proposition 2.7 (2), the Minkowski dimension of J(p) is h < 2. Hence for ε > 0 sufficiently small we have

N ε < ε -h . Choosing ε = µ -m
Exp , for m sufficiently large, K m is covered by at most µ mh Exp disks of radius µ -m Exp . Thus the measure of K m is at most πµ m(h-2) Exp , and the conclusion follows.

In the following we assume p satisfies TCE+WR or Positive Lyapunov. Let ∆ = D(0, r 0 ). The set

W = ∆ × W 0 is contained in F (f ), and f (W ) ⊂ W , let K = (∆ × C) \ W . Let W ′ be the ε-neighborhood of W , W ′ := {x ∈ ∆ × C : dist(x, W ) < ε} .
For ε sufficiently small W ′ is forward invariant.

Let K ′ = (∆ × C) \ W ′ . By using Proposition 3.16, we show that for most vertical fibers, the critical points on the fiber move to the Fatou set fairly quickly. The argument is similar to Peters-Smit [START_REF] Peters | Fatou components of attracting skew-products[END_REF] who treated the sub-hyperbolic case.

Let us choose a critical value variety V passing through v ∈ CV (p) ∩ J(p), parametrized as before: V = (t l , ψ(t)) : t ∈ ∆ . Let φ n be the renormalization map defined in subsection 3.4. For every integer s ≥ 0, we define j(s) to be the maximal integer such that |λ s | ≤ ρ j(s) , where ρ n is as in Definition 3.14. Lemma 4.2. For every critical value variety V passing through v ∈ CV (p) ∩ J(p) that does not coincide with the stable manifold at v, there is a full Lebesgue measure subset E v ⊂ ∆ such that for every u ∈ E v there exist an integer N u and β > 0 independent of u such that for every s ≥ N u , we have f j(s)+βs (γ(λ s u)) ∈ W ′ . where γ(t) = (t l , ψ(t)).

We note that there is some abuse in notation. For the simplicity when we write a noninteger number s as an iteration number of a map f , we mean the iteration of ⌊s⌋ times.

Proof. Fix β > 0 arbitrary for the moment. For every integer s ≥ 0, let A s be the set

A s = u ∈ ∆ : f j(s)+βs (γ(λ s u)) ∈ K ′ .
By the definition of the renormalization map φ n , we have

A s = u ∈ ∆ : f βs (λ s u) l , φ j(s) 1 ρ j(s) λ s u ∈ K ′ . Let M = sup x∈Ω ∂f ∂z , ∂f ∂t
, where Ω is a compact subset such that ∆ × C \ Ω is in the basin of ∞. By a shadowing argument there exists C > 0 such that for every integer m ≥ 0, if

f m (x) ∈ K ′ and |π 1 (x)| < CM -m , then π 2 (x) ∈ K m . It is equivalent to say that if x ∈ ∆ × C satisfies |π 1 (x)| < CM -m and π 2 (x) ∈ W m , then f m (x) ∈ W ′ .
We choose β sufficiently small, such that for large enough s we have

(λ s u) l < CM -βs β < -log |λ| log M is enough .
Thus we get

A s ⊂ u ∈ ∆ : φ j(s) 1 ρ j(s) λ s u ∈ K βs .
Next we estimate the measure of the slightly bigger set

A s := u ∈ ∆ : φ j(s) 1 ρ j(s) λ s u ∈ K βs = ρ j(s) λ -s φ -1 j(s) (K βs ).
By Lemma 4.1 the Lebesgue measure of K βs decreases exponentially with s. Next we prove that we can choose ε 0 and ι in Lemma 3.10 (2) sufficiently small so that the ratio

meas(K βs ∩ 1 2 V j(s) ) r 2 j(s) is exponentially small. Since meas(K βs ∩ 1 2 V j(s) ) r 2 j(s) ≤ meas(K βs ) r 2 j(s)
, it is sufficient to show meas(K βs )/r 2 j(s) is exponentially small.

If we choose β = -log |λ| 2 log M , then by Lemma 4.1 (again for large enough s)

(4.1) meas(K βs ) ≤ πµ s log |λ|(2-h) 2 log M Exp .
On the other hand, by Proposition 3.16 we have

ρ j(s) ≤ Cµ -j(s) CE r j(s) ≤ µ -j(s) CE (since r 0 small).
Then by the definition of j(s) we have

j(s) ≤ -s log |λ| log µ CE .
Then by Lemma 3.10 (2) we have

(4.2) r j(s) ≥ C 3 r 0 e -(α+2)j(s)ι (1 + ε 0 ) -2j(s) ≥ C 3 r 0 e (α+2) log |λ| log µ CE ιs (1 + ε 0 ) 2 log |λ|s log µ CE .
By (4.1) and (4.2) we can choose ε 0 and ι sufficiently small such that meas(

K βs ∩ 1 2 V j(s) )/r 2 j(s)
is exponentially small.

It is proved in Proposition 3.16 that the map φ n has uniformly bounded degree, so by Lemma 2.4 (2.4) the measure of φ -1 j(s) (K βs ∩ 1 2 V j(s) ) also decreases exponentially with s. Finally since ρ j(s) λ -s is uniformly bounded with s, the Lebesgue measure of A s also decreases exponentially with s. Thus ∞ s=1 meas(A s ) ≤ ∞ s=1 meas( A s ) < ∞. By the Borel-Cantelli lemma, there is a full measure subset E v ∈ ∆ such that for every u ∈ E v , there exist an integer N u such that when s ≥ N u , u / ∈ A s . In other words, f j(s)+βs (γ(λ s u)) ∈ W ′ and the conclusion follows.

In the case where the critical value variety V passing through v ∈ CV (p) ∩ J(p) coincides with the stable manifold of v, every y ∈ V will shadow v forever. Thus we get an estimate of the returning time to Crit' of y, simply by Przytycki's lemma (Lemma 2.1) Lemma 4.3. Assume that the critical value variety V passing through v ∈ CV (p) ∩ J(p) coincides with the stable manifold of v. Let γ : ∆ → C 2 be the parametrization of this stable manifold such that γ(0) = v, γ(t) = (t, ψ(t)) where ψ is holomorphic.

Let C be the critical variety of f such that f (C) = V. Then for every fixed α > 0 there exists a constant K(α) > 0 such that for every n ≥ 0, 0 ≤ s ≤ n and u ∈ ∆, if dist v (f n (γ(u)), C) ≤ e -αn and dist v (f n-s (γ(λ s u)), C) ≤ e -αn , then s ≥ Kn.

Proof. We let L u to be the vertical line {t = u}. By the construction of bi-disks in Proposition 3.11 , there exist constants C 0 > 0, λ 1 < 1 such that for every n ≥ 0 and u ∈ ∆ we have dist

v (f n (γ(u)), p n (v)) ≤ C 0 λ n 1 . Together with dist v (f n (γ(u)), C) ≤ e -αn we get dist v (p n (v), C ∩ L λ n u ) ≤ e -αn + C 0 λ n 1 . For similar reasons we have dist v (f n-s (γ(λ s u)), p n-s (v)) ≤ C 0 λ n 1 . Together with the in- equality dist v (f n-s (γ(λ s u)), C) ≤ e -αn we get dist v (p n-s (v), C ∩ L λ n u ) ≤ e -αn + C 0 λ n 1 . On the other hand there exist C 1 > 0, l ′ > 0 such that dist v (C ∩ L λ n u , c 0 ) ≤ C 1 |λ| n/l ′ (l ′
is related to the multiplicity of C at c 0 ), where c 0 = C ∩ L is the unique intersection point of C and the invariant line L. Then by the triangle inequality we have dist(

p n (v), c 0 ) ≤ e -αn + C 0 λ n 1 + C 1 |λ| n/l ′ and also dist(p n-s (v), c 0 ) ≤ e -αn + C 0 λ n 1 + C 1 |λ| n/l ′ . Thus s is a return time of p n-s (v) into the small neighborhood D(c 0 , e -αn + C 0 λ n 1 + C 1 |λ| n/l ′ ) of c 0 . By Przytycki's lemma (Lemma 2.1) we get s ≥ -C log(e -αn + C 0 λ n 1 + C 1 |λ| n/l ′ ) := Kn, the conclusion follows.
The main result of this section is the following equivalent form of Theorem 1.2, Theorem 4.4. There is a full Lebesgue measure subset E ⊂ ∆ such that for every u ∈ E, for Lebesgue a.e. x in the fiber L u : {t = u}, x slowly approach Crit'.

Proof. It is enough to prove that for each fixed α > 0 and u ∈ E, the set of points in L u satisfing dist v (f n (x), Crit') ≥ e -αn for all large n has full Lebesgue measure in L u . We let E be the intersection of all E v , where E v is in Lemma 4.2, and v ranges on the set of critical values. Thus E has full Lebesgue measure in ∆. For every u ∈ E we consider the sets

E n := c∈Crit' ∩L λ n u f -n D v (c, e -αn ) , and E ′ n := c∈Crit' ∩L λ n u f -n D v (c, e -2αn
) .

(Recall that D v stands for vertical disk). For an arbitrary critical point c ∈ Crit' ∩L λ n u , we let Γ be an arbitrary connected component of f -n (D v (c, e -αn )).

Step 1, we show that the cardinality # {0 ≤ s ≤ n : f s (Γ) ∩ Crit = ∅} is uniformly bounded with respect to n.

For n large enough f s (Γ) has no intersection with any critical variety C such that C ⊂ Crit'. The reason is the following. Take the radius of ∆ sufficiently small to make sure that

C ⊂⊂ F (f ). Thus if c ′ ∈ C ∩ f s (Γ) for some 0 ≤ s ≤ n, then dist v (f n-s (c ′ ), J(p)) > δ for some uniform constant δ. On the other hand dist v (c, J(p)) ≤ C|λ| n l
, where c is as in the definition of E n and E ′ n . Thus dist v (f n-s (c ′ ), c) > δ ′ for some uniform constant δ ′ , this is impossible when n large since f n-s (c ′ ) ∈ D v (c, e -αn ). Thus it is sufficient to show that the cardinality # {0 ≤ s ≤ n : f s (Γ) ∩ Crit' = ∅} is uniformly bounded with respect to n. For this it is sufficient to show that # {0 ≤ s ≤ n : f s (Γ) ∩ C = ∅} is uniformly bounded with respect to n for every local component of critical variety C ⊂ Crit'. Now there are two cases. Let V = f (C) be a critical value variety, and let v be the unique intersection point of V and L, v ∈ CV (p) ∩ J(p). In the first case we assume that V does not coincide with the stable manifold at v as in Lemma 4.2. We claim that if n is large, s satisfies s + 1 + j(s + 1) + β(s + 1) ≤ n and s ≥ N u , then we have

f s (Γ) ∩ C = ∅. For otherwise if c ′ ∈ f s (Γ)∩C then v ′ := f (c ′ ) ∈ f s+1 (Γ)∩V. Then by Lemma 4.2 we have f j(s+1)+β(s+1) (v ′ ) ∈ W ′ . Since W ′ is forward invariant, when n -s -1 ≥ j(s + 1) + β(s + 1) implies f n-s-1 (v ′ ) ∈ W ′ , thus f n-s (c ′ ) ∈ W ′ .
By the definition of Γ we also have dist v (f n-s (c ′ ), C) ≤ e -αn , which is a contradiction when n large. To summarize, there exist a uniform constant 0

< θ < 1 such that f s (Γ) ∩ C = ∅ implies s ≥ θn or s ≤ N u .
We need the following Lemma. Lemma 4.5. There is a constant N = N (θ) > 0 such that

# {θn ≤ s ≤ n : f s (Γ) ∩ Crit = ∅} ≤ N.
Proof. We first show that the cardinality # {(1κ)n < s ≤ n : f s (Γ) ∩ C = ∅} is uniformly bounded with respect to n, where κ is the constant defined by,

κ = min -θ log |λ| 4l log M , 1 2 with M = sup x∈Ω ∂f ∂z , ∂f ∂t .
Note that by the definition of κ

(4.3) M κ ≤ |λ| -θ 4l . Assume s 1 < s 2 satisfy f s i (Γ) ∩ C = ∅ and (1 -κ) n < s i ≤ n, i = 1, 2. Let c i ∈ f s i (Γ) ∩ C, i = 1, 2. Let c 0 = C ∩ L be the unique intersection point. Then we have (4.4) dist v (f n-s 2 (c 2 ), c 0 ) ≤ dist v (f n-s 2 (c 2 ), C ∩ L λ n-s 2 u ) + dist v (C ∩ L λ n-s 2 u , c 0 ) ≤ e -nα + C|λ| n l . Similarly dist v (f n-s 1 (c 1 ), c 0 ) ≤ dist v (f n-s 1 (c 1 ), C ∩ L λ n-s 1 u ) + dist v (C ∩ L λ n-s 1 u , c 0 ) ≤ e -nα + C|λ| n l .
By the definition of θ we also have dist

v (c 1 , c 2 ) ≤ C|λ| nθ l . Thus by (4.3) we have dist v (f n-s 2 (c 1 ), f n-s 2 (c 2 )) ≤ M n-s 2 C|λ| nθ l ≤ C|λ| 3nθ 4l . (By the choice of s 2 ). Let y = π 2 (f n-s 2 (c 2 )). Using (4.3) again we have dist v (f n-s 1 (c 1 ), p s 2 -s 1 (y)) ≤ M s 2 -s 1 dist v (f n-s 2 (c 1 ), f n-s 2 (c 2 )) ≤ C|λ| nθ 2l . Thus we have (4.5) dist(c 0 , p s 2 -s 1 (y)) ≤ dist v (f n-s 1 (c 1 ), c 0 ) + dist v (f n-s 1 (c 1 ), p s 2 -s 1 (y)) ≤ e -nα + C|λ| nθ 2l .
Combining (4.4) and (4.5) we infer that s 2s 1 is a return time of y in the small disk D(c 0 , e -nα + C|λ| nθ 2l ), by Przytycki's lemma (Lemma 2.1) there exist a constant

K(α) such that s 2 -s 1 ≥ Kn. Thus # {(1 -κ)n < s ≤ n : f s (Γ) ∩ C = ∅} ≤ κ K(α) + 1 . By Lemma 2.4 (2.3) there exist α 1 > 0 such that diam f (1-κ)n (Γ) ≤ e -α 1 n . Next if we consider the cardinality # {(1 -2κ)n < s ≤ (1 -κ)n : f s (Γ) ∩ C = ∅}, we re- place f n (Γ) by f s (Γ) where s satisfies (1 -2κ)n < s ≤ (1 -κ)n, f s (Γ) ∩ C = ∅ and is maximal. Repeating the same argument we know there is a constant K(α 1 ) > 0 such that # {(1 -2κ)n < s ≤ (1 -κ)n : f s (Γ) ∩ C = ∅} ≤ κ K(α 1 ) + 1.
After finitely many iteration of the argument we get # {θn < s ≤ n : f s (Γ) ∩ C = ∅} is uniformly bounded with respect to n.

We continue the proof of Theorem 4.4. Since we have # {0 ≤ s ≤ θn : f s (Γ) ∩ C = ∅} ≤ N u . Thus by Lemma 4.5 we get that # {0 ≤ s ≤ n : f s (Γ) ∩ Crit = ∅} is uniformly bounded with respect to n.

In the second case we assume that V = f (C) coincides with the stable manifold at v as in Lemma 4.3. We also want to show that # {0 ≤ s ≤ n : f s (Γ) ∩ Crit = ∅} is uniformly bounded with respect to n. Let as before γ be the parametrization of the stable manifold. Assume 0 ≤ s 1 < s 2 ≤ n satisfy

f s i (Γ) ∩ C = ∅, let c i ∈ f s i (Γ) ∩ C, i = 1, 2. Let f (c 1 ) = γ(u 1 ), then f (c 2 ) = γ(λ s 2 -s 1 u 1 )
. By the definition of Γ we have dist v (f n-s 1 (γ(u 1 )), C ∩ L λ n-s 1 u 1 ) ≤ e -αn , and dist v (f n-s 2 (γ(λ s 2 -s 1 u 1 ), C∩L λ n-s 1 u 1 ) ≤ e -αn . By Lemma 4.3 we have s 2 -s 1 ≥ Kn. Thus # {0 ≤ s ≤ n : f s (Γ) ∩ Crit = ∅} ≤ 1/K + 1 which is uniformly bounded.

Step 2, By Step 1 we already know that f n : Γ → D v (c, e -αn ) has uniformly bounded degree. Now we show that the conclusion of the theorem holds.

Let Γ ′ be the component of f -n (D v (c, e -2αn )) contained in Γ. By Lemma 2.4 (2.4) there exist a constant α ′ > 0 such that meas Γ ′ / meas Γ ≤ e -α ′ n . Since ∞ is an attracting fixed point, the set E n is uniformly bounded. Thus meas E n < A for some constant A > 0. Thus we have meas

E ′ n meas E n = meas Γ ′ meas Γ ≤ e -α ′ n ,
where the sum ranges over all possible critical points and connected components.

Finally we have shown that meas E ′ n ≤ Ae -α ′ n . Thus n i=0 meas E ′ i < ∞, and by the Borel-Cantelli lemma for Lebesgue a.e. point in x ∈ L u , x / ∈ E ′ n for large n, which means dist v (f n (x), Crit') ≥ e -2αn . In other words, x slowly approach Crit'. The conclusion follows.

Positive vertical Lyapunov exponent

In this section we prove Theorem 1.4: if x slowly approach Crit' and ω(x) ⊂ J(p) then χ -(x) ≥ log µ Exp , where µ Exp > 1 is the constant appearing in the definition of the TCE condition.

Definition 5.1. Let x ∈ ∆ × C satisfies dist v (f n (x), J(p)) ≤ r
4 for every n ≥ 0, where r is as in the definition of TCE condition. We say a positive integer n is a expanding time of x if for every 0 ≤ m ≤ n, the connected component

Γ of f -m (D v (f n (x), r 4 )) containing f n-m (x), satisfies diam(Γ) ≤ µ -m
Exp . Lemma 5.2. There exist a uniform constant θ > 0 such that if x ∈ ∆ × C satisfies dist v (f n (x), J(p)) ≤ r 4 , then every n ≤ -θ log |π 1 (x)| is an expanding time, provided |π 1 (x)| is small enough.

Proof. Let n be an arbitrary integer, and for 0 ≤ m ≤ n let Γ be the connected component of

f -m (D v (f n (x), r 4
)) containing f n-m (x). Let M = sup x∈Ω ∂f ∂z , ∂f ∂t as before. Then for arbitrary x 1 = x 2 ∈ Γ, let y 1 = π 2 (x 1 ), y 2 = π 2 (x 2 ). Then for i = 1, 2 we have

dist v (p m (y i ), f m (x i )) ≤ M m |π 1 (x i )| ≤ M m |π 1 (x)|. Let z ∈ J(p) such that dist v (z, f n (x)) ≤ r 4 , If n satisfies M n |π 1 (x)| ≤ r 4 , then we have dist(p m (y i ), z) ≤ dist v (p m (y i ), f m (x i )) + dist v (f m (x i ), z) ≤ dist v (p m (y i ), f m (x i )) + dist v (f n (x), f m (x i )) + dist v (z, f n (x)) ≤ r 4 + r 4 + r 4 < r. for i = 1, 2.
Thus p m (y i ) is in the disk D(z, r), i = 1, 2. By TCE we know for every connected component Γ ′ of f -m D(z, r), we have diam

(Γ ′ ) ≤ µ -m Exp , thus dist(y 1 , y 2 ) ≤ µ -m Exp . In other words, dist v (x 1 , x 2 ) ≤ µ -m
Exp . Thus diam(Γ) ≤ µ -m Exp , which implies n is an exponential expanding time of x providing M n |π 1 (x)| ≤ r 4 . Thus for any 0 < θ < 1 log M , the condition n ≤ -θ log |π 1 (x)| implies n is an expanding time.

The main result of this section is the following.

Theorem 5.3. If x ∈ W s (J(p)) slowly approach Crit', then χ -(x) ≥ log µ Exp .
Proof. Without loss of generality we may assume that x satisfies dist v (f n (x), J(p)) ≤ r 4 for every n ≥ 0. For fixed sufficiently small α > 0, there exist N > 0 such that for n ≥ N , dist v (f n (x), Crit') ≥ e -αn by slow approach. Since x is in W s (J(p)) the orbit of x will stay away from any component of the critical variety which does not belongs to Crit'. Thus we have dist v (f n (x), Crit) ≥ e -αn as well. We may also assume that |π 1 (x)| < 1.

Set δ := 2α log M log µ Exp , where M = sup x∈Ω ∂f ∂z , ∂f ∂t . Let 0 < θ ′ < 1 be a constant that will be determined later. For (1θ ′ )n < s ≤ n, let Γ s be the connected component of f s-n D v (f n (x), e -δn ) containing f s (x). Set δ 0 = δ 2 log M , and note that for all large n, r 4 M -δ 0 n ≥ e -δn . In particular

f δ 0 n (D v (f n (x), e -δn )) ⊂ D v (f n+δ 0 n (x), r/4), hence D v (f n (x), e -δn ) ⊂ Comp f n (x) f -δ 0 n (D v (f n+δ 0 n (x), r 4
)).

Here Comp y denotes the connected component containing y.

We claim that there exist 0 < θ ′ < 1 such that for large n, θ ′ n + δ 0 n is an expanding time of f n-θ ′ n (x). Indeed by Lemma 5.2, for every m ≤ n large, -mθ log |λ| is an expanding time of f m (x). Provided α is sufficiently small we have δ 0 is sufficiently small as well, thus θ ′ = θ log |λ|+δ 0 θ log |λ|-1 is a positive number. We conclude that θ ′ n + δ 0 n is an expanding time of f n-θ ′ n (x). Thus for every nθ ′ n < s ≤ n, we have

diam Comp f s (x) f s-n (D v (f n (x), e -δn )) ≤ diam Comp f s (x) f s-(n+δ 0 n) (x)(D v (f n+δ 0 n (x), r 4
))

≤ µ s-(n+δ 0 n) Exp ≤ µ -δ 0 n Exp = e -αn .
(By the choice of δ 0 .)

Thus from dist v (f s (x), Crit) ≥ e -αs we get that

Comp f s (x) f s-n (D v (f n (x), e -δn )) ∩ Crit = ∅. This implies that f θ ′ n restricted to Comp f n-θ ′ n+1 (x) f -θ ′ n+1 (D v (f n (x), e -δn )) is univalent.
Since θ ′ n is an expanding time of f n-θ ′ n (x), we have

(5.1) diam Comp f n-θ ′ n+1 (x) f -θ ′ n+1 (D v (f n (x), e -δn )) ≤ µ -θ ′ n
Exp . Thus by (5.1) and Koebe distortion, there is a uniform constant C > 0 such that

Df θ ′ n f n-θ ′ n+1 (x) (v) ≥ Ce -δn µ θ ′ n Exp ,
where v is the unit vertical vector.

Next we replace f n (x) by f n-θ ′ n (x), and repeat the argument above, we get an estimate

Df θ ′ n 1 f n 1 -θ ′ n 1 +1 (x) (v) ≥ Ce -δn 1 µ θ ′ n 1 Exp ,
where

n 1 = n -θ ′ n.
We define n m := n m-1θ ′ n m-1 , m ≥ 1, we set n 0 = n. We can repeat this procedure until for some k, n -k i=0 n i θ ′ ≤ N . In this final time we can not define n k as

n k = n k-1 -θ ′ n k-1 , instead we choose the final n k to satisfying N < n -k i=0 n i θ ′ ≤ 2N .
Combining these estimates, take the product of this derivatives, we have

|(Df n ) x (v)| ≥ ε 1 Df θ ′ n k f n k -θ ′ n k +1 (x) (v) • • • Df θ ′ n f n-θ ′ n+1 (x) (v) ≥ ε 1 Ce -δn k µ θ ′ n k Exp • • • Ce -δn µ θ ′ n Exp ≥ ε 1 C k+1 e -δ(n-2N )/θ ′ µ n-2N
Exp , where we take ε

1 = min 0≤j≤2N |Df j | x (v)|.
It is not hard to give an upper bound of k. Indeed we let S

m := n -m i=0 n i θ ′ , then S m satisfies S m = (1 -θ ′ )S m-1 , for 1 ≤ m ≤ k. Thus we get S k = (1 -θ ′ ) k+1 n. Now S k > N implies k < log N -log n log(1-θ ′ ) -1.
Thus C k+1 is a sub-exponentially large term with respect to n. Taking the limit in the above inequality we get

χ -(x) = lim inf n→∞ 1 n log |Df n | x (v)| ≥ log µ Exp - δ θ ′ .
Letting α → 0 then δ/θ ′ → 0 as well, and we get χ -(x) ≥ log µ Exp .

Corollary 5.4. There are no wandering Fatou components in ∆ × C, W s (J(p)) = J(f ), and Fatou set F (f ) is equal to the union of basins of attracting cycles. Moreover Lebesgue a.e. point x ∈ J(f ) slowly approach Crit' and χ -(x) ≥ log µ Exp .

Proof. By Theorem 4.4 and Theorem 5.3, for Lebesgue a.e. point x ∈ W s (J(p)), x slowly approach Crit' and χ -(x) ≥ log µ Exp . We also know W s (J(p)) is the union of J(f ) and the wandering Fatou components.

It is clear that points in the Fatou set can not have a positive vertical Lyapunov exponent, thus there are no wandering Fatou component in ∆ × C, and W s (J(p)) = J(f ). Since every attracting basin of p bulges to an attracting basin of f , for every point x such that x / ∈ W s (J(p)) we get that x is in a basin of attracting cycle. Thus the Fatou set is the union of basins of attracting cycles. 6. The Julia set J(f ) has Lebesgue measure zero

In this section we prove Theorem 1.5, thus finishing the proof of the main theorem. In the following we assume that x ∈ J(f ) is both slowly approaching Crit' and satisfies χ -(x) ≥ log µ Exp . We begin with a definition. Definition 6.1. Let 1 < σ < log µ Exp , Let m be a positive integer. We say m is a σ-

hyperbolic time for x if Df m-i f i (x) (v) ≥ σ m-i holds for each 0 ≤ i ≤ m -1,
and v is the unit vertical vector.

We fix once for all 1 < σ < σ ′ < µ Exp . Since χ -(x) ≥ log µ Exp , the hyperbolic times have positive density by Pliss's Lemma [START_REF] Pliss | On a conjecture due to Smale[END_REF], in the following sense: Lemma 6.2. There is a constant θ > 0 such that if we consider the set

H n = m ∈ {1, • • • , n} : m is a σ ′ -hyperbolic time for x , then for large n we have #H n n > θ.
For the proof see [START_REF] Levin | The Lyapunov exponent of holomorphic maps[END_REF]Theorem 3.1].

Next for a positive integer n we define φ(f n (x)) :=log dist v (f n (x), Crit'). Multiplying the metric by a constant we can further assume φ is a positive function. We show that Lemma 6.3. There exists a constant C = C(x) > 0 such that for every n ≥ 0,

n-1 k=0 φ(f k (x)) ≤ Cn.
Proof. Fix α > 0 small, by slow approach for large n we have dist v (f n (x), Crit') ≥ e -αn , or in other words φ(f n (x)) ≤ αn.

We claim that there exist constant 0 < θ < 1 and C 1 > 0 such that such for large n we have n-1 k=θn φ(f k (x)) ≤ C 1 n. To show this, let z = π 2 (f θn (x)). By Lemma 2.3 we have

(1-θ)n-1 j=0 except M terms φ(p j (z)) ≤ Q(1 -θ)n.
In particular φ(p j (z)) ≤ Q(1θ)n holds for j appearing in above sum. On the other hand there is a constant K > 0 so that dist v (p j (z), f θn+j (x)) ≤ K j |λ| θn . We choose θ sufficiently close to 1 so that e Q(θ-1)n -K (1-θ)n |λ| θn ≥ e 2Q(θ-1)n . Thus we have dist v (f θn+j (x), Crit') ≥ dist(p j (z), Crit')dist v (f θn+j (x), p j (z)) ≥ e -φ(p j (z)) -K (1-θ)n |λ| θn ≥ e -φ(p j (z)) + e 2Q(θ-1)ne Q(θ-1)n ≥ e -2φ(p j (z)) , which implies φ(f θn+j (x)) ≤ 2φ(p j (z)). Then we get

n-1 k=θn except M terms φ(f k (x)) ≤ 2Q(1 -θ)n.
Together with slow approach φ(f n (x)) ≤ αn we have

n-1 k=θn φ(f k (x)) ≤ (2Q(1 -θ) + M α) n.
Setting C 1 := (2Q(1θ) + M α) we get the conclusion.

Repeat the above argument we get the estimate in the time θ 2 n to θn we get

θn-1 k=θ 2 n φ(f k (x)) ≤ C 1 θn.
Keep repeating the above argument in the time θ j n to θ j-1 n until for some j, the slow approach property dist v (f θ j+1 n (x), Crit') ≥ e θ j+1 n does not holds. The final step is a bounded time, and the sum of φ(f k (x)) in this bounded time is bounded by a constant depending on x. Summing up there is a constant C = C(x) > 0 such that n-1 k=0 φ(f k (x)) ≤ Cn.

We now introduce some notions from [START_REF] Levin | The Lyapunov exponent of holomorphic maps[END_REF]Theorem 3.1]. Given K > 0 we define the shadow S(j, K) of a positive integer j to be the following interval of the real line: S(j, K) := (j, j + Kφ(f j (x))].

For a positive integer N , let A(N, K) be the set of all positive integer n such that at most N integers j satisfy n ∈ S(j, K). The following lemma and Theorem are proved in [27, Theorem 3.1], and both rely on the one-dimensional DPU lemma (Lemma 2.3). In our case we can replace the DPU lemma by Lemma 6.3, and get exactly the same statements. Lemma 6.4. For any N and K, for n sufficiently large we have

{A(N, K) ∩ {1, • • • , n}} n ≥ 1 - CK N + 1 .
Theorem 6.5. Suppose m is an σ ′ -hyperbolic time and m ∈ A(N, 1 log σ ), then there exist a constant δ > 0 such that if we let V m be the connected component of f

-m D v (f m (x), δ) containing x, then f m : V m → D v (f m (x), δ) has degree at most N .
If we choose N sufficiently large, by Lemma 6.4 the density of A(N, 1 log σ ) is close to 1. Together with Lemma 6.2, we get for m large, the intersection H ′ m := H m ∩ A(N, 1 log σ ) has uniform positive density when m → ∞. Now by Theorem 6.5 we have Corollary 6.6. For large n there exist a subset H ′ n ⊂ {1, • • • , n} and constants α > 0, δ > 0 such that #H ′ n ≥ αn and for every m ∈ H ′ n , f m : V m → D v (f m (x), δ) has degree at most N .

Now we are able to establish the main result of this section.

Theorem 6.7. The Julia set J(f ) in the basin of L has Lebesgue measure zero.

Proof. Let δ and N be as in Corollary 6.6. First we observe that the Fatou set F (p) in the invariant fiber L has full Lebesgue measure, as a consequence of TCE. Let Ω be a relatively compact subset of F (p), then there exist a constant ε > 0 such that for x satisfying |π 1 (x)| < ǫ, π 2 (x) ∈ Ω, we have x ∈ F (f ). Thus for y ∈ ∆ × C with π 1 (y) sufficiently small we have meas D v (y, δ/2) ∩ J(f ) meas D v (y, δ/2) ≤ ε,

here meas denote the one-dimensional Lebesgue measure, and ε is a constant to be determined in the next paragraph. Now we argue by contradiction. Suppose J(f ) has positive Lebesgue measure, then by the Lebesgue density theorem and the Fubini theorem there exist x ∈ J(f ) such that x is a Lebesgue density point in the vertical line containing x. We may also assume that x has positive Lyapunov exponent and slowly approach Crit'. By Corollary 6.6 there is a sequence of positive integers {n 0 , • • • , n k , • • • } such that f n k : V n k → D v (f n k (x), δ) has degree bounded by N for all k ≥ 0. Let V ′ n k be the connected component of f -n k D v (f n k (x), δ/2) containing x. Let ε sufficiently small such that C 4 ε 2 -N << 1, where C 4 is the constant in Lemma 2.4 (2.4). By Lemma 2.4 (2.4) we have

meas V ′ n k ∩ J(f ) meas V ′ n k ≤ C,
where C < 1 does not depend on k. Again by Lemma 2.4 (2.1) diam V ′ n k is exponentially small, and also by Lemma 2.4 (2.2), V ′ n k has uniformly good shape (the ratio of the diameter and the inradius of V ′ n k is uniformly bounded). This contradicts that x is a Lebesgue density point. Thus J(f ) must have Lebesgue measure zero. Remark 6.8. The main theorem also holds in a slightly more general setting. Let ∆ be a disk. Let f : ∆ × P 1 → ∆ × P 1 be a skew product holomorphic map in the following form: f (t, z) = (λt, h(t, z)), where |λ| < 1 and h(t, z) is a rational map in z for fixed t. We assume moreover that the degree of h(t, z) in z is a constant for t ∈ ∆. Let L = {t = 0} be the invariant fiber and let p = f | L . Assume p has non-empty Fatou set and p satisfies either 1.TCE+WR or 2.Positive Lyapunov. Then the Fatou set of f is the union of the basins of attracting cycles and the Julia set of f has Lebesgue measure zero.

We notice that ∆ × P 1 can not be embedded into P 2 since any two projective lines in P 2 have non-trivial intersection. However ∆ × P 1 can be embedded into P 1 × P 1 , and the f above can be realized as a semi-local restriction of a globally defined meromorphic map from P 1 × P 1 to P 1 × P 1 . To construct such examples, we start with a skew product meromorphic self map f : P 1 × P 1 → P 1 × P 1 , f (t, z) = (g(t), h(t, z)), where g is a one-variable rational function and h is a two-variable rational function. The function h has finite number of indeterminacy points, and f is holomorphic outside these indeterminacy points. We choose g such that g has an attracting fixed point t 0 , and there are no indeterminacy points in the line {t 0 } × P 1 . Thus there exist a small neighborhood Ω of {t 0 } × P 1 such that f : Ω → Ω is holomorphic, thus f is a skew product holomorphic map. We note that indeterminacy points are necessary since a globally holomorphic self map of P 1 × P 1 must be a product map, see [START_REF] Favre | Dynamique des applications rationnelles des espaces multiprojectifs[END_REF]Remark 1.6].

Thus H δ is a continuous function, and H δ →log | • | when δ → 0 in L 1 (ω c ). According to [START_REF] Graczyk | Lyapunov exponent and harmonic measure on the boundary of the connectedness locus[END_REF] Thus the proof is complete.

  π 2 ) be the projection map to the first (resp. second) coordinate. Definition 3.2 (Hubbard and Oberste-Vorth). The map f is called a crossed mapping of degree d from B 1 to B 2 if there exists

Lemma 4 . 1 .

 41 If p satisfies TCE, then the Lebesgue measure of K m decreases exponentially fast with m.

H 1 i=0F-log δ n n- 1 i=0F( 1 n-(-

 111 section 1.1, for a.e. c ∈ ∂M d in the sense of harmonic measure, the critical value c of f c is typical with respect to ω c . Here typical means for every continuous function H on J(f c ), i c (c)) = H dω c .Applying (B.1) to H δ , together with that fact thatH δ →log | • | in L 1 (ω c ) as δ → 0 δ (f i c (c)) =log |z| dω c = 0.On the other hand, for every δ > 0 let F δ be a positive continuous function such that supp F δ ⊂ D(0, 2δ), F δ ∞ = 1 and F δ ≥ χ D(0,δ) . Then for for a.e. c ∈ ∂M d in the sense of harmonic measure we have(B.3) lim n→∞ log δ n nδ (f i c (c)) =log δ F δ dω c ≤log δ ω c (D(0, 2δ)).By[START_REF] Przytycki | Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps[END_REF] Lemma 4] (or by the fact that the dynamical Green function is Hölder continuous, see[START_REF] Sibony | Dynamique des applications rationnelles de P k[END_REF] Theorem 1.7.3), for every c ∈ C there exist constants C = C(c) > 0, α = α(c) > 0 such that for every r > 0 we have ω c (D(0, r)) ≤ Cr α . Thus for c satisfying (B.3) we have lim n→∞ δ (f i c (c)) ≤log δC(2δ) α . .2) and (B.4) we get for a.e. c ∈ ∂M d in the sense of harmonic measure (c) / ∈D(0,δ) log |f i c (c)| = 0. Finally by the main theorem of [21], for a.e. c ∈ M d in the sense of harmonic measure lim n→∞ log |(f n c ) ′ (c)| = log d, which is equivalent to (since |f ′ c (z)| = d|z d-1 | ) log |f i c (c)| = 0. Combining (B.5) and (B.6) we get for a.e. c ∈ M d in the sense of harmonic measure 0,δ) (f i c (c)) = 0. Since |f ′ c (f i c (c))| = d|f i c (c)| d-1 , combining (B.7) and (B.8) we get lim log |f ′ c (f i c (c))| = 0.

Appendix A. Relations between non-uniformly hyperbolic conditions

In this Appendix we study the relations between non-uniform hyperbolic conditions given in section 2. In the following we assume f is a rational map on P 1 and distance are relative to the spherical metric.

Definition A.1. A rational map f satisfies Slow Recurrence condition with exponent α (SR(α) for short) if for every critical point c ∈ J(f ), there exist an α > 0 such that dist(f n (c), Crit') ≥ e -nα for n large.

Lemma A.2. WR(η,ι) implies SR(α) for some α(ι) > 0, and α → 0 when ι → 0.

Proof. By the definition of WR(η,ι) in particular we have

then it is straightforward that there exist an α > 0 such that dist(f n (c), Crit') ≥ e -nα for large n, and α → 0 when ι → 0. Thus for every β > 0 we have |f ′ (f n (c))| > e -βn for large n . Similarly to Lemma A.2, there exist an α > 0 such that dist(f n (c), Crit') ≥ e -nα for large n, and α → 0 when β → 0. Thus f satisfies SR(α) for every α > 0.

Lemma A.4. TCE+WR(η,ι) with η small or TCE+SR(α) with α small implies CE.

Proof. This lemma was proved by Li in [START_REF] Li | Topological invariance of the Collet-Eckmann condition for one-dimensional maps[END_REF] for real maps, and Li's argument can also apply to rational maps. Here we give a simple proof for rational maps. This kind of argument has already appeared in [START_REF] Przytycki | Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps[END_REF].

Note that for all large n, rM -εn ≥ e -α 1 n , here r is the constant appearing in the definition of the TCE condition. We note that

here Comp y means the connected component containing y For every 0 ≤ s ≤ n and n large we have

Exp = e -αn By SR(α), for all large n and all 0 ≤ s ≤ n we have

Exp . Since α is small we get f is CE.

Appendix B. Genericity of non-uniformly hyperbolic conditions

In this Appendix we give some families of polynomials satisfying the consitions in our main theorem (i.e. TCE+WR or Positive Lyapunov).

In real dynamics, the WR condition was first introduced in the Tsujii's paper [START_REF] Tsujii | Positive Lyapunov exponents in families of one dimensional dynamical systems[END_REF]. Avila and Moreira proved that CE+WR condition generic (has full Lebesgue measure) in every non-trivial analytic family of S-unimodal maps [START_REF] Avila | Statistical properties of unimodal maps: smooth families with negative schwarzian derivative[END_REF]. The condition CE+WR was also studied by Luzzatto and Wang [START_REF] Stefano | Topological invariance of generic non-uniformly expanding multimodal maps[END_REF], and also by Li [START_REF] Li | Topological invariance of the Collet-Eckmann condition for one-dimensional maps[END_REF] in relation to topological invariance. For the Positive Lyapunov condition, Avila and Moreira proved that this condition is generic (has full Lebesgue measure) in every non-trivial analytic family of quasi-quadratic maps [START_REF] Avila | Statistical properties of unimodal maps[END_REF]. The quadratic family f t (x) = tx 2 for -1 4 ≤ t ≤ 2 is obviously a non-trivial analytic family of quasi-quadratic maps, and of S-unimodal maps. So our theorem also applies for these real polynomials (seen as complex dynamical systems).

In the rational map case it was shown by Astorg, Gauthier, Mihalache and Vigny that the CE and WR(η, ι) with arbitrarily small ι are robust [5, Lemma 5.5] (in the sense that there is a positive Lebesgue measure set in the parameter space satisfying both these two conditions).

Next we consider the family of uni-critical polynomials, i.e. the family f c (z) = z d + c , c ∈ C and d ≥ 2 an integer. We let M d be the connectedness locus, and let ∂M d be the bifurcation locus. There is a harmonic measure (with pole at ∞) supported on ∂M d . It is shown by Graczyk and Swiatek in [START_REF] Graczyk | Lyapunov exponent and harmonic measure on the boundary of the connectedness locus[END_REF] that for a.e. c ∈ ∂M d in the sense of harmonic measure the Lyapunov exponent at c exist and is equal to log d.

For the WR condition, the author is told by Jacek Graczyk that WR condition is actually generic in the sense of harmonic measure in the family of uni-critical polynomials. We thank Jacek Graczyk for kindly let us write down his argument here.