Non-density of stability for holomorphic mappings on <span class="mathjax-formula">$\protect \mathbb{P}^k$</span>

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Non-density of stability for holomorphic mappings onP Tome 4 (2017), p. 813-843.

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NON-DENSITY OF STABILITY FOR HOLOMORPHIC MAPPINGS ON P

^k

by Romain Dujardin

Abstract. — A well-known theorem due to Mañé-Sad-Sullivan and Lyubich asserts that J-stable maps are dense in any holomorphic family of rational maps in dimension 1. In this paper we show that the corresponding result fails in higher dimension. More precisely, we construct open subsets in the bifurcation locus in the space of holomorphic mappings of degreedofP^k(C)for everyd>2andk>2.

Résumé(Non-densité de la stabilité pour les applications holomorphes surP^k)

Un théorème célèbre dû à Mañé-Sad-Sullivan et Lyubich affirme que les paramètresJ-stables forment un ouvert dense de toute famille holomorphe de systèmes dynamiques rationnels en dimension 1. Dans cet article nous montrons que ce résultat ne subsiste pas en dimension supérieure. Plus précisément nous construisons des ouverts contenus dans le lieu de bifurcation des applications holomorphes de degréddeP^k(C)pour toutd>2etk>2.

Contents

1. Introduction. . . 813

2. Preliminaries. . . 815

3. Robust bifurcations from topology. . . 818

4. Robust bifurcations from fractal geometry. . . 825

5. Further considerations and open problems. . . 837

References. . . 842

1. Introduction

Let(f_λ)_λ∈Λbe a holomorphic family of holomorphic self-mappings on the complex projective spaceP^k. When the dimensionkequals1, the stability/bifurcation theory of such families was developed in the beginning of the 1980’s independently by Mañé, Sad and Sullivan [MSS83] and Lyubich [Lyu83, Lyu84], who designed the seminal notion ofJ-stability, that is, structural stability on the Julia set, which almost implies structural stability onP¹ (see [MS98]). A salient feature of their work is that the set

Mathematical subject classification (2010). — 37F45, 37F10, 37F15.

Keywords. — Holomorphic dynamics in higher dimension, J-stability, bifurcations, blenders.

Research partially supported by ANR project LAMBDA, ANR-13-BS01-0002 and a grant from the Institut Universitaire de France.

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of J-stable mappings is (open and) dense in any such family, which is ultimately a consequence of the finiteness of the critical set.

In higher dimension we denote byJ^∗the “small Julia set” off, which by definition is the support of its measure of maximal entropy. Typically,J^∗ is smaller than the usual Julia setJ, and concentrates in a sense the repelling part of the dynamics off, and most of the entropy.

In a remarkable recent paper, Bianchi, Berteloot and Dupont [BBD17] put forward J^∗-stability as the correct generalization of the Mañé-Sad-Sullivan-Lyubich theory in higher dimension. As these authors point out, there is no reason to expect that the density ofJ^∗-stability should hold in this setting, and leave the existence of persistent bifurcations as an open problem [BBD17, §6.2].

At this stage it is worth mentioning that for invertible dynamics in dimension2, the classical work of Newhouse [New70] shows that persistent bifurcations can be obtained by constructing persistent (generic) homoclinic tangencies. In the holomorphic context, Buzzard [Buz97] proved that persistent homoclinic tangencies exist in the space of polynomial automorphisms of C² of sufficiently high degree. Furthermore such automorphisms can easily be “embedded” inside holomorphic mappings of P² (see e.g. [HP94, §6] and also [Gav98]). It seems however that for these examples, the Newhouse phenomenon is somehow unrelated toJ^∗-(in)stability, which has to do with repelling periodic orbits (see Section 5.1 for a more precise discussion).

Our main goal in this paper is to address the density problem forJ^∗-stability. We denote byH_d(P^k)the space of holomorphic mappings ofP^k of degreed. By choosing a set of homogeneous coordinates on P^k, we can express f ∈ Hd(P^k) in terms of a family of(k+1)homogeneous polynomials in(d+1)variables without common factor.

In this way we get a natural identification betweenHd(P^k)and a Zariski open subset ofP^N, forN= (k+ 1)

(d+k)!/d!k!

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Main Theorem. — The bifurcation locus has non-empty interior in the spaceHd(P^k) for everyk>2andd>2.

As we shall see, our constructions are rather specific, since we start with the sim- plest possible mappings, namely products, and construct robustly bifurcating examples by taking small perturbations. More interestingly, we isolate some mechanisms leading to open sets of bifurcations. The next step would be to understand the preva- lence of these phenomena in parameter space.

While he was working on this paper, the author learned about the work of Bianchi and Taflin [BT17], in which the authors construct a 1-dimensional family of rational maps on P² whose bifurcation locus is the whole family, thus offering an alternative approach to the question of [BBD17].

Throughout the paper, to avoid overwhelming the ideas under undue technicalities, we focus on dimension k= 2 which is the first interesting case, and explain in Section 5.2 how to adapt the arguments to higher dimension.

By the analysis of Berteloot, Bianchi and Dupont, to obtain open subsets in the bifurcation locus, it is enough to create a persistent intersection between the

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post-critical set and a hyperbolic repeller contained in J^∗. We present two mechanisms leading to such persistent intersections. The first one is based on topological considerations: the idea is to construct a kind of topological manifold contained inJ^∗, which must intersect the post-critical set for homological reasons. We apply this strat- egy for mappings that are small perturbations of product maps of the form(p(z), w^d) (see Theorem 3.1). Of course what is delicate here is to manage dynamically defined sets with inherently complicated topology.

The second mechanism is based on ideas from fractal geometry. It relies on the construction of Cantor sets with a very special geometry: namely they are “fat” in a certain direction and admit persistent intersections with manifolds that are sufficiently transverse to the fat direction. The use of this property in dynamical systems originates in the work of Bonatti and Díaz [BD96] and has played an important role in C¹ dynamics since then (see the book by Bonatti, Díaz and Viana [BDV05] for a broader perspective). According to the real dynamics terminology we will refer to these Cantor sets asblenders.

We implement this idea for product mappings of the form(p(z), w^d+κ), wherep admits a repelling fixed pointz0which is almost parabolic, that is1<|p⁰(z0)|<1.01, andκis large (see Theorem 4.7). The parabolic case follows by taking the limit (see Corollary 4.11).

Despite the specific nature of these examples, our thesis is that the interior of the bifurcation locus is quite large. We point out a few explicit questions and conjectures in Section 5.3. In particular since by [BBD17] bifurcations happen when a multiplier of a repelling periodic point crosses the unit circle, it is likely that blenders are often created in this process.

Note also that blenders already appear in complex dynamics in the work of Biebler [Bie16] to construct polynomial automorphisms ofC³with persistent homoclinic tangencies.

The plan of the paper is as follows. In Section 2 we collect a few facts from J^∗-stability theory. The topological mechanism for robust bifurcations is detailed in Section 3, while Section 4 is devoted to complex blenders. Finally in Section 5 we explain how to extend the construction of Section 4 to higher dimensions and also show thatJ^∗-stability is compatible with the Newhouse phenomenon. We also state a number of open problems and directions for future research. The main theorem ultimately follows from Theorems 3.1 and Theorem 4.7 for the case k = 2, d > 3, Theorem 4.12 fork= 2, d= 2 and Theorem 5.4 fork>3.

Acknowledgements. — Thanks to Fabrizio Bianchi, Lasse Rempe-Gillen and Johan Taflin for useful comments, and to the anonymous referee for a careful reading of the paper.

2. Preliminaries

Notation. — When a preferred parameter λ0 is given in Λ, to simplify notation we denote f_λ₀ by f₀. We use the convention to mark with a hat the objects inΛ×P^k, likefb: (λ, z)7→(λ, f_λ(z)), etc.

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2.1. Stability and bifurcations for endomorphisms ofP^k. — Letf be a holomorphic map of degreedonP^k. A classical result of Briend and Duval asserts that it admits a unique invariant measure of maximal entropyµ_f, whose (complex) Lyapunov exponents χ_i, i = 1, . . . , k satisfy χ_i > ¹2logd, and which describes the asymptotic distribution of (repelling) periodic orbits [BD01]. Recall from the introduction that for a holomorphic mapf onP^k, we denote byJ^∗ the support ofµf.

Repelling periodic points are dense inJ^∗. In general the closure of repelling periodic orbits can be strictly larger thanJ^∗, which is a source of technicalities inJ^∗-stability theory. Note however that this phenomenon does not happen for regular polynomial skew products ofC²(see [Jon99]), which are basic to all constructions in this paper.

Let now(f_λ)_λ∈Λ be a holomorphic family of holomorphic maps of degree donP^k (or equivalently, a holomorphic map Λ → Hd(P^k)). A notion of stability for such a family was recently introduced in [BBD17]. The relevance of this notion is justified by its number of natural equivalent definitions. The following is a combination of Theorems 1.1 and 1.6 in [BBD17], with a slightly different terminology (see below for the notion of Misiurewicz bifurcation).

Theorem2.1(Berteloot, Bianchi & Dupont). — Let(fλ)_λ∈Λbe a holomorphic family of holomorphic maps of degreedonP^k. Then the following assertions are equivalent:

(i) The function on Λ defined by the sum of Lyapunov exponents of µf_λ: λ 7→

χ₁(λ) +· · ·+χ_k(λ)is pluriharmonic onΛ.

(ii) The sets(J^∗(f_λ))_λ∈Λ move holomorphically in a weak sense.

(iii) There is no (classical) Misiurewicz bifurcation inΛ.

If in addition Λ is a simply connected open subset of H_d(P^k) or if Λ is any simply connected complex manifold andk= 2, these conditions are equivalent to:

(iv) Repelling periodic points contained in J^∗ move holomorphically overΛ.

If these equivalent conditions hold we say that the family isJ^∗-stableoverΛ. For an arbitrary family we thus have a maximal open set of stability, thestability locusand its complement is by definition thebifurcation locus. We denote byBif the bifurcation locus.

Condition (ii) will not be used below so we don’t need to explain what the “weak sense” in (ii) exactly is. By “classical Misiurewicz bifurcation” in (iii) we mean a proper intersection in Λ×P^k between a post-critical component offband the graph of a holomorphically moving repelling periodic pointγ: Ω→P^k over some open set Ω⊂Λ. We will treat (a generalized form of) this notion in detail in Section 2.2 below (see in particular Definition 2.4 for the notion of proper intersection).

Let us also quote the following result, which is contained in Proposition 2.3 and Theorem 3.5 in [BBD17].

Proposition2.2. — Let (f_λ)_λ∈Λ be a holomorphic family of holomorphic mappings of degree d onP^k, with Λ simply connected. Assume that there exists a holomorphic map γ: Λ→P^k such that for every λ∈Λ,γ(λ) is disjoint from the post-critical set of f_λ. Then(f_λ)isJ^∗-stable.

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2.2. Hyperbolic repellers in J^∗. — Recall that if E0 is an invariant hyperbolic compact set for f0, then forf ∈ Hd sufficiently close tof0, there exists a hyperbolic set E(f) for f and a conjugating homeomorphism h = h(f) : E₀ → E(f). The conjugacy can be chosen to depend continuously on f, with h(f₀) = id, in which case it is unique and is actually a holomorphic motion. We will refer to it as the continuationofE0.

According to the usual terminology, a basic repeller E is a locally maximal and transitive hyperbolic repelling invariant set for f0. It is classical that ifE is a basic repeller, then repelling periodic points are dense inE.

Lemma2.3(see also [Bia16, Lem. 2.2.15]). — Let f0 be a holomorphic map on P^k of degree d > 2. Let E₀ be a basic repeller for f₀ such that E₀ ⊂ J^∗(f₀). Then there exists a neighborhoodΩof f0 inHd such that forf ∈Ωthe continuationE(f)ofE0

is a well-defined basic repeller forf, contained inJ^∗(f).

Proof. — The existence of E(f) and the fact that it is a basic repeller follows from general hyperbolic dynamics, so we concentrate on the last conclusion. By assumption there exists a neighborhoodN ofE₀ such thatT

k>0f^−k(N) =E₀, where we restrict to the preimages remaining in N. We can assume thatN is ar-neighborhood ofE0

for somer= 2r0>0. SinceE0is contained inJ^∗(f0), for every x∈E0we have that µ_f₀(B(x, r₀)) >0. Furthermore by compactness there exists δ₀ such that for every x∈E0,µf(B(x, r0/2))>δ0.

Now, forf close enough tof0,N =S

x∈E0B(x,2r0)is also an isolating neighborhood for E(f). In addition, the uniform continuity of the potential of the maximal entropy measure implies that ifx∈E₀,µ_f(B(x, r₀/2))>δ₀/2. By construction, iff is close enough tof0, everyyinEf is the intersection of a sequence of nested preimages:

{y}=T

k>0f_y⁻ⁿ(B(fⁿ(y), r0)), wheref_y⁻ⁿdenotes the inverse branch offⁿmapping fⁿ(y) to y, and fⁿ(y) is r₀/2 close to E₀. Therefore B(fⁿ(y), r₀) ⊃ B(x, r₀/2) for some x∈E₀, henceµ_f(B(fⁿ(y), r₀))>δ₀/2, and finallyy∈Supp(µ_f).

We now introduce a notion of proper intersection between a hyperbolic set and the post-critical set.

Definition2.4. — Let(fλ)λ∈Λ be a holomorphic family of holomorphic maps onP^k of degreed>2 onP^k and letλ₀∈Λ. Let E₀be a basic repeller forf₀. We say that a post-critical component V0 ⊂ f₀ⁿ(Crit(f0)) intersects E0 properly if there exists x0∈V0∩E0such that the graphxbof the continuation ofx0as a point of E0 is not contained inVb, where Vb is the irreducible component offbⁿ(Crit(fb))containingV0.

The terminology is justified by the fact that sincefbⁿ(Crit(fb))is of codimension1 in Λ×P^k, the assumption on bximplies that the graph bxand fbⁿ(Crit(fb))intersect properly at(λ₀, x₀). Note that this terminology is slightly inadequate since the notion depends on the family and not only onf0.

The mechanism leading to robust bifurcations will be based on the following lemma, which is implicitly contained in [BBD17] (see the proof of Proposition 3.8 there).

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Proposition-Definition2.5. — Let(fλ)_λ∈Λ be a holomorphic family of holomorphic maps on P^k of degree d > 2 on P^k. Assume that for some λ0 ∈ Λ, there exists a basic repeller E₀, contained in J^∗(f₀) and an integer n > 1 such that f₀ⁿ(Crit(f₀)) intersects E₀ properly.

Thenλ0 belongs to the bifurcation locus of the family. If such a situation happens, we say that a Misiurewicz bifurcation occurs atλ0.

Proof. — Fix a sequence of repelling periodic pointsxj inE0, converging tox0. The stability of proper intersections in analytic geometry [Chi89, §12.3] implies that there exists a sequence λ_j →λ₀ such thatx_j(λ_j)properly intersects f_λⁿ

j(Crit(f_λ_j)). Then by Theorem 2.1 we infer thatλj belongs toBif, and so does λ0. It follows that in condition (iii) of Theorem 2.1 we can replace classical Misiurewicz bifurcations by Misiurewicz bifurcations in this sense.

3. Robust bifurcations from topology

As outlined above, to construct robust bifurcations, we need to find situations in which the post-critical set has a robust intersection with a basic repeller. Our first method to produce such a robust intersection is based on topological arguments.

The starting parameter is a product map in C² of the form f(z, w) = (p(z), w^d), wherepis of degreed. Thenf extends toP² as a holomorphic map. The small Julia set off isJ^∗(f) =Jp×S¹, whereS¹ denotes the unit circle inC.

Theorem3.1. — Let f(z, w) = (p(z), w^d)be a product map withdeg(p) =d. Assume that psatisfies the following properties:

(A1) there exists a basic hyperbolic repellerE forpthat disconnects the plane;

(A2) pbelongs to the bifurcation locus inP_d(C).

Thenf belongs to the closure of the interior of the bifurcation locus in Hd(P²).

More precisely there exists a sequence of polynomialspj of degreedconverging top, a polynomial q of degree 62 and a sequenceε_j →0 such that for every j, the map fj ∈ Hd(P²)defined byfj(z, w) = (pj(z) +εjq(w), w^d)belongs toBif˚.

Notice that the assumption onprequires⁽¹⁾d>3, so the perturbationεh(w)does not affect the highest degree part off, hencef_ε∈ Hd(P²).

We denote byP_d(C)the space of polynomials of degreedinC. Examples of polynomials p ∈ Pd(C) satisfying the assumptions of Theorem 3.1 are abundant. One may be tempted to think that as soon as padmits one active critical point and an attracting periodic orbit, then there is a nearby pesatisfying the assumptions of the theorem. The next two corollaries are results in this direction. They will be proven at the end of this section.

(1)Indeed, we essentially need both an attracting orbit and an active critical point.

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Corollary3.2. — Let f(z, w) = (p(z), w^d), withdeg(p) =d>3. Assume thatpis a bifurcating polynomial in Pd(C), such that (d−2) of its critical points are attracted by periodic sinks. Thenf belongs to the closure of the interior of the bifurcation locus inH_d(P²).

When several critical points are active it is convenient to use the formalism of bifurcation currents (see [Duj14] for an introduction to this topic; the bifurcation current is denoted byTbif).

Corollary 3.3. — Let f(z, w) = (p(z), w^d), with deg(p) = d > 3. Assume that in Pd(C), p∈ Supp(T_bif^k ) for some1 6k 6d−1 and that (d−1−k) critical points are attracted by sinks. Thenf belongs to the closure of the interior of the bifurcation locus inHd(P²).

Notice that this holds in particular whenpbelongs to the Shilov boundary of the connectedness locus (corresponding to the casek=d−1).

Proof of Theorem3.1. — Observe first that it is enough to prove the result under the assumption

(A2’) there exists a simple critical point c for p and an integer k > 1 such that p^k(c)∈E.

Indeed let p0 be a polynomial satisfying (A1) and (A2). Since p0 belongs to the bifurcation locus, it has an active critical pointc. Taking a branched cover of parameter space, we may always assume thatc can be followed holomorphically asp7→c_p. Also,E locally persists as a repelling setEp in a neighborhood ofp0. A well-known and elementary normal families argument shows that there exists an arbitrary small perturbationp₁ofp₀and an integerksuch thatp^k₁(c_p₁)∈E_p₁. Furthermore the set of polynomialsp∈ Pdpossessing a multiple critical point is algebraic. SinceEis infinite, the set of polynomialsp∈ Pd such thatp(cp)∈E is not locally analytic nearp1.

Altogether, it follows that there is a sequence of polynomialspjsatisfying (A1) and (A2’) and converging to p₀. Thus, if the theorem has been shown to hold under the assumptions (A1) and (A2’) we simply pick such a sequence and get that the result holds forp0as well.

In a first stage let us prove the theorem in the case wherek= 1.

Step 1: topological stability of the hyperbolic set. — For ε= 0, f =f₀ admits a basic repelling set E =E×S¹. Notice thatE has empty interior in the plane. If g∈ Hd

is sufficiently close to f, E = E(f) admits a continuation E(g) as a hyperbolic set, that is, there exists a continuous (even holomorphic ing) family of equivariant home- omorphismshf,g:E(f)→ E(g). Actually there is more:

Lemma3.4. — The conjugating homeomorphism hf,g can be extended to a homeo- morphism ofP²(not compatible with the dynamics) depending continuously on gand such that hf,f = id.

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Proof. — The result is more or less part of the folklore. For completeness we sketch the argument. Let U be an isolating neighborhood of E, which can be assumed to have smooth boundary, and N be a neighborhood off in Hd such that for g ∈ N, T

n>0g⁻ⁿ(U) =E(g)(here and in the rest of the proof only preimages remaining inU are taken into account). By the hyperbolicity assumption (and reducingU and N a bit if necessary), there existsδ >0andλ >1 so that ifx, y∈U are such thatf^k(x) and f^k(y)remain inU andd(f^k(x), f^k(y))6δ for 06k6nthen d(x, y)< δλ⁻ⁿ, and the same is true for everyg∈ N. ReducingU andN again if necessary, we may assume that for everyg∈ N andx∈Udifferent preimages ofxundergstay far apart, so that each such preimage can be followed unequivocally asgvaries inN. In addition we can assume that for everyx, the continuationg₋₁(x)of any given preimagef₋₁(x) ofxsatisfiesd(f₋₁(x), g₋₁(x))< εforg∈ N, whereε < δ(1−1/λ)/3.

Given suchg we will construct the conjugacy h=hf,g by starting from ∂U. Fix h= idon∂U, then defineh:f⁻¹(∂U)→g⁻¹(∂U)by assigning to everyy∈f⁻¹(∂U) the corresponding preimage off(y)underg, and finally extendhto a homeomorphism from Urf⁻¹(U)to U rg⁻¹(U). In addition we can ensure that hisε-close to the identity. Now, for everyx∈UrE(f)we seth(x) =g−k◦h◦f^k(x), wherekis the last integer such thatf^k(x)∈U, andg_−k is the inverse branch off^katf^k(x)obtained by continuation off_−k :f^k(x)7→x. In this way we get an equivariant homeomorphism h:UrE(f)→UrE(g).

The point is to show that this conjugacy extends continuously toE(f). Then the resulting extension will be a homeomorphism by reversing the roles off andg, which simply extends to P² by declaring that h = id outside U, thereby concluding the proof.

To prove that such an extension exists, we need to prove that for every x∈ E, if (x_n)∈(UrE)^Nis any sequence converging toxthen(h(x_n))converges and its limit does not depend on(xn).

As a first step towards this result, let us show that for everyx∈UrE, we have that d(x, h(x))< δ/3. For this, letk be the last integer such thatf^k(x)∈U and let us show by induction onqthatd(f^k−q(x), g^k−q(h(x)))6εPq

j=0λ^−j. Then the result follows from our choice of εby putting q=k. Forq = 0, g^k(h(x)) =h(f^k(x)), and d(f^k(x), h(f^k(x)))6εby definition ofh. Now assume the result has been proved for some 06q6k−1. Then

d(f^k−q−1(x), g^k−q−1(h(x)))

6d(f₋₁(f^k−q(x)), g₋₁(g^k−q(h(x)))),

wheref₋₁, g₋₁ are appropriate inverse branches, 6d(f₋₁(f^k−q(x)), g₋₁(f^k−q(x))) +d(g₋₁(f^k−q(x)), g₋₁(g^k−q(h(x)))) 6ε+1

λd(f^k−q(x), g^k−q(h(x))),

where the last inequality follows from the definition ofεand the induction step. The estimated(x, h(x))< δ/3follows.

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We are now in position to conclude the argument. If(xn)and(yn)are two sequences converging to x, then for large n, the orbit of xn (resp.yn) shadows that of x for a long time. More precisely there exists k = k(n) −→

n→∞ ∞ such that for j 6 k, d(f^j(xn), f^j(yn))< δ/3. Therefore

d(h(f^j(xn)), h(f^j(yn))) =d(g^j(h(xn)), g^j(h(yn)))< δ,

and we conclude that d(h(xn), h(yn))< δλ^−k(n). The same argument applied toxn

andxmfor largen, mimplies that(h(xn))is a Cauchy sequence. Altogether, we infer that the sequence(h(xn))converges to a limit which depends only onx. This implies that the conjugacy extends toE and finishes the proof of the lemma.

Step 2: analysis off_ε. — In this paragraph we putf_ε(z, w) = (p(z) +εq(w), w^d)as in the statement of the theorem (an explicit expression forq will be given afterwards), and work only within the family(fε), whereεranges in a small neighborhood of0∈C. For notational ease, all dynamical objects will be indexed by ε (like E_ε, etc.). The problem is semi-local aroundE so we work inC². Also, we normalize thez-coordinate so thatp(c) = 0.

For every ε, the vertical line C ={c} ×Cis a critical component of fε. We set V_ε=f_ε({c} ×C), which is parameterized by

(1) V_ε=

(εq(w), w^d), w∈C .

The solid torusC×S¹ is totally invariant under the dynamics and containsEε. For small enoughε, the hyperbolic setE_εis homeomorphic to and close toE×S¹. Then from (A1) and Lemma 3.4 (applied tof

C×S¹) its complement inC×S¹admits both bounded connected components (theinsidepart ofEε) and a unique unbounded one (theoutside).

The following lemma provides a form of “topological transversality” between V_ε andEε.

Lemma3.5. — There exists a polynomialqof degree62and a sequenceεj →0such that forε=εj, the real analytic curve

vε=Vε∩(C×S¹) =

(εq(w), w^d),|w|= 1

intersects both the inside and the outside ofEε.

Proof. — Takeqof the formq(w) = (w−1)eq(w). Then the fiber{w= 1}is invariant under fε. Since in addition fε(z,1) = (p(z),1) does not depend on ε, we get that Eε∩ {w= 1}=E× {1}(indeed, ifx∈E× {1}thef_ε-orbit ofxtrivially shadows its f-orbit). The curvevε intersects{w= 1} in dpointsεq(ζ^k), where ζ=e^2iπ/d (here we identify{w= 1} withCand drop the second coordinate). Notice that fork= 0, εq(1) = 0 belongs toE. Putζ^±=ζ^±1.

Denote byOut(E)(resp.Inn(E)) the unbounded component of (resp. the union of bounded connected components of) the complement ofEin C, which by assumption are both non-empty. Remark that the uniform hyperbolicity and transitivity of E imply that everyx∈E is accumulated both byOut(E)andInn(E).

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Claim. — There exists αso that for q(w) = (w−1)(w−α), there exist εj → 0 so that εq(ζ⁻) (resp.εq(ζ⁺)) belongs to Inn(E)(resp.Out(E)).

Indeed letα∈R be such that(ζ⁺−α)/(ζ⁻−α) = e^iθ withθ /∈πQand assume that the claim is false. Then for every small enoughε,εq(ζ⁻)∈Out(E)implies that εq(ζ⁺)∈/ Inn(E). Using the above remark we reformulate this as

εq(ζ⁻)∈Out(E) =⇒εq(ζ⁺)∈C rInn(E) = Out(E)∪E= Out(E), so by continuity we conclude that

εq(ζ⁻)∈Out(E) =⇒εq(ζ⁺)∈Out(E).

Of course the reverse implication holds by symmetry, so we get that εq(ζ⁻)∈Out(E) ⇐⇒ εq(ζ⁺)∈Out(E).

Taking the complement and putting δ = εq(ζ⁻), we conclude that for every small enough δ∈C,

(2) δ∈Inn(E) ⇐⇒ δq(ζ⁺)

q(ζ⁻) ∈Inn(E).

Observe that q(ζ⁺)

q(ζ⁻)= ζ⁺−1 ζ⁻−1

ζ⁺−α

ζ⁻−α =−e^2iπ/dζ⁺−α

ζ⁻−α =−e^2iπ/de^iθ=e^iθ⁰, withθ⁰∈/πQ, therefore by (2),Inn(E)is invariant under an irrational rotation about0. Now, ifΩ is an open ball inInn(E)close to0, its orbit under this irrational rotation contains a circle about 0, which contradicts the fact that0 ∈Out(E), and concludes the proof of the claim.

Note that when E is a Jordan curve, which is often the case in practice (see Lemma 3.6), the argument can be simplified a bit. In particularEcannot be invariant under multiplication bye^2iπ/d at0so we can simply chooseq(w) =w−1.

Summing up, we have shown that for ε = ε_j, v_ε∩ {w= 1} intersects different connected components of{w= 1}rE. Thus to conclude the proof of the lemma it is enough to prove that for εsmall enough the map x7→Component(x)induces a 1-1 correspondence between the connected components of{w= 1}rEand the connected components of(C×S¹)rE_ε.

Indeed this is obvious for ε = 0. Now, when we vary ε, Lemma 3.4 shows that there is a homeomorphism hε of P² such that hε(E) = Eε, which is homotopic to the identity. Sincef_ε

{w=1} does not depend onε, the construction ofh_εshows that we can choose hε= id on this fiber. Thus for every x∈ {w= 1}rE, hε sends the connected component of (C×S¹)rE containing xto the connected component of (C×S¹)rE_ε containing x, which was the desired claim. This finishes the proof of

Lemma 3.5.

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Step 3: robustness of the Misiurewicz phenomenon inHd(P²). — Consider an arbitrary holomorphic map f⁰ ∈ Hd(P²) close tof0. Let us first analyze how the post-critical component V₀ can be continued forf⁰ (this has been done explicitly before for the family(f_ε)). Again, the problem is semi-local aroundE so for convenience we work in coordinates in C². The vertical line {c} ×C is a component of multiplicity 1 of Crit(f0), and moreoverCrit(f0)is smooth along{c} ×Coutside{w= 0}. Therefore every compact piece of ({c} ×C)r{w= 0}can be followed as a part of the critical set forf⁰ close to f0. More specifically, ifris so small thatD(c, r)contains no other critical point of p, and if we let U = D(c, r)× {1/10<|w|<10}, there exists a neighborhoodN(f0)such that for everyf⁰ ∈N(f0), Crit(f⁰)∩U is of multiplicity1 and consists in an annulusA(f⁰)which is a graph overA={c} × {1/10<|w|<10}, and converges to A as f⁰ → f0. Finally we define V(f⁰) to be the component of f⁰(Crit(f⁰))containingf⁰(A(f⁰)).

Fix a continuous function ϕ₀ : C×S¹ → R (depending only on z ∈ C) such that E₀ ={ϕ₀= 0}, Inn(E₀) ={ϕ₀<0} andOut(E₀) ={ϕ₀>0}. Extend it toC² by putting ϕ0(z, w) := ϕ0(z). Define also ψ0 by ψ0(z, w) = |w| −1, and let Φ0 = (ϕ0, ψ0) :C² →R², so that E0={Φ0= 0}. By Lemma 3.4, forf⁰ close tof0, there exists a continuous map Φ_f⁰ = (ϕ_f⁰, ψ_f⁰) : C² → R² such that E(f⁰) = {Φf⁰ = 0}.

Notice that forf⁰ =fε, sinceEε⊂C×S¹, we can chooseψf_ε =:ψε=ψ0.

Let now f⁰ =fε_j be as in Lemma 3.5. We will use elementary algebraic topology to show that V(f⁰) intersects E(f⁰) for f⁰ close to fεj. Working in the natural pa- rameterization ofV_εgiven in (1), the real valued functionϕ_ε_j changes sign along the segment [ζ⁻, ζ⁺] of the real curve vε_j, say ϕε_j(ζ⁻) <0 < ϕε_j(ζ⁺). In other words, ϕε_j◦fε_j(c,·)changes sign along the segment [ζ⁻, ζ⁺]of the unit circle.

Consider a simple loop`in the complex plane, starting atζ⁻, then joining it toζ⁺ outside the unit disk, and then returning back toζ⁻inside the unit disk, and staying in the annulus A (e.g. we can take the exponential of the oriented boundary of the rectangle[1−ρ,1 +ρ]×

−^2π_d,^2π_d

for some smallρ >0.) Then the loopfε_j({c} ×`) is disjoint fromE_ε_j and the winding number ofΦ_ε_j alongf_ε_j({c} ×`)is equal to1.

Forf⁰ sufficiently close tofε_j, we can lift{c} ×`to a loop`⁰ contained inA(f⁰)⊂ Crit(f⁰). By continuity, the winding number of Φf⁰ alongf⁰◦`⁰ is1. Since`⁰ bounds a disk in A(f⁰), we infer that Φf⁰ must vanish on f⁰(A(f⁰)). In other words, V(f⁰) intersectsE(f⁰), which was the desired result.

To conclude that a robust Misiurewicz bifurcation occurs at fε_j, it remains to check that the intersection betweenV(f⁰)andE(f⁰)that we have produced is proper.

For this, it is enough to observe that there are product maps (p₁(z), w^d), with p₁ arbitrary close top, such that ifc1∈Crit(p1)is the critical point continuingc, then p1(c1)∈/E(p1). Indeedcis active atpinPd(C)so under arbitrary small perturbations, it can be sent into the basin of a sink. This finishes the proof of the theorem in the case wherek= 1.

It remains to treat the case wherekis arbitrary. Observe that we can assume that p^j(c)is not critical for16j6k, otherwise we replacecby the last appearing critical

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point in this orbit segment. The structure of the proof is the same. Step 1 doesn’t need to be modified.

For Step 2, since in the case k = 1 we have used the explicit parameteriza- tion of f_ε({c} ×C), here rather than working with f_ε^k(Crit(f_ε)), we show that fε(Crit(fε)) intersects fε^−(k−1)(Eε) in topologically transverse manner for a well- chosen sequence(εj). The only slight difficulty to keep in mind when pulling back is thatEε can intersect other components of the post-critical set.

We put F =p^−(k−1)(E)and normalize the zcoordinate so thatp(c) = 0∈F. As in the previous case letqbe of the formq(w) = (w−1)(w−α). LetFε=fε^−(k−1)(Eε) which is contained in C×S¹. For ε = 0, F0 = F ×S¹ but now Fε needn’t be homeomorphic toF0. Sincef is proper and dominantFεis a compact set with empty interior. Thus if Ωis a bounded connected component of(C×S¹)rEε,fε^−(k−1)(Ω) is a union of bounded components of (C×S¹)rF_ε. It follows that in C×S¹, Inn(Fε) =fε^−(k−1)(Inn(Eε))and likewise forOut(Fε).

We have thatFε∩ {w= 1}=F, and if(z,1)∈Inn(F)inC, then(z,1)∈Inn(Fε) in C×S¹, and likewise for the outer part. Indeed f^k−1(z,1) = (p^k−1(z),1) and p^k−1(z)∈Inn(E)so(p^k−1(z),1)∈Inn(E_ε)and the result follows by pulling back the corresponding inner component. So we can argue exactly as in Lemma 3.5 to conclude that there exists a sequenceεj→0such that the real curveve=fε({c}×C)∩(C×S¹) intersects bothInn(Fε)andOut(Fε).

This analysis being done, we can push forward by f_ε^k−1

j to get a segment of the real analytic curvef_ε^k_j({c} ×C)∩(C×S¹)whose endpoints belong toInn(Eε_j)and Out(Eε_j) respectively and apply the argument of Step 3 without modification. The

proof is complete.

Corollary 3.2 is a direct consequence of the following lemma.

Lemma 3.6. — Let p be a polynomial of degree d > 3 with marked critical points c1, . . . , cd−1. Assume that c2, . . . , cd−1 are attracted by attracting cycles and thatc1 is active. Then there existspearbitrary close topsatisfying assumptions (A1) and (A2).

Proof. — Observe first that ifpis an arbitrary polynomial and B is the immediate basin of attraction of an attracting cycle of period`, such that∂Bcontains no critical point nor parabolic periodic points, then by Mañé’s lemma [Mañ93]∂Bis a hyperbolic set, which obviously disconnects the plane. We claim that it is basic, that is, transitive and locally maximal. Local maximality follows from general arguments: namely by [PU10, Lem. 6.1.2] it follows from the fact that f

∂B is an open map (due to the absence of critical points).

To prove transitivity, letΩbe a component ofB(f^`(Ω) = Ω). It is enough to show that f^`

_∂Ωis transitive. Uniform hyperbolicity implies that ∂Ω is locally connected, therefore the Riemann mapφ:D→Ωextends by continuity toφ:D→Ω. In addition φ

D must be injective otherwise contradicting the polynomial convexity of Ω.

Thus∂Bis a Jordan curve. (This result actually holds in much greater generality, see [RY08]).

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The formulaφf^`φ⁻¹defines a self-map of the unit disk, which can be extended by Schwarz reflexion to a rational map onP¹preserving the unit circle, hence a Blaschke product. Now the Julia set of a Blaschke product is either the unit circle or a Cantor set inside the unit circle, and the second case occurs only if there is an attracting or parabolic fixed point on the circle, which is not the case here. We conclude that the Julia set is the circle, hence topologically transitive (even topologically mixing).

Coming back to f we conclude that f^`

_∂Ω is topologically mixing, which yields the desired claim.

Under the assumptions of the lemma, letB be the immediate attracting basin of the cycle attracting, say, c2, which is robust under perturbations, and consider the active critical pointc₁. Sincepis a bifurcating polynomial, one can make a cycle of period`mbifurcate close to p, so there existsp1 close to pwith a parabolic pointα of period`mand multiplier different from±1. Then:

(1) α /∈∂B: indeed p^`m₁ behaves like a rational rotation of order greater than 2 at α, so there cannot exist an invariant Jordan curve through α. On the other hand by [RY08]∂Bis a Jordan curve invariant underp^`.

(2) αmust attractc1 under iteration ofp^`m₁ , thereforec1∈/∂B.

This implies that for this polynomialp1,∂Bis a basic repeller disconnecting the plane so (A1) holds. Since in additionp₁has a parabolic cycle, (A2) holds and the corollary

follows.

Proof of Corollary3.3. — We freely use the formalism of bifurcation currents, see [Duj14] for an account. By passing to a branched cover, it is no loss of generality to assume that the critical points are marked as c₁, . . . , c_d−1. Let T_i, 16i6d−1, be the associated bifurcation currents. Since p ∈ Supp(T_bif^k ), reordering the critical points if necessary, we can assume that p ∈ Supp(T1 ∧ · · · ∧Tk) and that c_k+1, . . . , c_d−1 are attracted by sinks. Then, using the continuity of the potentials of the Ti, we infer that there exists a sequence of subvarieties[Wn] of codimension k−1 in Pd along whichc2, . . . , ck are periodic and a sequence of integers(dn)such thatT1∧ · · · ∧Tk= limn→∞T1∧d⁻¹_n [Wn](see [DF08, Th. 6.16]). Therefore arbitrary close topthere are polynomials with one active critical point and the remaining ones attracted by periodic sinks. The result then follows from Corollary 3.2.

4. Robust bifurcations from fractal geometry

In this section we consider product mappings of the formf(z, w) = (p(z), w^d+κ), wherepadmits a repelling fixed pointz0that is almost parabolic, i.e., the multiplier p⁰(z0)satisfies1<|p⁰(z0)|<1.01, andκis large. We show that certain perturbations off exhibitblenders, in the sense of Bonatti and Díaz [BD96], after proper rescaling.

The degree2case requires a slightly different treatment so we handle the casesd>3 and d= 2 separately. Also we strive for a uniformity inκwhich allows to cover the parabolic case as well (see Corollary 4.11).

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4.1. Estimates for Iterated Functions Systems in C. — It is well-known [DK88, HS16] that the limit set of the IFS generated by two affine maps of the form`±(z) = µz±1 has empty interior whenµis non real and|µ| is sufficiently close to 1. Since we will need precise estimates as well as variants of this result, we give a detailed treatment. It will be convenient for us to work with mappings that are contractions on the unit disk, so we write them in a slightly different form.

LetL= (`_j)^d_j=1 be a family of holomorphic maps`_j :D→`_j(D)bD. Since the`_j contract the Poincaré metric they define an IFS whoselimit set is

E= T

n>0

S

|J|=n

`j_n◦ · · · ◦`j₁(D)

(we use the classical multi-index notation J = (j₁, . . . , j_n)). We say that the IFS L satisfies thecovering propertyon an open set∆⊂DifSd

j=1`_j(∆)⊃∆. It then follows that ∆ ⊂ E and this property obviously persists for any small (C¹) perturbation of(`j).

Our first model situation concerns an IFS withd>3branches roughly directed by d^th roots of unity.

Lemma4.1. — Let d>3 andL= (`j)^d_j=1 be the IFS generated by the affine maps

`_j :z7−→mz+α_j(1− |m|)e^(2πi/d)j,

where0.98<|m|<1 andα_j is such that3/5<|αj|<1 and|argα_j|< π/20. ThenL satisfies the covering property on the diskD(0,1/10).

Proof. — Since|α|<1we have that`_j(D)bDso we are in the above situation. Let nowz in D(0,1/10), we have to prove that`⁻¹_j (z)∈D(0,1/10)for somej. We will show that if arg(z) ∈[−π/3, π/3] then j =d (or equivalently j = 0) is convenient.

Then the other cases follow, since the unit circle is covered by translates of[−π/3, π/3]

byd^th roots of unity.

We compute`⁻¹₀ (z) = _m¹ (z−α0(1− |m|)). We have to show that forz=ρe^iθ, with ρ61/10and|θ|6π/3, then

`⁻¹₀ (z)

<1/10, or equivalently |z−α0(1−|m|)|<|m|/10.

Now,

|z−α₀(1− |m|)|²=ρ²+|α₀|²(1− |m|)²−2<(zα₀(1− |m|))

6ρ²+ (1− |m|)²−2ρ|α0|cos(arg(z)−arg(α0))(1− |m|) 6ρ²+ (1− |m|)²−2

5ρ(1− |m|),

where in last line we use|α0|>3/5 andcos(arg(z)−arg(α₀))>1/3. So we are left to showing that

ρ²+ (1− |m|)²−2

5ρ(1− |m|)<|m|²

10² , forρ6 1

10 and0.98<|m|<1.

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For the sake of this computation, put1− |m|=δ <1/50. The previous equation can be rewritten as

ρ²+δ²−2

5ρδ < (1−δ)²

10² ⇐⇒ ρ ρ−2

5δ

<(1−δ)² 10² −δ². Sinceρ61/10, to prove the last inequality it is enough to show that

ρ−2

5δ < (1−δ)²

10 −10δ² ⇐⇒ ρ− 1 10 <1

5δ−99 10δ²=1

5δ 1−99

2 δ , and the latter is true because the left hand side is non-positive while the right hand

side is positive.

Remark4.2. — Notice that conjugating by a rotation implies that for every fixedθ, the same result holds whenαj(1− |m|)is replaced bye^iθαj(1− |m|)in the expression of`_j.

In view of2-dimensional applications, it is useful to point out a slight reformulation of this result. LetAbe the range of allowed values forαin Lemma 4.1, i.e.,

(3) A=n

α∈C, 3

5 <|α|<1, |arg(α)|< π 20

o.

Lemma4.3. — Let d>3 andL= (`_j)^d_j=1 be the IFS generated by the affine maps

`j :z7−→mz+αj(1− |m|)e^(2πi/d)j.

where 0.98 < |m| < 1. Fix η > 0 and assume that for every j = 1, . . . , d, the disk D(αj, η)is contained inA.

Then for everyz0∈D(0,1/10), there existsj such that

`⁻¹_j (D(z0, η(1− |m|)))⊂D(0,1/10).

Proof. — Notice that whenηtends to0this is precisely the statement of Lemma 4.1.

In the general case, we simply observe that for every z, when αvaries in a disk of radius η, `⁻¹_j (z) ranges in a disk of radius η(1− |m|)/|m|. For η > 0, put Aη = {α∈A, D(α, η))⊂A} (beware that this is the opposite of aη-neighborhood). If for some z,

`⁻¹_j (z)∈D(0,1/10) for everyα∈A

(this is what we have proved forzin an angular sector of width2π/3andj= 0) then

`⁻¹_j (z)∈D

0, 1

10−η1− |m|

|m|

for everyα∈Aη. Now, since `⁻¹_j (D(z, r)) = D `⁻¹_j (z), r/|m|

, we infer that for α ∈ A_η, the subset

`⁻¹_j (D(z, η(1−|m|)))is contained inD(0,1/10), which was the result to be proved.

For an IFS L, we define Lⁿ to the IFS generated by n-fold compositions of the generators ofL. It has the same limit set asL. Here is an analogue of Lemma 4.1 for an IFS with two branches. In this case the multiplier needs to be chosen close to the imaginary axis.

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Lemma4.4. — Let L=`_± be the IFS generated by the affine maps

`_±:z7−→mz±α(1− |m|),

where 0.99 < |m| < 1, |arg(m)−π/2| < π/50 and α is a complex number with 0.9<|α|<1. ThenL² satisfies the covering property on the diskD(0,1/10).

Proof. — To establish the result, we show thatL²= (`²₊, `+◦`₋, `₋◦`+, `²₋)satisfies the assumptions of Lemma 4.1 (up to conjugating by an appropriate rotation). Indeed, observe first that conjugating by a rotation, we can assume that αis a positive real number. Now, let us compute the expression of`²₊:

`²₊(z) =m²z+ (m+ 1)α(1− |m|) =m²z+β++(1− |m|²), whereβ++= (m+ 1)α

|m|+ 1 . Observe first that0.98<|m|²<1. Next, we see thatβ++ is approximately equal to

1+i

2 α. More precisely, given the assumptions onm it can be shown that:

• 1.8<|m+ 1|²<2.2, therefore0.6<|(m+ 1)α|/(|m|+ 1)<0.8;

• |arg (m+ 1)−π/4|< π/20.

For`₊◦`₋,`₋◦`₊and`²₋, we have analogous estimates, withπ/4replaced by−π/4, 3π/4 and5π/4 respectively. Thus, conjugating by a rotation of angleπ/4, we are in

position to apply Lemma 4.1, and the result follows.

4.2. Blenders inC². — We now study a 2-dimensional version of the phenomenon studied in Section 4.1. Similarly to (3) we define the angular sector

(4) A⁰ =n

α∈C, 0.7<|α|<0.9, |arg(α)|< π 40

o. It is easily shown that for every α ∈ A⁰, the disk D α,1/40

is contained in A, therefore if αj ∈A⁰, the conclusion of Lemma 4.3 holds for anyη <1/40.

Lemma4.5. — Letd>3andL= (L_j)^d_j=1be an IFS inD²generated by biholomorphic contractions of the form

Lj(z, w) = (`j(z), ϕj(z, w)),

and letE be its limit set. Assume that`j is of the form

`j :z7−→mz+αj(1− |m|)e^(2πi/d)j,

where0.98<|m|<1 andαj∈A⁰, and that ϕj :D²→Dis a holomorphic map such that

∂ϕ_j

∂z

<1 and

∂ϕ_j

∂w

< 1 2.

Then any vertical graph Γ intersecting D(0,1/10)×D, whose slope is bounded by

1

100(1− |m|)must intersectE, that isΓ∩ E 6=∅.

Furthermore, the same holds for any IFSL generated by(Lj)^d_j=1, whenever

(5)

L_j−L_j

_C₁< 1

1000(1− |m|).

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Note that by the Cauchy estimates a control on theC⁰norm ofLj−Ljin a slightly larger domain is enough to achieve (5).

Proof. — Let us work directly with the perturbed situation. LetGbe the set of vertical graphsΓin D²of the formz=γ(w)with|γ⁰|<₁₀₀¹ (1− |m|)and|γ(w0)|<1/10for some w0∈D.

We want to show that if Γ∈ G thenΓ∩ E(L)6=∅. For this we must prove that there exists an infinite sequence(jk)such that⁽²⁾ Γ∩ Lj₁◦ · · · ◦Lj_k(D²)

6=∅, or in other words, that L⁻¹_j_k ◦ · · · ◦L⁻¹_j₁ is well-defined on some piece ofΓ.

Arguing by induction, to establish this property it is enough to show that ifΓ∈ G there existsj ∈ {1, . . . , d} such that L⁻¹_j (Γ)∩D²∈ G. WriteL_j =L_j+ε_j. We first observe that the assumption on the slope implies that the diameter of the first pro- jection ofΓis smaller than ₅₀¹(1− |m|), hence contained in a diskD z0,₅₀¹(1− |m|) for some z0 ∈D(0,1/10). Let j ∈ {1, . . . , d} be as provided by Lemma 4.3 for this value ofz0, and considerL⁻¹_j (Γ)(for notational ease we drop the restriction toD²).

This is a subvariety contained inL⁻¹_j D z0,₅₀¹(1− |m|)

×D

. We claim that it is contained inD(0,1/10)×D.

Indeed, writing in coordinates Lj = (`j +ε1, ϕj +ε2), we see that if (z, w) is such that Lj(z, w) belongs to D z0,₅₀¹(1− |m|)

×D, then `j(z) +ε1(z, w) ∈ D z₀,₅₀¹(1− |m|)

. Since by assumption, |ε1(z, w)| <10⁻³(1− |m|)it follows that

`_j(z)∈D z₀,₄₅¹(1− |m|)

. By Lemma 4.3 this implies thatz∈D(0,1/10), thereby establishing our claim.

The equation ofL⁻¹_j (Γ)is of the form

(6) `j(z) +ε1(z, w) =γ(ϕj(z, w) +ε2(z, w)), which can be rewritten as

(7) z=`⁻¹_j (γ(ϕ_j(z, w) +ε₂(z, w))−ε₁(z, w)).

The z-derivative of the right hand side of this equation is smaller than 1 so the contraction mapping principle tells us that it has a unique solution for each w. In other wordsL⁻¹_j (Γ)is a vertical graph. Finally, the implicit function theorem applied to (6) implies that the slope of this graph is bounded by

|γ⁰| ¹₂+|∂ε2/∂w|

+|∂ε1/∂w|

|m| − |γ⁰|(1 +|∂ε₂/∂z|)− |∂ε₁/∂z| < 1

100(1− |m|)

and we are done.

Remark 4.6. — A similar result holds for an IFS with two branches of the form (`_±(z), ϕ_±(z, w), for`_±(z) =mz±α(1− |m|)and mclose to the imaginary axis, as in Lemma 4.4. Indeed, exactly as in the proof of that lemma, it is enough to take two iterates of the IFS to get back to the setting of Lemma 4.5.

(2)Notice the order of compositions.

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4.3. Blenders for endomorphisms ofP²and the post-critical set Theorem4.7. — Let d>3 andf :P²→P² be a product map of the form

f(z, w) = (p(z), q(w)) = (p(z), w^d+κ),

with deg(p) =d. Assume that the polynomialpbelongs to the bifurcation locus inPd

and admits a repelling fixed pointz0 of low multiplier:1<|p⁰(z0)|<1.01.

Then there exists a constantκ0depending only ondsuch that if|κ|> κ0,f belongs to the closure of the interior of the bifurcation locus in H_d(P²).

It is easy to construct polynomialspsatisfying the assumptions of the theorem for every d > 3. A variant of this construction is given in Theorem 4.12 below which allows to treat the cased= 2as well.

Proof. — We translate thez coordinate so thatz₀ = 0, and put m = (p⁰(0))⁻¹, so that m is a complex number such that 0.99 < |m| < 1. To fix the ideas we work in the case d= 3, the adaptation to the general case is easy and left to the reader.

Furthermore to ease notation we replace κ by −κ so that the second coordinate becomes q(w) =w³−κ.

Step 1: analysis ofJq. — Whenκis large, the Julia set ofqis a Cantor set; here we describe its geometry. Consider the disk Dκ := D(0,2|κ|^1/3). The critical point 0 satisfiesq(0) =κ /∈Dκsoq⁻¹(Dκ)is the union of 3 topological disks, invariant under a rotation of angle 2π/3. Fix a cube rootκ^1/3 of κ. If x∈Dκ, solving the equation w³−κ=xyields

w³=κ+x=κ 1 +x

κ

, hencew=κ^1/3ζ 1 + x

3κ+O |κ|^−4/3 , whereζ∈

1, j, j² ranges over the set of cube roots of unity. Therefore we see that q⁻¹(Dκ)is approximately the union of 3 disks of radius2/3|κ|^1/3centered at the cube roots ofκ. Defineq⁻¹_ζ to be the inverse branch ofq onDκsuch thatq_ζ⁻¹(0) =ζκ^1/3. We introduce the new coordinate we = ¹₂κ^−1/3w. After coordinate change, the expression ofq becomes q(ew) =e ¹₂κ^2/3(8we³−1), so that for large enough κ, qe⁻¹(D) is made of 3 components respectively contained in D ¹₂ζ,¹₂|κ|^−2/3

, ζ ∈

1, j, j² . Similarly, from the Cauchy estimate we infer that|(qe⁻¹_ζ )⁰|6|κ|^−2/3onD.

Step 2: rescaling and construction of a blender. — Close to the origin,pbehaves like the multiplication by p⁰(0), more precisely we havep(z) =p⁰(0)z+O(z²). In particular there is a unique inverse branchp⁻¹₀ defined in a neighborhood of0of sizeδ0(p)and such thatp⁻¹₀ (0) = 0.

For 0 < δ δ0(p) we rescale the disk D(0, δ) to unit size by introducing the new coordinate ez = δ⁻¹z. In the new coordinate, p becomes p(·) =e δ⁻¹p(δ·), thus p(ez) =e p⁰(0)ez+O(δ)where theO(·)is uniform forz∈D, and likewise for the inverse map pe⁻¹₀ (z) =e mze+O(δ). Similar results hold for the derivatives by applying the Cauchy estimates in a slightly larger disk.