Dynamics semi-conjugated to a subshift for some polynomial mappings in C2

Publ. Mat. 51 (2007), 201–222 DYNAMICS SEMI-CONJUGATED TO A SUBSHIFT FOR SOME POLYNOMIAL MAPPINGS IN C2 Gabriel Vigny Abstract We study the dynamics near infinity of polynomial mappings f in C2. We assume that fhas indeterminacy points and is non constant on the line at infinity L∞. If L∞is f-attracting, we decompose the Green current along itineraries defined by the indeterminacy points and their preimages. The symbolic dynamics that arises is a subshift on an infinite alphabet. 1. Introduction We are interested in the dynamics of polynomial mappings fin C2 whose meromorphic extensions to P2admit indeterminacy points and for which the line at infinity (which we denote by L∞) is f-attracting (that is: there exists C > 1 such that for p∈C2with kpklarge enough, one has kf(p)k ≥ Ckpk). In particular, given any large ball Bin C2, these maps are polynomial-like in the sense of [DS03] from f−1(B) to B. The dynamics is studied there: there exists an invariant probability measure which is K-mixing and of maximal entropy. Our goal is to study the dynamics near infinity, especially the structure of the Green current, which is a positive closed current of bidegree (1,1) invariant under the action of f∗. In [DDS05], the authors consider the case where f∞, the restriction of fto L∞, is constant and they decompose the Green current into pieces associated to an itinerary defined by indeterminacy points. On the basin of attraction of the indeterminacy set, the itinerary map semiconjugates fto a shift. Another case which has been studied is when fadmits a holomorphic extension to P2: in [BJ00], the authors showed the Green current admits a local laminar decomposition consisting of the stable manifolds of fat 2000 Mathematics Subject Classification. 37F10, 37B10. Key words. Polynomial mapping, Green current, subshift. 202 G. Vigny the points of the Julia set of f∞. Applying one dimensional theory, one also obtains in this case a dynamics semi-conjugated to a shift. We study here a mixed situation. We assume that fadmits indeterminacy points on L∞and that f∞is not a constant function. In order to describe clearly the new phenomena happening here, we consider the case where f∞is hyperbolic. The method we use allows to study more general cases. We will complete our study by giving several examples. In the hyperbolic case, we show that the Green current decomposes along some itineraries defined by the indeterminacy points and their preimages. Surprisingly, the local stable manifolds associated to the points of the Julia set of f∞are not charged by the Green current. Furthermore, the symbolic dynamics we obtain is a subshift (a Markov chain), which is new for polynomial mappings. The main tools we use are horizontal-like maps and a theorem of convergence of currents proved in [Duj04] and [DDS05]. Roughly speaking, such applications are contracting in the vertical direction and expanding in the horizontal one in some bidisk. For the reader’s convenience, we give the basic properties of these objects. Next, we define and study the basic properties of the family Gof maps we consider. We give a simple sufficient condition for a map f to be in Gand we prove the algebraic stability. Then, by a theorem of Sibony [Sib99], one can associate to fa natural invariant current (Green current). We give an easily computable formula for the trace of the Green current at infinity. This trace is a probability measure which is a combination of Dirac masses at the indeterminacy points and their preimages. Under some additional hypothesis, we also compute the topological degree. We then study the decomposition of the Green current on a neighborhood of infinity under the hypotheses that the indeterminacy set is located in the Fatou set of f∞, with no indeterminacy point being periodic for f∞and that f∞is hyperbolic. This set of maps contains an open subset of G. The decomposition of the Green current semi-conjugates f to a subshift on an infinite alphabet. A classic object in this setting is the escape rate which measures the asymptotic speed at which a point goes to infinity (see [DS04] and [FJ05] for some interesting examples and results on this topic). Under some additional hypothesis, we show that the range of the escape rate is a full interval which is new for polynomial maps and we compute a mean escape rate. We will explain briefly how to obtain a weaker decomposition of the Green current in a more general case. Finally, we study examples, in particular the case where the indeterminacy points are located in the exceptionnal set of f∞, in Some Subshifts in Complex Dynamics 203 this case the support of the Green current is strictly contained in the Julia set of f. 2. Polynomial maps with dynamics at infinity 2.1. Horizontal-like maps. We recall here the facts we use on horizontal-like maps. Proofs and details can be found in [Duj04] and [DDS05]. Let D(resp. Dr) be the unit disk (resp. the disk of radius rcentered at 0) in C. Let ∆ be the unit bidisk in C2, we denote its vertical boundary by ∂v∆, and its horizontal boundary by ∂h∆. Namely: ∂v∆ = {(z, w)∈C2,|z|= 1,|w|<1}and ∂h∆ = {(z, w)∈C2,|z|<1,|w|= 1}. We have the following definitions: Definition 2.1. Let ∆i⊂Mibe an open subset biholomorphic to ∆ in the complex surface Mifor i= 1,2. Let fbe a dominating meromorphic map defined in some neighborhood of ∆1with values in M2. The triple (f, ∆1,∆2) defines a horizontal-like map if: •fhas no indeterminacy points in ∂v∆1and f(∂v∆1)∩∆2=∅; •f(∆1)∩∂∆2⊂∂v∆2; •f(∆1)∩∆26=∅. Definition 2.2. A positive closed (1,1)-current Tin ∆ is vertical if: Supp T⊂D1−ε×Dfor some ε > 0. Similarly, we can define horizontal currents. We can define the (horizontal) slice measures mw0of a vertical positive closed (1,1)-current Tby T∧[w=w0]. These measures have the same mass, which we call the slice mass of T. The current Tis zero if and only if its slice mass is zero. The main fact is that we can define the pull-back of such a current by a horizontal-like map, and we have the following: Let (f, ∆1,∆2)be a horizontal-like map then there exists a positive integer d≥1such that for every vertical positive closed current Tin ∆2 of slice mass 1,1 df∗(T)is a vertical positive closed current in ∆1of slice mass 1. We call this integer the degree of f, it can be computed as the number of intersections of the preimage of a vertical line with a horizontal line 204 G. Vigny (with multiplicity). The following result is our main tool to obtain the convergence in Theorem 3.5: Theorem 2.3 ([DDS05]).Let {(fi,∆i,∆i+1)}i≥1be a sequence of horizontal-like maps of degree disuch that (fi)−1(∆i+1)⊂D1−ε×D⊂∆i for a fixed ε > 0. Assume that K=Tn≥1f−1 1...f−1 n(∆n+1)has zero Lebesgue measure. For each n, let Tnbe a vertical positive closed (1,1)-current of slice mass 1in ∆n. Then, the sequence of iterated pull-back 1 d1f∗ 1... 1 dnf∗ nTn+1nconverges to a vertical positive closed current τof slice mass 1in ∆1which is independent of (Tn). 2.2. The class G. We say that the line L∞is attractive for a polynomial mapping fof C2 if there are constants C > 1 and Mlarge enough such that for kpk ≥ M, we have kf(p)k ≥ Ckpk. We can consider the meromorphic extension of fto P2, which can have indeterminacy points, we still denote by f that extension and by I(f) the indeterminacy set. Let f∞be the unique map extending f|L∞\I(f). The case where f∞is constant was studied in [DDS05], so we will consider the following set of mappings G: Definition 2.4. Let Gthe set of mappings fsatisfying the following properties: •The line L∞is attractive. •The meromorphic extension of fto P2admits indeterminacy points. •The map f∞is not constant. Let f= (f1, f2) of algebraic degree Dbe in G. We denote by f+ 1 and f+ 2the homogeneous parts of maximal degree. After a linear change of coordinates, we can assume deg f+ 1=Dand deg f+ 2=D′≤D. The meromorphic extension of fto P2is given by [TDf1(Z/T, W/T ) : TDf2(Z/T, W/T ) : TD] and the restriction of fto L∞= (T= 0) is f∞[Z:W] = [f+ 1(Z, W ) : 0D−D′f+ 2(Z, W )]. Thus, in order to have f∞ not constant, we need D=D′and f+ 1not proportional to f+ 2(otherwise, fsends L∞to [1 : 0 : 0] or [1 : λ: 0]). The indeterminacy set I(f) of fis the common zeros of f+ 1and f+ 2: if the line Dof equation ajz−bjw= 0 satisfies f+ 1(D) = {0}and f+ 2(D) = {0}then [bj:aj: 0] is in I(f). Some Subshifts in Complex Dynamics 205 One deduces from above that all the mappings of Gcan be written as: f(z, w) =  Y j≤m (ajz−bjw)αjP1(z, w) + Q1(z, w), Y j≤m (ajz−bjw)αjP2(z, w) + Q2(z, w) , (1) where the ajand bjare complex numbers satisfying (aj, bj)6= (0,0), mand the αjare positive integers, P1and P2are homogeneous polynomials of degree d′≥1 with no common factor and the Qjare polynomials of degree strictly smaller than the degree of f. We denote by dthe sum Pj≤mαj, so that fhas degree d+d′=D. So we have that f∞([Z:W]) = [P1(Z, W) : P2(Z, W )]. We define the multiplicity of an indeterminacy point Ias the intersection multiplicity at Iof L∞and f−1(L) where Lis a generic line. The indeterminacy points of fare the Ij= [bj:aj: 0] with multiplicity αj. We assume of course that the (aj, bj) are not proportional. The following proposition shows that we can find fwith any given set of indeterminacy points with multiplicity and any given restriction at infinity. Furthermore, it shows that for D≥3, Gcorreponds to a Zariski open set of the space of parameters of (1). Proposition 2.5. Let f= (f1, f2)be as in (1). Assume that the polynomial Φ = f1P2−f2P1has degree ≥2 + d′. If for all j,ajz−bjwdoes not divide the homogeneous part of maximal degree of Φ, then L∞is f-attracting. Proof: Let fbe as above and Nbe a small neighborhood of infinity. Observe that for any neighboorhood Vof I(f), there exists a constant C such that if p= (z, w)∈N\Vwe have CkpkD≤ kf(p)k. So we just have to prove the estimate on V. Since P1and P2have no common factor, there is λ > 0 such that max(|P1(z, w)|,|P2(z, w)|)≤λk(z, w)kd′on N. The hypothesis implies that |Φ(z, w)|&k(z, w)kdeg Φ near I(f), hence: 2kf(z, w)k ≥ |Φ(z, w)| max(|P1(z, w)|,|P2(z, w)|)&k(z, w)k2. The proposition follows. Observe that for a generic map g∈ G, we have deg Φ = 2d′+d−1. The criterion is not optimal, but it is generic for D≥3 and easy to check. If D= 2, we obtain in the same way that kf(z, w)k&k(z, w)k, 206 G. Vigny so we may have to multiply fsatisfying the above criterion by a large enough constant in order to have that L∞is attractive. Recall that a meromorphic mapping f:P2→P2is said to be algebraically stable if no algebraic curve is sent to an indeterminacy point after some iterations, equivalently, if fis of algebraic degree Dthen fnhas degree Dnfor all n≥1 . It is clear that the mappings of Gare algebraically stable because no algebraic curve can be sent on an indeterminacy point. So by [Sib99], we can define the Green current for a map fin G(see the discussion before Proposition 2.8 for details). We use the notation of (1) in the following proposition. Proposition 2.6. Let fand gbe in Gthen f◦g∈ G. More precisely, if f= (PQ1+R1, PQ2+R2)has degree Dand g= (P′Q′ 1+R′ 1, P′Q′ 2+R′ 2) has degree D′then f◦g= (P′′Q′′ 1+R′′ 1, P′′Q′′ 2+R′′ 2)where: P′′ = (P′)DP(Q′ 1, Q′ 2), Q′′ 1=Q1(Q′ 1, Q′ 2),and Q′′ 2=Q2(Q′ 1, Q′ 2). In particular, f◦ghas degree D+D′and for n∈N∗we have (f∞)n= (fn)∞. Proof: With the above notations, the homogeneous part of maximal degree of the components of f◦gare equal to: P(P′Q′ 1, P′Q′ 2)Q1(P′Q′ 1, P′Q′ 2) = (P′)DP(Q′ 1, Q′ 2)Q1(Q′ 1, Q′ 2) and P(P′Q′ 1, P′Q′ 2)Q2(P′Q′ 1, P′Q′ 2) = (P′)DP(Q′ 1, Q′ 2)Q2(Q′ 1, Q′ 2). We only have to check that Q1(Q′ 1, Q′ 2) and Q2(Q′ 1, Q′ 2) have no common factor: if not, since two homogeneous polynomials have no common factor if and only if they have no common non trivial zero and since Q1 and Q2have no common factor, we would have that Q′ 1and Q′ 2have a non trivial common zero. 2.3. Multiplicity of the indeterminacy points, trace of the Green current at infinity. Let Edenote the set Sn≥0f−n(I(f)) = Sn≥0I(fn). For p∈E, we denote by λp,n the real number equal to the multiplicity at pof fn as an indeterminacy point divided by Dn, that is: λp,n =multp(fn) Dn (these numbers will appear in the symbolic dynamics of f). We have the following lemma: Some Subshifts in Complex Dynamics 207 Lemma 2.7. For all p∈E,(λp,n)is an increasing sequence bounded by 1. Let λpbe its limit. Then: X p∈E λp= 1. Proof: Write fn= (PnQ1,n +R1,n, PnQ2,n +R2,n). Recall that I(fn) is the intersection of L∞with the zero set of Pn. By Proposition 2.6: Pn+1 = (Pn)DP(Q1,n, Q2,n). Hence, (λp,n) is increasing since (Pn)Dis a factor of Pn+1. Set dn= deg(Pn) and d′ n= deg(Qi,n). We deduce from Proposition 2.6: d′ n= (d′)nand dn=Dn−(d′)n. So, Pp∈Eλp,n =dn Dn→1. This completes the proof. Remarks. 1. In a way, the indeterminacy points of fntake asymptotically all the available degree, so they carry the main part of the dynamics near L∞(cf. Proposition 2.8). 2. The sequence (λp,n)ncan be strictly increasing as we will see in the last example of Section 3.6. One can check that (λp,n)nis strictly increasing after some rank if and only if pis preperiodic. Recall that, on C2, for falgebraically stable of degree D, the sequence of positive functions un=1 Dnlog+kfn(z, w)kalmost decreases (i.e. (un+cn)nis decreasing for some sequence of constant (cn)ndecreasing to zero) to the Green function uof fwhich is a potential of the Green current Tof f(ddcu=T). Furthermore, the function ˜u(z, w) = u(z, w)−1 2log(|z|2+|w|2+1) is a bounded quasi-plurisubharmonic function on C2, thus it extends to P2, and this extension satisfies ddc˜u= T−ωF S where ωF S is the Fubini-Study form on P2(see [Sib99]). We will see in Proposition 2.8 that ˜u|L∞is not identically equal to −∞ so we can define the measure m∞=T∧[L∞] which is the trace of the Green current at infinity. Since the sequence of functions ˜un(z, w) := un(z, w)−1 2log(|z|2+|w|2+ 1) is almost decreasing, m∞is the limit in the sense of current of the sequence ((ddc˜un(z, w) + ωF S )∧[L∞]). In particular, we have m∞=ddc(˜u|L∞) + (ωF S)|L∞. The next proposition shows that m∞is a combination of Dirac masses at the points of E, with computable coefficients. 208 G. Vigny Proposition 2.8. Let fbe in Gand ˜ube as above. For p∈E, we denote by [ap:bp: 0] its homogeneous coordinates. Then: ˜u([z:w: 0]) = log  Y p∈E|apw−bpz|λp −1 2log(|z|2+|w|2). In particular, we have the formula: m∞=X p∈E λpδp where δpis the Dirac mass at p. Proof: With the above notations, we have that in C2: ˜un(z, w) = 1 Dnlog+k(PnQ1,n +R1,n)(z, w),(PnQ2,n +R2,n)(z, w)k −1 2log(|z|2+|w|2+ 1). So, first outside of E, and hence everywhere on L∞by semi-continuity, the extension is given by: ˜un([z:w: 0]) = 1 Dnlog k(PnQ1,n)(z, w),(PnQ2,n)(z, w)k −1 2log(|z|2+|w|2). By definition of the λp,n, there is a constant Cndepending on the choice of the coordinates of the elements of Esuch that: ˜un([z:w: 0]) = X p∈E λp,n log |apw−bpz| +1 Dnlog kfn ∞[z:w]k+Cn−1 2log(|z|2+|w|2). From one-dimensional theory, we know that 1 d′nlog k(f∞)n[z:w]k− 1 2log(|z|2+|w|2) converges to a continuous function on L∞and Pp∈Eλp,n log(|apw−bpz|) converges thanks to the previous lemma. The last identity and the fact that d′< D imply the first formula in the proposition. The formula giving m∞is then clear by the Poincar´e formula. Remark. The previous proof can be applied to all the algebraically stable polynomial maps of C2with indeterminacy points on L∞. Some Subshifts in Complex Dynamics 209 The computations in this section are very similar to those in [Dem05]: the author iterates “mappings” in P1of the form h= [Hp :Hq] where H,p, and qare homogeneous polynomials in two variables. A measure µ is introduced and the author proves that it depends continuously on the coefficients of h. Considering h= [PQ1, PQ2], we see that here µis m∞, so we deduce that m∞depends continuously on the coefficients of P,Q1 and Q2. 2.4. Topological degree. Let Nbe a small enough neighborhood of L∞and Vbe a neighborhood of I(f), then there are constants Cand C′such that for pin N\V, we have: CkpkD≤ kf(p)k ≤ C′kpkD. Let us assume here that the considered mapping satisfies in addition: for all I∈I(f), there exist a number lI, a neighborhood V(I) of I, a neighborhood V(f∞(I)) of f∞(I) and constants C1and C2such that for all p∈V(I) with f(p)/∈V(f∞(I)), we have: (2) C1kpklI≤ kf(p)k ≤ C2kpklI. This condition is easy to check in practice. Under these assumptions, we can compute the topological degree of fwhich is the mass of the pullback of any probability measure by f. The difference with the case with no dynamics on L∞is that we have to count the number of preimages of a generic line by f∞. We have the following proposition: Proposition 2.9. Let f∈ G satisfying (2). Then the topological degree of fis given by: dt=X I∈I(f) lIαI+d′D. In particular, we have dt> D. Proof: Let Lbe a generic line, we consider the probability measure [L∞]∧[L] (which is the Dirac mass at the intersection of Land L∞). By definition, its pull back by fis of mass dt. After some change of coordinates, we can assume that the point [1 : 0 : 0] is not on Land f−1(L). So we work in the coordinates (u, v) = (Z/W, T/W) where a potential of L∞= (v= 0) is ϕ(u, v) = log |v|. We must compute: ZP2 f∗([L∞]∧[L]) = Zf−1(L) ddc(ϕ◦f). For each Iin I(f), let BIbe a bidisk in V(I) for the (u, v) coordinates, and for each pin f−1 ∞(L∞∩L) let Bpbe a bidisk around p. Since Lis a 216 G. Vigny are disjoints, so we can write uniquely f∗Sp=Pq∈f−1(p)∪I(f)S′′ qwhere S′′ q=f∗ q,pSpis a vertical positive closed current in ∆q. We iterate: 1 Dk(fk)∗S=X β∈ΣkQk−1 i=0 dβ(i),β(i+1) DkLβ(0),β(1) ...Lβ(k−1),β(k)Sβ(k) =X β∈Σk ν(Cβ)Lβ(0),β(1) ...Lβ(k−1),β(k) Sβ(k) λk . The left hand side goes to the Green current T. By the first part of the theorem, the general term of the right hand side tends to Tβfor βgeneric so we get the result by dominated convergence. Remark. The dynamics of fnear infinity is semi-conjugated to the subshift σin the sense that f(Kβ)⊂ Kσ(β). We see that the current gives full mass to Kwhich does not meet J∞. So, as announced in the introduction, the local stable manifolds to the points of the Julia set of f∞do not carry any part of the Green current, but they are contained in its support. 3.4. Escape rate. We take f∈ G satisfying the condition (2), we also suppose that f∞(I(f)) ∩E=∅(else, we would have pin Esuch that f(p)∈ f(V(I(f)))). We want to compute the possible values of the upper escape rate ¯ lwhere log(¯ l) = lim sup 1 nlog+log+kfnkwhich becomes lim sup 1 nlog log(kfnk) in N. In the same way, we define the lower escape rate land we are interested in knowing where these two functions match up, in which case we note ltheir common value which we simply call the escape rate. For p∈E\I(f), we set lp=D. We have the following lemma: Lemma 3.6. Let β∈Σand q∈ Kβ. We have: 1 nlog log kfn(q)k=1 nlog(lβ(0)lβ(1) . . . lβ(n−1)) + Olog n n. Proof: We have constants c1and c2such that: c1≤log kfj+1(q)k−lβ(j)log kfj(q)k ≤ c2. Taking a combination of these inequalities for j≤n−1 gives: c1  n−1 X j=0 lβ(j+1) . . . lβ(n−1) +lβ(0) ...lβ(n−1) log kqk ≤ log kfn(q)k, Some Subshifts in Complex Dynamics 217 with a similar inequality for the right hand side. Taking the logarithm and dividing by ngive:  1 nlog log kfn(q)k− 1 nlog(lβ(0) ...lβ(n−1)) ≤1 nlog  log kqk+C n−1 X j=0 1 lβ(0) . . . lβ(j−1)  . The sum in the right hand side is a O(n) which concludes the proof. Choosing a suitable β, we deduce from the lemma that the range of the escape rate in Nis [min lI, D] (the details are left to the reader). In this case, it is interesting to observe that the set of possible escape rates is an interval which is a new property for polynomial mappings. Let λdenote the slice mass 1 −PI∈I(f)λIof Toutside a neighborhood of I(f). We have the following theorem: Theorem 3.7. For kTk-almost every point qin N, the escape rate l(q) exists and is equal to DλQI∈I(f)lλI I. Proof: Since the left shift σis ergodic for ν, the Birkhoff’s ergodic theorem yields that for ν-almost every β: exp 1 n n−1 X i=0 log lσi(β)(0)!−→ exp ZΣ log lβ(0) dν=DλY I∈I(f) lλI I. And the theorem follows from the previous lemma and Theorem 3.5. 3.5. Generalization. In the case where some indeterminacy points are on J∞(possibly periodic), we can obtain a decomposition of the Green current by building a cover of J∞by disks such that for all Din this cover, there exist disjoint disks D1,D2,...,Dd′in the cover with f−1 ∞(D)⊂D1∪D2∪···∪Dd′ and D⋐f∞(Di) for all i≤d′. The trick is to have two disks DIand D′ I around each indeterminacy point I∈I(f) so that ∂f∞(DI)∩D=∅ or ∂f∞(D′ I)∩D=∅. Finally, we follow the construction of Section 3.1 with Ubeing replaced by the union of all those disks. This time we only have a finite number of bidisks and when we pull back the Green current near some point of Eto an indeterminacy point I in J∞, we may have to choose between the two bidisks centered at Iin order to have a horizontal-like map. We only get a finite subshift, but taking a finer cover, we get more precision on the decomposition (only 218 G. Vigny on a smaller neighborhood of L∞). Somehow the decomposition is not intrinsic because we do not pull back according to the itinerary but it assures that the Green current is not extremal in a neighborhood of L∞. 3.6. Examples. First let us explain our results in two examples where the dynamics at infinity is linear. Example 1. Consider the case where f∞is given by u7→ 2uand where the indeterminacy set is reduced to (1,0) with multiplicity 1 in the (u, v) coordinates (thanks to Proposition 2.5, we know this case exists, take for example f(z, w) = C(2z(z−w) + z, w(z−w)) for Clarge enough). Then, using Proposition 2.8, we find that: •E=pn=1 2n,0, n ≥0. •λn=1 2n. •The matrix of the subshift is:      1 2 1 4 1 8... 100... 010... . . .. . .. . ....      . An element β∈Σ can be written (pn1, pn1−1,...,p0, pn2,...,p0,...) for some sequence (ni) in N. The dynamics in the space of itineraries is simple: a point in Kβwhere β0=pn1is sent near pn1−1then near pn1−2 and so on until it arrives near p0, in which case it can be sent near any element of Esince p0is an indeterminacy point. Example 2. This time, we still take f∞given by u7→ 2uand we suppose that the indeterminacy points are I0= (2,0) and I1= (1,0) with multiplicity 1 in the (u, v) coordinates, so D= 3 (for example: f(z, w) = (2z(z−w)(z−2w) + z2, w(z−w)(z−2w))). In this case, we have that f−1 ∞I0=I1. Again, using Proposition 2.8, we find that: •E=pn=1 2n−1,0, n ≥0. •We have λ0=λI0=1 3,λ1=λI1=4 9,λpn=4 3n+1 . •The matrix of the subshift is:        1 3 4 9 4 27 4 34... 1 2 1 3 1 9 1 33... 0 1 0 0 ... 0 0 1 0 ... . . .. . .. . .. . ....        . Some Subshifts in Complex Dynamics 219 The interesting fact here is that the entries of the second row are not proportionnal to the the slice mass, indeed a point near I1will have “more chances” to be sent on ∆I0by fsince f∞(I1) = I0. Example 3. Now, we consider the case were the indeterminacy points are in the exceptionnal set of f∞(namely f−1(I) = I). Observe that this case does not meet the hypothesis of Theorem 3.5 since the indeterminacy points are periodic. For example, let f: (z, w)7→ (z3+w2, zw2). By Proposition 2.5, L∞is f-attracting. We even have kf(z, w)k ≥ k(z, w)k2for k(z, w)klarge enough. The meromorphic extension of f to P2is given by: f([Z:W:T]) = [Z3+T W 2:ZW 2:T3]. The indeterminacy set of fis reduced to I0= [0 : 1 : 0] and the dynamics at infinity is given by f∞: [z:w: 0] 7→ [z2:w2: 0] (so f−1 ∞(I0) = I0). Thus fis in Gand is algebraically stable. The topological degree dtof f, which is by definition the number of preimages of a generic point, is equal to 8 (solve f(z, w) = (0,1)). It is greater than the algebraic degree. We use the coordinates (u, v) = Z W,T Win which L∞is given by (v= 0). The map fbecomes: f: (u, v)7−→ u3+v u,v3 u. In these coordinates, the point I0becomes (0,0). The map f∞is given by u7→ u2for which the Julia set J∞is the unit circle (|u|= 1). We have the following lemma: Lemma 3.8. Let V=(u, v),|u|<1 2and |v|<1 4|u|3, then f(V)⊂V. Proof: Observe that (0,0) is not in Vsince fis not defined there. Let (u, v) be in V. We check: |u3+v| |u|≤ |u|2+|v| |u|<1 4+|u|2 4<1 2. We also have the inequalities: |u3+v| |u|≥ |u|2−|v| |u|>|u|2−|u|2 4>1 2|u|2 |v|3 |u|<1 43|u|8. 220 G. Vigny It is then sufficient to check that: 1 43|u|8<1 41 2|u|23 which is obvious. We deduce from the lemma that Vis in the Fatou set since the sequence of iterates is normal there. Let then D0⊂D1be disks on L∞ centered on I0, small enough to be contained in V, with f−1 ∞(D0)⋐D1. Let D2be a disk centered on [1 : 0 : 0] containing the Julia set of f∞ with ∂D2⊂V. We have that f−1(D2)⋐D2. We can shrink those disks to have D1∩D2=∅. As in Proposition 3.1, we want to “thicken” those disks in order to have bidisks such that fdefines by restriction horizontal-like maps between them. Close to I, the norm of a point (in the (z, w) coordinates) is given by |v|−1, but next to [1 : 0 : 0], it is controled by |u| |v|so we use the coordinates (u′, v′) = T Z,W Zthere. Then, we define ∆0=D0×(|v|< ε), ∆1=D1×(|v|< ε) and ∆2=D2×(|u′|< ε′). Take εand ε′small enough so that the vertical boundaries of the bidisks are relatively compact in V. Observe that ∆1\∆0⊂Vis in the Fatou set of f. Recall that since I0is an indeterminacy point, any neighborhood of I0is sent on the whole L∞. Since L∞is f-attracting, and by uniform continuity of faway from any neighborhood of I0, we can chose εand ε′small enough so that: •f: ∆1→∆0defines a horizontal-like map of degree 3 denoted by f1,0. •f: ∆1→∆2defines a horizontal-like map of degree 1 denoted by f1,2. •f: ∆2→∆2defines a horizontal-like map of degree 2 denoted by f2,2. Next, we consider the Green current Tof f. We know that its support is contained in the Julia set of f(see [Sib99]). So we know that in some neighborhood of infinity, Tcan be written as T1+T2where T1and T2are vertical positive closed currents in ∆0⊂∆1and in ∆2. Pulling-back T1 and T2and using the invariance of T, we see that: 1 3f∗T=T=T1+T2. Some Subshifts in Complex Dynamics 221 So: T1=1 3f∗ 1,0T1+1 3f∗ 1,2T2 T2=1 3f∗ 2,2T2. Calling m1and m2the slice masses of T1and T2, we can compute them using the previous equation and the fact that the pull-back of a vertical current of slice mass mby a horizontal-like map of degree dis of slice mass dm. So, we have: m1=m1+1 3m2 m2=2 3m2. Hence, m2= 0 and so T2= 0. In particular, the support of the Green current of fis strictly contained in the Julia set Jsince the stable manifolds associated to the Julia set J∞of f∞are in Jbut supp(T) does not meet J∞. In [FS95], there is a different example of such phenomenon. For ε > 0, we consider the small perturbation fεdefined by: fε: (z, w)7−→ ((z+εw)z2+w2,(z+εw)w2). We check that fεgives the same map at infinity than fand that the indeterminacy point is now Iε= [−ε: 1 : 0]. We see that the preimages of Iεaccumulate on the Julia set of fε∞and we have seen that they are on the support of the Green current which contrasts with what happens for f. Here the support of the Green current does not vary continuously with εwhereas the Green current itself varies continuously for the weak topology. The method used to study that example can be generalized to the case where the indeterminacy set is contained in the exceptionnal set. For the case where f∞=znand I(f) = {[1 : 0 : 0],[0 : 1 : 0]}we also need to use the method of Section 3.5 (we take two bidisks around each indeterminacy point). The symbolic dynamics that arises here is a finite subshift at two elements. References [BJ00] E. Bedford and M. Jonsson, Dynamics of regular polynomial endomorphisms of Ck,Amer. J. Math. 122(1) (2000), 153–212. 222 G. Vigny [Dem05] L. DeMarco, Iteration at the boundary of the space of rational maps, Duke Math. J. 130(1) (2005), 169–197. [DDS05] T. C. Dinh, R. Dujardin and N. Sibony, On the dynamics near infinity of some polynomial mappings in C2,Math. Ann. 333(4) (2005), 703–739. [DS03] T. C. Dinh and N. Sibony, Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9) 82(4) (2003), 367–423. [DS04] T. C. Dinh and N. Sibony, Dynamique des applications polynomiales semi-r´eguli`eres, Ark. Mat. 42(1) (2004), 61–85. [Duj04] R. Dujardin, H´enon-like mappings in C2,Amer. J. Math. 126(2) (2004), 439–472. [FJ05] C. Favre and M. Jonsson, Eigenvaluations, available at www.arxiv.org/pdf/math.DS/0410417 (2005). [FS95] J. E. Fornæss and N. Sibony, Complex dynamics in higher dimension. II, in: “Modern methods in complex analysis” (Princeton, NJ, 1992), Ann. of Math. Stud. 137, Princeton Univ. Press, Princeton, NJ, 1995, pp. 135–182. [KH95] A. Katok and B. Hasselblatt,“Introduction to the modern theory of dynamical systems”, With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, Cambridge, 1995. [Mil99] J. Milnor,“Dynamics in one complex variable. Introductory lectures”, Friedr. Vieweg & Sohn, Braunschweig, 1999. [Sib99] N. Sibony, Dynamique des applications rationnelles de Pk, in: “Dynamique et g´eom´etrie complexes” (Lyon, 1997), Panor. Synth`eses 8, Soc. Math. France, Paris, 1999, pp. ix–x, xi–xii, 97–185. Math´ematiques - Bˆat. 425, UMR 8628 Universit´e Paris-Sud 91405 Orsay France E-mail address:[email protected] Primera versi´o rebuda el 12 de juliol de 2006, darrera versi´o rebuda el 16 de novembre de 2006.