Distribution of postcritically finite polynomials II: speed of convergence
In the moduli space of degree polynomials, we prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, we prove the exponential speed of convergence for -observables. This improves results obtained with arithmetic methods by Favre and Rivera-Letellier in the unicritical family
and Favre and the first author in the space of degree polynomials.
We deduce from that the equidistribution of hyperbolic parameters with distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for -observables. As an application, we prove the equidistribution (up to an explicit extraction) of parameters with distinct cycles with prescribed multiplier toward the bifurcation measure for any multipliers outside a pluripolar set.
DISTRIBUTION OF POSTCRITICALLY FINITE POLYNOMIALS II: SPEED OF CONVERGENCE
In a holomorphic family of degree rational maps, Mañé, Sad ans Sullivan [MSS] studied quite precisely the notion of -stability. In particular, they related the instability of critical orbits under small perturbations of the dynamics and instability of periodic cycles (see also [lyubich]).
This description has been enriched by DeMarco [DeMarco1] who introduced a current which is supported exactly on the bifurcation locus. This current and its self-intersections reveal to be an appropriate tool to study bifurcations from a measurable viewpoint. Now, consider the particular case of the moduli space of critically marked complex polynomials of degree modulo conjugacy by affine transformations. In that space, the maximal self-intersection of the bifurcation current induces a bifurcation measure , introduced by Bassanelli and Berteloot [BB1], which may be considered as the analogue of the harmonic measure of the Mandelbrot set when . The support of this measure is where maximal bifurcation phenomena occur. Recall that a polynomial is postcritically finite if all its critical points have finite orbit, it is postcritically finite hyperbolic if all its critical points are periodic cycle (the Julia set of such polynomials is then hyperbolic). Recall that a polynomial is postcritically finite if all its critical points have finite orbit. Using the bifurcation measure, it is proved in [BB1] that its support is accumulated by postcritically finite hyperbolic parameters (which are in a certain way the most stable parameters) and that it coincides with the closure of parameters having a maximal number of neutral cycles. Still using the bifurcation measure, the first author also showed in [Article1] that its support has maximal Hausdorff dimension.
As the bifurcation locus of the moduli space is a complicated fractal set, a natural approach is to study it on some dynamical slices. In particular, one can study the maps having a superattracting orbit of fixed period . The study of such a set is difficult and involves naturally arithmetic, combinatorics, topology and complex analysis (see e.g. [MilnorCubic]). Furthermore, to understand the global geography of the moduli space , it is useful to approximate the bifurcation current (resp. the bifurcation measure) by dynamically defined hypersurfaces (resp. finite sets). Following the topological description of the bifurcation locus given by [MSS], one can try to approximate the bifurcation currents by different types of phenomena: existence of critical orbit relations or periodic cycles of given nature.
Let us focus first on the simplest case of the unicritical family, i.e. the family defined by , . Consider the set of parameters that admit a superattractive periodic point of period dividing . Recall that the Multibrot set is defined by . Beware that is a non-polar connected compact set. Finally, let be the harmonic measure of . Then, the first result in this direction goes back to Levin [Levin1]. In the quadratic family (), he proved that the measure equidistributed on the set converges to the harmonic measure of the Mandelbrot set, as . Favre and Rivera-Letelier [FRL] gave a quantitative version of Levin’s in the unicritical family, using arithmetic methods. Namely, they proved that there exists such that for any and :
Using complex analytic potential theory, we prove here the following.
Let . Then, there exists a constant depending only on such that for any and any ,
A classical interpolation argument immediately gives Favre and Rivera-Letelier’s above result. We shall also give a similar estimate for non-postcritically finite hyperbolic parameters in Section Remark..
More recently, this subject has been intensively studied in the moduli space for . As the critical points are marked, i.e. can be followed holomorphically in the whole moduli space , the bifurcation current decomposes as (see §Distribution of postcritically finite polynomials II: speed of convergence for the definition of ). For , let
Refining Levin’s approach, Dujardin and Favre prove in [favredujardin] that for any sequence satisfying , the sequence converges to in the weak sense of currents on . Recently, Okuyama [okuyama:distrib] gave a simplified proof of their result in the case .
Compared to the unicritical family, a significant difficulty comes from the fact that two distinct critical points can very well be, at some parameter, in the immediate Fatou component of the same attractive periodic point. To overcome that, for and , we let
The set happens to be a complex hypersurface of degree , where is defined by induction by and for . It is known that, for a fixed , the sequence converges weakly to on the moduli space . The case has been established by Bassanelli and Berteloot [BB2] using an approximation formula for the Lyapunov exponent (see also [BB3]). The more delicate case has been proved recently by the first author in [DistribTbif] (see also [multipliers] for the case of quadratic polynomials with changing multipliers).
Building on arithmetic methods, Favre and the first author [favregauthier] proved that postcritically finite hyperbolic parameters with distinct super-attracting cycles (resp. strictly postcritically finite parameters with given combinatorics) equidistribute towards the bifurcation measure. The proof developped in that work is only qualitative, since there exists no effective version of Yuan’s arithmetic equidistribution Theorem. It also raises the question to know whether the result is of purely arithmetic nature or not. We consider the present work as a continuation of [favregauthier].
Our main goal here is twofold. First, we want to establish a quantitative equidistribution theorem for postcritically finite hyperbolic parameters. Second, we aim at giving a simpler proof than the one of [favregauthier], relying only on pluripotential theoretic and complex analytic arguments. To our purposes, as in the recent works [favredujardin, BB2, DistribTbif, favregauthier], we shall use the following “orbifold” parametrization of the moduli space . For , we let
where is the monic symmetric polynomial of degree in . Observe that the canonical projection is a finite branched cover of degree and that the critical points of are exactly with the convention that (see Section Distribution of postcritically finite polynomials II: speed of convergence for details). For an integer , we let be the sum of its divisors . The function is known to be bounded from above by for some constant independent of .
Our main result may be stated as follows.
Let . Then there exists a constant depending only on such that for any -tuple of pairwise distinct positive integers with and every test function , if is the probability measure equidistributed on the set of parameters in for which the critical point is periodic of exact period for all , we have
The first ingredient of the proof is a (slight generalization) of a very general dynamical property established by Przytycki: if a critical point of a rational map does not lie in an attracting basin , the points and can not be to close (see Lemma 3.1). The idea to use Przytycki estimate in this context has been introduced by Okuyama [okuyama:distrib] (see also [okuyama:speed]) and constitutes the starting point of our work. Combined with known global properties of the family , this allows us to have precise pointwise estimates outside some specific “bad” hyperbolic components. The second important tool is a transversality result for critical periodic orbit relations proved in [favregauthier] and relying on Epstein’s transversality theory (see [epstein2]). The last important ingredient we use is a estimate for specific solutions of the Laplacian in a bounded topological disk of an affine curve of , which proof crucially relies on length-area estimates (see Theorem 3.2). This allows us to replace an estimate involving the diameter of hyperbolic components with their volume. This actually is a key step, since even in the quadratic family, estimating the diameter of hyperbolic components is a very delicate problem related to the famous hyperbolicity conjecture.
Nevertheless, notice that, in the context of the unicritical family, the equation is known to have simple roots so we do not need to exclude parameters with a periodic critical point of period dividing . As a consequence, we do not use transversality statements à la Epstein. Hence, we will start by the proof of Theorem A which is simpler and more efficient than in the general case. We may regard this as a model for the general case.
Following the strategy of the proof of [favregauthier, Theorem 3], we can deduce from Theorem B a speed of convergence for the measure equidistributed on the (finite) set of parameters admitting distinct attracting cycles of given respective multipliers and of given mutually distinct periods towards . Let us be more precise and pick a -tuple of mutually distinct positive integers and . When the set is finite, let
Notice that is a probability measure and that, when , the measure is exactly the measure equidistributed on the set of parameters in having cycles of respective exact period and multipliers .
The precise result we prove may be stated as follows.
Pick . Then there exists a constant such that for every and every -tuple of pairwise distinct positive integers with , if is as above, we have
for any test function .
Combining Theorem C with techniques from pluripotential theory (see e.g. [ThelinVigny1] and [dinhsibony2]), we can actually prove that for all -tuples of multipliers lying outside a pluripolar set of , the measure equidistributed on parameters having cycles of respective multipliers converges towards the bifurcation measure, as soon as the periods of the given cycles grow fast enough:
Pick any sequence of -tuples of pairwise distinct positive integers such that the series converges. Then there exists a pluripolar set of such that for any , the set is finite for any and the sequence converges to in the sense of measures.
As an obvious corollary, we can deduce that if we only assume , then, for outside a pluripolar set, up to extraction converges to in the sense of measures. Here is another immediate and interesting consequence of Theorem D.
Pick any sequence of -tuples of pairwise distinct positive integers such that the series converges. Then, for almost any , if , the sequence converges to in the sense of measures.
Notice that Bassanelli and Berteloot [BB3] proved a weaker version of Corollary E: they prove that the average measures converge weakly to the bifurcation measure. Contrary to ours, their proof also works in any codimension.
We view these results as parametric analogues of important dynamical phenomena. Indeed, Theorem B is an analogue of the equidistribution of repelling periodic points of a holomorphic endomorphism of towards its maximal entropy measure , and Theorem D is an analogue of the equidistribution of preimages of a generic points, again towards the measure (see [lyubich:equi, briendduval]).
The questions we discuss here may be addressed in a more general setting. A first natural generalization is the case when critical points can have the same period. In that case, the transversality theory à la Epstein fails at parameters admitting multiple critical points and we a priori have no control of the multiplicity of intersection at those parameters.
A second natural question is concerned with the case of the moduli space of degree rational maps. Even in the case of quadratic rational maps which is much better understood that the general case, important difficulties occur. The main problem comes from the fact that, contrary to the case of polynomials, the support of the bifurcation measure is not compact in the moduli space of quadratic rational maps and that the collection of relatively compact hyperbolic components cluster at infinity (see [Mod2]).
We shall study both cases in future works.
On the other hand, in a more recent preprint ([GV2]), instead of focusing on the distribution of hyperbolic postcritically finite parameters, we study the distribution of Misiurewicz parameters (as is also done in [favregauthier]). In this preprint, we study this problem using this time combinatorial tools developped by Kiwi [kiwi-portrait] and Dujardin-Favre [favredujardin], enlightening slightly different, though related, phenomena.
Section Distribution of postcritically finite polynomials II: speed of convergence is devoted to needed material. In Section 2.8, we establish two preliminary results: Przytycki distance estimate and the estimate for solutions of the Laplacian. We then present the proof of Theorem A and its corollaries in Section Remark.. The initial estimates relying on Przytycki Lemma are established in Section Distribution of postcritically finite polynomials II: speed of convergence. Section Distribution of postcritically finite polynomials II: speed of convergence is dedicated to the proof of the main Theorem B and Section Distribution of postcritically finite polynomials II: speed of convergence to proving Theorems C and D. Finally, in Section Distribution of postcritically finite polynomials II: speed of convergence, we investigate other approximation phenomena. We try here to understand the distribution of maps for which the critical points are sent to some prescribed target (and not necessarily themselves). We are especially interested in a theorem of Dujardin [Dujardin2012] that proves the convergence outside some pluripolar set. We give here some convergence estimates and show that in some cases, the pluripolar set can be described explicitly.
The research of both authors is partially supported by the ANR project Lambda ANR-13-BS01-0002. We would like to thank Vincent Guedj for very helpful discussions concerning estimates for solutions of the Laplacian and François Berteloot and Charles Favre for useful comments on preliminary versions.
In this section, we want to recall briefly background material on bifurcation currents and on classical complex analytic tools we will rely on in the whole paper.
Let us recall classic facts concerning holomorphic families of rational maps.
A holomorphic family of degree rational maps parametrized by is a holomorphic map
such that the map is a degree rational map, or equivalently if the map is holomorphic.
We say that a holomorphic family is with a marked critical point if there exists a holomorphic map such that for all .
We say that a marked critical point is passive at if there exists a neighborhood of such that the sequence of holomorphic maps defined by is a normal family on . We say that is active at if it is not passive at . The activity locus of is the set of parameters such that is active at .
Let be the Fubini-Study form of normalized so that .
Theorem 2.2 (Dujardin-Favre).
The sequence converges in the weak sense of currents to a closed positive -current on which is supported by the activity locus of .
More precisely, there exists a locally uniformly bounded sequence of continuous function such that , see e.g. [favredujardin, Proposition-Definition 3.1] or [DeMarco1]. It is also known that (see [Article1, Theorem 6.1]).
Let us also recall that, when a holomorphic family is with marked critical points , the current is supported by the bifurcation locus in the sense of Mañé, Sad and Sullivan (see [DeMarco2]).
We define the bifurcation measure of a family as
This measure detects, in a certain sense, the strongest bifurcations which occur in . Finally, when is a holomorphic family of polynomials with marked critical point , we let
for and . The current is then given by (see e.g. [favredujardin]). We also can remark that it actually can be considered as equipped with marked critical points letting and that for all .
For the material of the present section, we refer to [Silverman, §4.1] and to [BB3, BB2, Milnor3] (see also [favregauthier, §6]). We follow the notations of [favregauthier].
We let be a holomorphic family of degree polynomials parametrized by a quasi-projective variety . For any , the -th dynatomic polynomial of is defined as
where stands for the Moebius function. This defines a polynomial map satisfying if and only if
either is periodic under iteration of with and its exact period is ,
or is periodic under iteration of with and its exact period is and is a primitive -th root of unity.
When is endowed with marked critical point , we may apply this construction to those marked critical points . We let
By the above, we have
Pick , and . Then if and only if is periodic under iteration of with exact period .
We also can describe the set of parameters admitting a cycle of given period an multiplier. For , set
This defines a polynomial . Again, we find
Pick and . Then if and only if one of the following occurs:
if , has a cycle of exact period and multiplier ,
if , there exists such that has a cycle of exact period and multiplier a primitive -root of unity.
Recall that the moduli space of degree polynomials is the space of affine conjugacy classes of degree polynomials with marked critical points. We define a finite branched cover of as follows. For and , let
where is the monic elementary degree symmetric polynomial in the ’s. This family is known to be a finite branched cover (see e.g. [favredujardin, §5]). Remark also that the critical points of are exactly , where we set , and that they depend algebraically on .
We define a continuous psh function by setting
It is known that the connectedness locus
is compact and satisfies , where is the Julia set of (see [BH]). Moreover, the bifurcation measure, in this actual family, coincides with the Monge-Ampère mass of , i.e.
as probability measures on . It is also known that
where we set , and that the function is the pluricomplex Green function of . In particular, is supported by the Shilov boundary of (see [favredujardin, §6]). In particular, the estimates of [favredujardin] give
There exists a constant independent of such that for any , any and any ,
First, let us show that
for some constant depending only on . As observed in the proof of [favredujardin, Lemma 6.4], there exists such that , hence
for all . On the other hand, if , one clearly has for some depending only on and
Finally, if , we directly find
and the conclusion follows.
Now, an immediate induction gives
for all and all and the conclusion follows making . ∎
We will denote in what follows the classical spherical distance on , normalized so that has diameter . For a selfmap , we denote the spherical derivative of :
Recall that if is an annulus and if is conformally equivalent to with , the modulus of A is the same as the modulus of :
We will rely on the following classical estimate (see [briendduval, Appendix]).
Lemma 2.7 (Briend-Duval).
For any , there exists a constant depending only on such that for any holomorphic disks , and any hermitian metric on ,
where is the annulus and areas and distances are computed with respect to .
We also rely on the following classical integration by part formula, which can be stated as follows (see [Demailly, Formula 3.1 page 144]).
Let be bounded open sets. Assume that has smooth boundary. Let be psh functions on and let be a closed positive -current on such that and are well-defined. Then
In this section, we establish the two main technical estimates we will rely on. The first one is of dynamical nature and follows very closely a classical result of Przytycki. The second is of more geometric nature and might be of independent interest.
We shall use the following estimate in a crucial way. Though we will need it only in the case of families polynomials, we state and prove the estimate for general familes of rational maps for sake of completeness. The proof follows very closely that of [Przytycki3, Lemma 1]. The idea to use this result for proving equidistribution phenomena in parameters spaces first appeared in the recent work [okuyama:distrib] of Okuyama.
Let be a holomorphic family of degree rational maps and let be a marked critical point of . Assume that does not lie persistently in a parabolic basin of , i.e. there exists such that is not attracted by a parabolic cycle of . There exists a universal constant and a continuous function such that, for any and any ,
either lies in the immediate basin of an attracting cycle of period dividing ,
In particular, when , then .
Notice that the functions defined by
are continuous on . Moreover, the map is Lipschitz with constant with respect to , i.e.
for any and any . We rely on the following.
There exists a constant such that for every and any if and , then
Fix and . We now assume all along the proof that does not belong to an attracting basin of period . In particular, . When , the above Claim implies
as required. Assume now . There are two distinct cases to treat. First, assume lies in a parabolic basin of . Since is not persistently in such a component, the critical point is active at , hence there exists with by Montel Theorem. By continuity of the function , we find
in that case, i.e. .
In any other case, by assumption, lies in a different Fatou component of than . Then, either which, by the Claim, implies
or and we have
and the proof is complete. ∎
We are left with proving the Claim.
Proof of the Claim.
In what follows, we let (resp. ) denote the spherical (resp. euclidean) ball of center and radius . Recall that the group of (holomorphic) isometries for the spherical metric acts transitivelly on . Pick some parameter and let and be such isometries. Then for :
Choosing and such that and gives
Let be the rational map of defined by so that . As and are isometries, we have so that . Hence is -Lipschtiz. Set now , then:
so that . As the spherical metric and the euclidean metric are comparable on , there exists such that for all ,
so that :