D Sarrazin

Blake

M Gailhanou
email: marc.gailhanou@univ-amu.fr

P Sarrazin

D Blake

Modeling of X-ray fluorescence full field imaging using planar square pore micro-channel plate optics

Modeling of x-ray fluorescence full field imaging using

planar square pore micro-channel plate optics M. Gailhanou, P.

INTRODUCTION

Elemental imaging using X-ray fluorescence is a non destructive technique capable of bringing out important information in many fields [START_REF] Tsuji | New developments of x-ray fluorescence imaging techniques in laboratory[END_REF], from materials science to cultural heritage [START_REF] Alfeld | Optimization of mobile scanning macro-xrf systems for the in situ investigation of historical paintings[END_REF][START_REF] Reiche | Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old phoenician ivories using sr-ff-microxrf with the color x-ray camera[END_REF][START_REF] Bertrand | Cultural heritage and archaeology materials studied by synchrotron spectroscopy and imaging[END_REF][START_REF] Walter | Full-field xrf instrument for cultural heritage: Application to the study of a caillebotte painting[END_REF] and planetary surface analysis [START_REF] Blake | The mapping x-ray fluorescence spectrometer (mapx)[END_REF] . A first class of methods by which X-ray fluorescence images can be obtained consists in the two dimensional scanning of a beam on the object and collection of the fluorescence X-rays at every point of the map [START_REF] Alberti | Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces[END_REF][START_REF] Romano | Real-time elemental imaging of large dimension paintings with a novel mobile macro x-ray fluorescence (ma-xrf) scanning technique[END_REF]. It requires a focusing device and a minimum of two-axis motorized scanner. A second class of methods is based on direct X-ray fluorescence imaging with no moving parts. These direct imaging methods are more suited to situations where mechanical simplicity is critical. Among the different imaging devices that can be used, square-pore micro-channel plate X-ray optics (MPO), sometimes referred to as "Square Multi channel plate optics" , "Multi-pore optics" or "lobster-eye optics" is one of the most attractive because of its efficiency given by the corner cube effect [START_REF] Angel | Lobster eyes as x-ray telescopes[END_REF][START_REF] Price | Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics[END_REF][START_REF] Chapman | X-ray focusing using square channel-capillary arrays[END_REF][START_REF] Sarrazin | Mapx: 2d xrf for planetary exploration -image formation and optic characterization[END_REF], in particular when compared with straight polycapillary optics [START_REF] Chapman | X-ray focusing using cylindrical-channel capillary arrays. i. theory[END_REF][START_REF] Romano | Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station[END_REF] used for 1:1 imaging. Planar MPOs, to which this study will be restricted, offer a magnification of 1 to cover a surface area commensurate with the size of the detector, and provide an additional degree of freedom, compared to spherical MPOs, with the possibility of changing the working distance without changing the magnification.

The understanding of MPO properties relies first on the modeling of perfect structures for which the Chapman et al. publication of 1991 [START_REF] Chapman | X-ray focusing using square channel-capillary arrays[END_REF] is a reference. The intention of the present publication is to bring several improvements to this model and examine the effect of short distances which are not considered in the work of Chapman et al. The effect of defects related to the manufacturing technology is beyond the scope of our publication with the exception of the modification of reflectivity by surface roughness.

In Chapman et al. [START_REF] Chapman | X-ray focusing using square channel-capillary arrays[END_REF] two approximations of the reflectivity curve are used, depending on the X-ray energy, and are defined with a single parameter. We will show that a single simplified description might be used with two parameters, the first corresponding to the real part of the reflecting material refractive index and the second related to both absorption (imaginary part of the refractive index) and surface roughness. Using this new simplified curve the behavior of the MPO is much better described in a large range of energies, in particular for heavy reflecting materials like platinum or iridium.

In addition the Chapman et al. model is a continuous model where the pore size is considered as small enough to replace summations by integrals. For this reason the only contributions to the central spot of the PSF considered by Chapman et al. are the ones corresponding to an odd number of reflections in each of the two perpendicular planes containing the pores faces. It will be shown here that two additional contributions cannot be neglected when the source-MPO distance is decreased. The first additional contribution is related to reflections on the central row and column of pores relative to the optical axis and is visible at medium distance. The second contribution comes from direct transmission by the central pores closest to the optical axis and becomes important only at very small working distances. A new semi-continuous model based on the use of the two-parameter reflectivity curve and taking into account the three contributions mentioned above will be described in the first part of this publication. Using this model the influence of geometrical parameters and X-ray energy on the point spread function (PSF) central spot integrated intensity and profile are shown. The modifications of other features of the PSF such as the two characteristic perpendicular wings and the pseudobackground related in part to the direct transmission through the pores are also investigated.

In the second part it will be shown that the PSF is not unique, it is a periodic function of the source coordinates reflecting the periodicity of the planar MPO itself. It will be shown that the influence of this periodicity is enhanced at small distances and can be reproduced in most cases using a discrete model. The semi-continuous model can however be used in the case of a modified point spread function where a small square with a size ( defined as the side length ) equal to the pore periodicity is used as a source instead of a point.

Finally in the third part the trends shown in part 2 for the PSF are illustrated by images simulated using ray tracing and improvements to the standard MPO are proposed.

Throughout this publication ray tracing simulations are used to support our modeling because the analysis carried out in this paper is based on geometrical optics. Using the very crude criterion of an angular diffraction contribution on the order of λ/D where λ is the X-ray wavelength and D the pore size we believe that it should be valid in the energy range of the fluorescence of most of the elements and for pore sizes of tens of microns. However it might be necessary to use a wave theory taking into account wave-guide effects, diffraction, eventual interference between pores and the effect of partial coherence for lower energy fluorescence and smaller pore sizes, but it is beyond the scope of this paper.

RAY TRACING

Ray tracing simulations ( sometimes referred to as Monte Carlo simulations ) are used to produce PSFs and quantities which are eventually compared with the results of analytic models. These simulations include the calculation of X-ray reflectivity which is done using the standard matrix method [START_REF]X-ray and neutron reflectivity, theory and applications[END_REF]. Anomalous scattering and absorption are taken into account through the calculation of the atomic scattering factor as well as surface and interface roughness through a static Debye-Waller factor with a single parameter which is the root mean square roughness σ. Most of the ray tracing simulations are carried out with a monochromatic point source to obtain the PSF. The remainder use a two dimensional object emitting monochromatic X-rays as a source. The results shown do not depend on the number of rays used, except for the shot noise. The initial number of rays ( between 10 million and 1 billion) was, for each kind of simulation, large enough to make the shot noise negligible, or at least not cumbersome. Emission was considered isotropic in the maximum ±0.1rad range used in the simulations.

SEMI CONTINUOUS MODEL

The model developed below will be called "semi-continuous" because at some places sums over the pores of the MPO are replaced by integrals (i.e. pores are considered as infinitely small) while at other places the discrete aspect of the MPO is taken into account.

A. Parameters

Figure 1 shows the main geometrical parameters used in the model. In the following D, T and t are respectively the pore size ( square side length), the period of the MPO square lattice and the MPO pore length. The distance between the source and the MPO is l s and is equal to l i the distance between the MPO and the image plane ( see also Figure 15 ). l s and l i are measured with respect to the plane which is half way from the MPO entrance plane and the MPO exit plane. The MPO in-plane dimensions and, as a result, the total number of pores are considered to be infinite, edge effects will not be discussed. Additional material parameters are used through the inner pore coating reflectivity properties.

B. Reflectivity

In order to enable analytical calculations, a two parameters simplified (linearized) reflectivity curve is used instead of the reflectivity curve calculated using the standard matrix method. (fig. 2). These two parameters are the total reflection critical angle γ c and R the reflectivity averaged between 0 and γ c . The total reflection critical angle is written as :

γ c = λ r e 2π N av ρ m K (1) ≈ 1.643 10 -3 λ ρ m K [γ c (rad), λ(Å), ρ m (g.cm -3 )] K = 2 ∑ i c i ( f 0i + f i (λ)) ∑ i c i M i
where λ is the X-ray wavelength, r e is the classical electron radius, N av the Avogadro number, ρ m the mass density, c i is the element i composition in the reflecting material, f 0i + f i (λ) the element i atomic scattering factor including the anomalous part f', and M i the element i atomic mass. For light elements the K coefficient is close to 1, for heavy elements such as Ir or Pt it is around 0.8 and of course for a mixture of light and heavy elements K will be between these values. We define R(λ) by :

R(λ) = 1 γ c (λ) γ c (λ) 0 R(γ, λ)dγ (2)
where R(γ, λ) is the reflectivity of the material or stack of materials at the grazing angle γ and X-ray wavelength λ. The value of R(λ) depends on the surface roughness and on the reflecting material elements atomic scattering factors which themselves depend on energy and can have singular points at X-ray absorption edges. It can be calculated directly by numerical integration of the reflectivity curve. The simplified reflectivity is then given by :

R(γ, λ) = 1 + 2(R(λ) -1)γ/γ c (λ) γ ≤ γ c (λ) 0 γ > γ c (λ) (3)
When absorption and roughness are low, R(λ) will be close to 1 and the reflectivity model will be close to the model used by Chapman et al. at high energy. For higher absorption R(λ) will be close to 0.5 and the simplified reflectivity will be similar to the model used by Chapman at low energy. In the case of medium absorption and roughness -see for example figure 2 -it should work better than the two asymptotic models. γ c (λ) and R(λ) are the two parameters defining a simplified reflectivity curve. In the calculations it will be however more convenient to use ∆R(λ) instead of R(λ), with ∆R(λ) = 2(R(λ) -1) ( figure 2 ). It might occur, for high absorption and because of the additional effect of roughness, that equation 2 gives a value of R lower than 0.5. In this case equation 3 gives negative values for R when γ is close to γ c . To avoid this, the γ c value used in the model is replaced by an effective value γ ceff and R by R eff : R < 0.5 R eff = 0.5

γ ceff = 2Rγ c (4)
This substitution applies for example in the 2keV-3keV region in the case of Ir for a root mean square (rms) roughness of 2nm, as will be shown later in figure 6.

It appeared that there was a problem with commonly used f' and/or f" tables, such as Henke or Cromer Liberman tables, in the region of M edges of heavy elements such as Ir or Pt, eventually giving negative values of the real part of the atomic scattering factor. For these elements the values published by C.T. Chantler in 2000 [START_REF] Chantler | Detailed tabulation of atomic form factors, photoelectric absorption and scattering cross section, and mass attenuation coefficients in the vicinity of absorption edges in the soft x-ray (z=30-36, z=60-89, e=0.1 kev-10 kev), addressing convergence issues of earlier work[END_REF] were used. Finally we would like to point out that the simplified two parameters analytical approximation of the X-ray reflectivity discussed in this paragraph might be used in grazing incidence applications, beyond the context discussed in this publication.

C. Point spread function (PSF) central spot integrated intensity

A typical simulated MPO point spread function (PSF) is shown on figure 3. The main features are a central spot which is the desirable part for image formation, two perpendicular lines forming a cross and weaker intensity in the quadrants delimited by the cross. All these characteristics parts can be found in experimental PSFs as well [START_REF] Sarrazin | Mapx: 2d xrf for planetary exploration -image formation and optic characterization[END_REF]. In this section we will calculate Ω eff , the effective solid angle acceptance of the MPO, which multiplied by the source intensity per unit solid angle gives the intensity in a square twice the size the MPO period, i.e. 2T. The intensity outside this square is not taken into account because it is not properly focused in the image plane. The central spot intensity is usually considered as coming from an odd number of reflections in the two (x,z) and (y,z) planes. At long working distance this contribution is the most significant, however there are two other contributions to the central spot. Figure 4 shows the distribution in the entrance plane of the MPO of the rays contributing to the PSF central spot as a function of the number of reflections in the (x,z) and (y,z) planes noted (n x , n y ). In this figure, the MPO parameters and the energy of X-rays are such that a maximum of one reflection can occur in each plane. The (odd,odd) reflections -(1,1) in the figure -come from 4 equivalent 2-dimensional regions of the MPO, so this contribution will be named Ω 2d . Note that these 4 regions correspond to the 4 regions visible on figure 6 of the Chapman et al. 1991 publication [START_REF] Chapman | X-ray focusing using square channel-capillary arrays[END_REF]. The second and newly considered contribution is coming from (odd,0) and (0,odd) reflections ((1,0) and (0,1) in the figure) and corresponds to the one or two rows and columns of pores which are the closest to the optical axis. This contribution will be noted Ω 1d as the MPO regions from which it is coming are linear. The last contribution, related to the one to four pores which are the closest to the optical axis, is noted Ω 0d as it comes from a very localized region. We have the following relation :

Ω eff = Ω 2d + Ω 1d + Ω 0d -0.
= 20µm, T = 26µm , t = 1.2mm, l s = 10cm, E = 6400eV,

reflective layer: Ir

Ω 2d is first calculated with a method close to, and largely inspired by, the method of Chapman et al. [START_REF] Chapman | X-ray focusing using square channel-capillary arrays[END_REF] using small angle approximations. If D/l s γ c the rays entering a particular pore of angular position with respect to the source (θ x , θ y ) ( see figure 1a ) are considered parallel and their direction is defined by the same angles as the pore angular position. The beam entering this particular pore and that is submitted to n x reflections in the (x,z) plane and n y reflections in the (y,z) plane will have a dimension δ nx (θ rx ) along x and δ ny (θ ry ) along y. When the number of reflections is greater than 0, δ nx (θ rx ) and δ ny (θ ry ) are given by the expression:

δ n s (θ rs ) D =          0 |θ rs | ≤ n s -1 |θ rs | -(n s -1) n s -1 < |θ rs | ≤ n s n s + 1 -|θ rs | n s < |θ rs | < n s + 1 0 |θ rs | ≥ n s + 1 (5)
where the index s will be either x or y and θ rs = θ s t/D. In the case where n s = 0:

δ 0 (θ rs ) D = 1 -|θ rs | 0 ≤ |θ rs | < 1 0 |θ rs | ≥ 1 (6)
The corresponding effective collected solid angle corresponding to (n x , n y ) reflection in the same single pore is:

Ω p (θ x , θ y ) = δ n x (θ rx )δ n y (θ ry ) (l s -t/2) 2 R n x (θ x )R n y (θ y ) (7)
For infinitely small pores the number of pores in the dθ x dθ y element of solid angle is :

d 2 N(θ x , θ y ) = η(l s -t/2) 2 D 2 dθ x dθ y ( 8
)
where η is the fraction of the MPO entrance surface occupied by pores. Considering a pore size D and a period of T in both x and y directions:

η = D 2 /T 2
The total effective solid angle corresponding to (n x , n y ) reflections, Ω(n x , n y ), is then obtained by integration :

Ω(n x , n y ) = +∞ θ x =-∞ +∞ θ y =-∞ Ω p (θ x , θ y )d 2 N(θ x , θ y ) = 4ηγ 2 c S n x S n y (9)
with

S n =        1 α α 0 δ n (θ rs ) D R n (θ rs t/D) dθ rs n = 0 1 2α n = 0 ( 10
)
where n is either n x or n y and:

α = t γ c D
Equation 10 is valid for any expression of the reflectivity for which the upper bound, above which reflectivity is equal to zero, is the total reflection critical angle γ c . It is the case with the simplified reflectivity model of equation 3 which is then used to calculate the integral of equation 10. Writing ∆R = 2(R -1) and after integration we obtain in the case where n > 0:

αS n =                      0 α ≤ n -1 F n (∆R) -F n ((n -1)∆R/α) n -1 < α < n F n (n∆R/α) -F n ((n -1)∆R/α)+ G n (∆R) -G n (n∆R/α) n ≤ α < n + 1 F n (n∆R/α) -F n ((n -1)∆R/α)+ G n ((n + 1)∆R/α) -G n (n∆R/α) α ≥ n + 1 (11)
where the functions F n (U) and G n (U) are defined as follows:

F n (U) = α ∆R 2 U (1 + U) n+1 n + 1 - (1 + U) n+2 (n + 1)(n + 2) -(n -1) α ∆R (1 + U) n+1 n + 1 G n (U) = α ∆R (1 + U) n+1 - α ∆R 2 U (1 + U) n+1 n + 1 - (1 + U) n+2 (n + 1)(n + 2)
Finally as this contribution to the central PSF spot corresponds to odd numbers of reflections in both xz and yz planes:

Ω 2d = 4ηγ 2 c S 2 odd (α, ∆R) (12)
with :

S odd (α, ∆R) = +∞ ∑ k=0 S 2k+1 (13)
Ω 2d (equation 12) is proportional to the product of the square of the critical angle and of a function which only depends on α and ∆R. It is important to note that this main contribution to the effective solid angle does not depend on l s ( as will be illustrated later in figure 9) . The reason is that the 1/(lst/2) 2 dependence of the single pore effective solid angle ( equation 7) is perfectly compensated by the (lst/2) 2 dependence of the number of pores involved ( equation 8). Figure 5 shows S odd (α, ∆R) for a set of values of ∆R going from the maximum value 0 to the minimum value -1, and a range of values of α. In the paper of Chapman et al. [START_REF] Chapman | X-ray focusing using square channel-capillary arrays[END_REF] only the extreme cases are considered, ∆R = 0 in the case of high energies and ∆R = -1 for low energies. The refinement brought by the new two parameters reflectivity curve separates the effect of absorption/roughness from the effect of the energy dependent critical angle. In the case of heavy reflecting materials for example, the absorption is neither low nor high in a wide energy range, hence using this 2-parameter reflectivity curve is a necessity.

When considering a particular reflecting material the dependence of ∆R and γ c with the X-ray energy can be calculated. The example of Ir is shown on figure 6. The refractive index has a simple 1/E dependence with the X-ray energy, with the exception of the energy region of the M absorption edges. ∆R is a more complex function of the X-ray energy: it is much more sensitive to the presence of absorption edges such as the Ir L edges in the 11-13 keV range and the M absorption edges in the 2-3 keV range. It is also clearly dependent on the surface root mean square roughness σ. In the 2-3 keV region it was necessary to use effective values of ∆R and γ c , in the case of Ir with a 2nm rms roughness, calculated using equations 4.

Using the data shown on figure 6, it is possible to plot Ω 2d as a function of the X-ray energy on one axis and the ratio t/D on the other axis. The map obtained in the case of an Ir layer with a rms roughness of 2nm is shown in figure 7. This map can be a guide for the choice of the MPO t/D ratio, which will depend on the spectral band of interest. For example large values might be chosen for higher fluorescence energies. As already mentioned there is no dependence of Ω 2d on the working distance l s so for a particular reflecting material all the information on Ω 2d is in this map.

The Ω 1d contribution to the PSF central spot coming from the row and column of pores shown in dark blue in figure 4. It corresponds to an odd number of reflections in one direction and a direct transmission in the other. A one dimensional integration is carried out in this case, giving for the row in the x direction:

Θ 1x (n x ) = +∞ θ x =-∞ √ η(ls -t/2) D δ n x (θ rx ) (l s -t/2) R n x (θ x )dθ x = 2 √ ηγ c S nx 0
Ω 1d = D l s + t/2 ∑ k Θ 1x (2k + 1) + Θ 1y (2k + 1) (14)
= 4D √ ηγ c l s + t/2 S odd (α, ∆R) = 2D l s + t/2 Ω 2d (15)
Ω 1d has a lower dependence to X-ray energy than Ω 2d because it depends linearly on γ c which is roughly proportional to the X-ray wavelength and also because S odd changes with X-ray energy are lower than the changes of S 2 odd . Ω 1d depends on l s , shorter distances will increase this contribution.

The last contribution is related to the pinhole-like transmission by the central pore:

Ω 0d = D 2 (ls + t/2) 2 (16)
which shows no spectral dependence and a stronger dependence with the working distance l s . Finally the total effective solid angle collection of the MPO is given by:

Ω eff = 4ηγ 2 c S 2 odd (α, ∆R) + 4D √ ηγ c l s + t/2 S odd (α, ∆R) + D 2 (ls + t/2) 2 = 2 ∑ n=0 g(n) [ √ ηγ c S odd (α, ∆R)] n D l s + t/2 2-n (17)
where g(n) is a kind of degree of degeneracy, g(n) = 4 for n = 1 or n = 2 and g(0) = 1. It corresponds to the number of equivalent regions for the three different contributions, which are visible on figure 4. Ray tracing results are compared in figure 8 with the model of equation 17 in a case where the dominant term in Ω eff is Ω 2d , because l s is large ( l s = 0.1m). For this reason the plots of figure 8 are similar to the ones of figure 5. The influence of the two other terms Ω 1d and Ω 0d is visible on figure 9. Ω 2d does not depend on l s so the l s dependency of Ω eff clearly visible on this figure is related to Ω 1d and Ω 0d .

= 2nm, t = 1.2mm, D = 20µm, T = 26µm, E = 6400eV.

D. PSF central spot profile

The general shape of the central spot is a pyramid, taking the intensity as a third dimension. However, this is the shape related specifically to Ω 2d , the two other contributions to the PSF central spot having different profiles. These three profiles can be calculated using the same simplified reflectivity model that was used to determine Ω eff . For this purpose two functions of x or y have to be calculated.

P 1x (x) =            √ ηγ c D |∆R| +∞ ∑ k=0 - (1 -U) 2(k+1) 2(k + 1) U max (k,x) U min (k,x) |x| D < 1 0 |x| D ≥ 1 P 2x (x) =    1 2ls |x| < D l s l s + t/2 0 otherwise (18)
where

D = 1 1 -t/(2l s ) D U min (k, x) = (2k + |x|/D )|∆R|/α U max (k, x) = min(U m (k, x), max(|∆R|, U min (k, x))) U m (k, x) = (2(k + 1) -|x|/D )|∆R|/α
The equivalent functions P 1y (y) and P 2y (y) are obtained by simple substitution of x by y. The three components P 2d , P 1d , P 0d of the profile P eff are then given by: P 2d (x, y) = P 1x (x)P 1y (y) P 1d (x, y) = P 1x (x)P 2y (y) + P 2x (x)P 1y (y)

P 0d (x, y) = P 2x (x)P 2y (y) (19)
The following equation can be used to determine the total effective solid angle in a small dxdy surface element ( a detector pixel for example ) :

d 2 Ω eff (x, y) = P eff (x
= 2nm, l s = 0.1m, D = 20µm, T = 26µm, E = 6400eV.
The profiles obtained using equation 20 in the x direction for y = 0 are compared with the results of ray tracing simulations in figure 10. This figure shows that the modifications induced in the central spot profile by varying the MPO thickness are well reproduced by the model.

Figure 11 shows the integral breadth ( the integrated intensity divided by the maximum intensity, close to the full width at half maximum here ) of the central spot in the x ( or equivalently y) direction as a function of α for different energies. For α > 1.5 the central spot integral breadth is almost constant and equal to the pore size D, while for values between 0 and 1 there is a linear increase of the integral breadth with α. The consequence is that, for a particular MPO thickness, pore size and material, there will be a critical energy below which the central spot size will be constant, and above which it will decrease. We have however to keep in mind that this study focuses on perfect MPO structures, while the central spot profile is sensitive to some MPO defects, in particular the pore orientation dispersion.

E. PSF cross arms and pseudo background

An important feature of the PSF is the presence of a cross centered on the main spot (figure 3). This cross having detrimental effects on image resolution, it is interesting to see how its intensity and reach, which should be both minimized, is influenced by instrumental parameters. It is however a complex task because the PSF cross arms result from the concentration of all the rays undergoing an odd number of reflections in one direction (x or y) and their dispersion in the other direction because of an even number of reflections (including 0). Furthermore the intensity and eventually the reach of the cross arms in the region around the PSF central spot depend on the direct transmission by the MPO, which is a purely geometrical factor, and on the reflectivity which changes with the material and X-ray energy.

To evaluate the effect of PSF cross arms, we will use three quantities: Ω arms the integrated cross arms intensity, Ω arms0 the intensity of the cross arms close to the PSF central spot corresponding to (odd,0) and (0,odd) reflections, and the local cross arm intensity within the central spot estimated using Ω 1d ( equation 14). The two first quantities can be calculated using equations 9, 10 and 11:

Ω arms = 8ηγ 2 c S even (α, ∆R)S odd (α, ∆R) -Ω 1d (21)
Ω arms0 = 4η γ 2 c α S odd (α, ∆R) -Ω 1d (22)
In these two expressions Ω 1d is subtracted from the cross arms intensity because it is considered as a part of the PSF central spot. This subtraction can be neglected if l s D √ Ω 2d . S odd and S even are defined by equations 13 and 23, respectively.

S even (α, ∆R) = +∞ ∑ k=0 S 2k (23)
The following part is restricted to the case where l s D √ Ω 2d . To evaluate the cross arms intensity relative to the intensity of the central spot, we can calculate the ratio : Figure 12(a) shows that, for α > 2, the ratio of the cross arms intensity over the central spot intensity is roughly constant while it increases sharply when α is reduced below 1. Given fixed values of t/D and a particular reflecting material, α and ∆R will only depend on the X-ray energy. As a result, it is possible to plot the cross arms relative intensity as a function of the energy. Figure 12(b) shows that in the case of an Ir reflecting surface with a 2nm RMS roughness a t/D ratio in the 200-400 range will give small changes in this relative intensity over a 3-20 keV range. It is a t/D range where Ω 2d changes are also minimized as it was shown on figure 7.

Ω arms Ω eff ≈ Ω arms Ω 2d = 2 S even S odd (24)
Ω arms0 Ω eff ≈ Ω arms0 Ω 2d = 1 αS odd (25)
The last contribution to the PSF is the direct transmission and rays undergoing (even,even) reflections, called pseudobackground for its 2-dimensional nature as opposed to the 0dimensional central spot and 1-dimensional cross arms. The ratio between its integrated intensity and the central spot intensity when l s D √ Ω 2d is written as:

Ω p.background Ω eff ≈ Ω p. background Ω 2d = S 2 even S 2 odd = 1 4
Ω arms Ω 2d 2 ( 26
)
Large values of Ω arms /Ω 2d will correspond to even larger values of Ω p. background /Ω 2d as this ratio is always greater than one. The reach of the cross and pseudo background is:

W cross = max(4l s D/t, 4l s γ c )
It is proportional to l s and depends on the X-ray energy below a critical energy influenced by reflecting material and D/t ( see figure 13 in the case of Ir). The size of the cross and pseudo- background region tends to be higher at low energy, but their intensity with respect to the central spot intensity tends to be lower. Figure 13 also shows that lower values of the cross arms and pseudo background reaches are obtained with higher values of the t/D ratio.

DISCRETE MODEL AND SEMI-CONTINUOUS MODEL IN THE CASE OF SHORT l s DISTANCES

For decreasing values of l s , some situations result in a limited number of pores contributing to the PSF and, as a consequence, integrations have to be replaced by discrete sums over the pores and integrations within each pore. For these lower l s values the periodic nature of the PSF is more visible as shown in an extreme case in figure 14. Let us consider the pore (i,j) at a distance (x i , y j ) from the optical axis, with x i = x 0 + iT and y j = y 0 + jT. -x 0 and -y 0 are the coordinates of the optical axis in the plane of the MPO with respect to the center of the nearest pore ( see figure 1 (b) ).

Restricting first the analysis to the (xz) plane ( see figure 15), its angular position θ i and angular aperture ∆θ can be written as:

θ i = |x i | l s -t/2 ∆θ = D l s -t/2
The minimum and maximum angles of the rays entering in the pore i are, as illustrated in figure 15 and in the case where i = 0:

θ min (i) = θ i -∆θ/2 θ max (i) = θ i + ∆θ/2
The case of i = 0 is a little bit more complicated. If |x 0 | ≥ D/2 the formula above apply. In the case where |x 0 | < D/2 the optical axis is inside the pore, and we split the pore in two parts, noted with the exponents + and -, with : From θ min (i) and θ max (i), the minimum and maximum num-bers of reflections of rays entering the pore i can be calculated:

θ + min (0) = 0 θ + max (0) = θ i + ∆θ/2 θ - min (0) = 0 θ - max (0) = -θ i + ∆θ/2
n min (i) = θ min (i)t D n max (i) = θ max (i)t D + 1
where the symbols indicates the floor integer value. For a particular position of pore x i and a particular number of reflections n = 0 in the (xz) plane, the effective source angular range going through the pore can be written as :

∆θ eff (n, x i ) = θ 2 θ 1 R n (λ, γ)dγ = γ c ∆R (1 + U) n+1 n + 1 ∆Rθ 2 /γ c ∆Rθ 1 /γ c (27)
where

θ 1 , θ 2 =      min(θ min , γ c ), min(θ x i (n min + 1), γ c ) n = n min min(θ x i (n), γ c ), min(θ x i (n + 1), γ c ) n min < n < n max min(θ x i (n max ), γ c ), min(θ max , γ c ) n = n max
with :

θ x i (n) = 1 l s + t/2 [|x i | + (n -1/2)D]
As opposed to the continuous model the ray angle and the reflectivity changes within a single pore are taken into account. Similar equations can be written for reflections in the (yz) plane and the pore position y i , simply replacing x by y.

Using the approximated expression of reflectivity of equation 3 for the integrals of equations 27, two quantities are then calculated :

Σ x,odd = +∞ ∑ i=-∞ +∞ ∑ k=0 ∆θ eff (2k + 1, x i ) Σ y,odd = +∞ ∑ j=-∞ +∞ ∑ k=0 ∆θ eff (2k + 1, y j )
In each of the sums over the pores ( indices i and j) the eventual splitting of the central pore in two parts, along x and/or y direction, has to be taken into account.

Two other quantities are necessary for the full calculation corresponding to the part of the direct transmission which reaches the image plane at (∆x, ∆y) from the optical axis, with |∆x| ≤ D and |∆y| ≤ D.

Σ x,0 = ∑ k, n minx =0 [min(θ x k (1), D/(2l s )) -min(θ min-xk , D/(2l s ))] Σ y,0 = ∑ k, n miny =0 min(θ y k (1), D/(2l s )) -min(θ min-yk , D/(2l s ))
The sums are carried out over the pores for which the minimum number of reflections in each plane is equal to 0, and take again into account the eventual splitting of the central pore in two or four parts. The effective collected solid angle then becomes :

Ω effective,d = (Σ x,odd + Σ x,0 )(Σ y,odd + Σ y,0 ) (28)
This discrete model was tested first by comparing the intensity profile obtained using ray tracing by moving the MPO along the x direction, changing x 0 , with y 0 equal to 0 to the one . Ω eff as a function of x for y = 0. The continuous line was calculated using equation 28. The first series of ray tracing simulations was carried out using the simplified reflectivity curve ( circles), the second series using the real reflectivity (squares). The x profile is quite sensitive to the exact reflectivity profile. Parameters : Ir, energy = 6400eV, σ = 2nm, l s = 5mm,t = 2.4mm, D = 20µm, T = 26µm.

obtained using equation 28 ( figure 16) . Two sets of ray tracing simulations were carried out, the first set ( ray tracing 1 on the figure) using the exact reflectivity curve while the second set ( ray tracing 2 on the figure) used the approximated curve. The points obtained using this second set are very close to the model demonstrating that the quantitative differences observed between the model and the first set are related to the shape of the reflectivity curve used in the model. As the approximated reflectivity is closer to the real reflectivity when absorption is lower, it is expected that the discrete model works better at higher energies. Figure 17 shows that the two kinds of ray tracing simulations give almost the same results above 10keV and are very close to the discrete model. Below that value there are visible differences. The difference between the discrete model and the ray tracing of type 2 comes from the fact that, at low values of l s and at low energy, other combinations of reflections than the ones considered in the model should be taken into account. At larger distances the rays undergoing these combinations of reflections are not reaching the central PSF spot. When l s = t/2 every ray that is not absorbed contributes to the central spot which is actually the whole PSF. For intermediate and short values of l s a part of these rays will be in the central spot. We did not try to make a model for these specific situations, leaving them to ray tracing.

A. Validity of the semi-continuous model at short distance l s when using a square source

Although simulations with a point source and the discrete model presented above are useful in showing the modulation of the PSF with the source position, it is interesting with respect to fluorescence imaging to know the PSF averaged over the MPO unit cell. It can be obtained using a source emitting uniformly over a square of size T ( the MPO period), and it can be shown that it is equivalent to the integration of the semi-continuous model. If N s is the total number of photons emitted by this source in the 4π solid angle, the number of photons emitted in the solid angle element dΩ by the surface element of the source dS is: 17. Ω eff as a function the X-ray energy in the case of a short l s distance. RT1 and RT2 are ray tracing simulations using two kinds of reflectivity curves ( see text). These ray tracing simulations were done using a point source at x = 0,y = 0 (square symbols), a point source at x = T/2,y = T/2 ( diamond symbols) or a uniform square source of side length T (circle symbols). The ray tracing simulations are compared with the discrete and the semi-continuous model (lines). Parameters : Ir, l s = 2.5mm, σ = 2nm, t = 2.4mm, D = 20µm, T = 26µm.

d 4 n = N s 4πT 2 dSdΩ = N s η 4πD 2 dSdΩ
Because of the source size, for any source-MPO distance l s and any pore size D, a full pore will be illuminated in any direction (θ x , θ y ). In this direction the element of surface of the source providing photons undergoing (n x , n y ) reflections will be exactly dS = δ nx (θ x )δ ny (θ y ) that can be calculated using equations 5 and 6. At the exit of the MPO, the number of photons, incoming in the solid angle dΩ = dθ x dθ y at angles (θ x , θ y ) and having experienced (n x , n y ) reflections, is given by:

d 2 n = δ nx (θ rx )δ ny (θ ry )R n x (θ x )R n y (θ y ) N s η 4πD 2 dθ x dθ y
If we calculate this quantity for N S /(4π) equal to 1, we find the product Ω p (θ x , θ y )d 2 N(θ x , θ y ) of the two quantities given by equations 7 and 8. We obtain the important result that even for a short working distance equation 17 remains valid with an extended square source having for size the MPO period T. The local variations of the PSF can only be predicted by a discrete model, but the average over a MPO period can be calculated using the semi-continuous model. The comparison of ray tracing simulations using a square source (figure 17, red circle symbols) with the semi-continuous model shows a good agreement at a low l s distance -except at low energy for the reason exposed above in the case of a point source.

IMAGING

We have examined in details and explained the influence of the parameters of an MPO imaging experiment on the MPO point spread function using two different models. This section aims to show that the trends outlined in this analysis are visible when imaging extended objects. For that purpose, a Siemens star and a regular grid are used as monochromatic extended X-ray sources in ray-tracing simulations.

Figure 18 shows images of a Siemens star obtained at different energies. The most visible feature is the decrease in intensity when increasing energy, which is predicted by equation 12 and illustrated in figure 7. At each energy a cross is visible in the central part of the image, it is directly connected to the PSF cross seen on figure 3. The effect of the cross is also noticeable in the region outside of the circle containing the Siemens star, creating a background intensity which does not change substantially when increasing energy. It means that the contrast is decreasing with energy, in agreement with equation 24 which predicts an increase with energy in the ratio between the cross integrated intensity and the central spot intensity.

Figure 19 shows the increase in intensity when decreasing l s = l i as predicted by equation 17 and illustrated by figure 9. For the lower distances the spatial dependence of the PSF explained by the discrete model ( equation 28) is more and more visible.

Finally figure 20 is an illustration of the dependence of the central spot size -and thus image resolution -with the MPO t/D ratio predicted by equation 20 and illustrated by figures 10 and 11. Two grid orientations are used : the numerical experiment shows that the grid starts to be visible for higher MPO thicknesses when the grid is parallel to the MPO square array than when it is at 45 degrees because of the anisotropy of the PSF. We remind the reader that MPO are considered as perfect in this publication and that the relatively high resolution predicted in some cases might be difficult to achieve due to pores imperfect orientations or slope errors. Figure 20 is also an illustration of the increase in the ratio between the PSF cross integrated intensity and the PSF central spot that might occur when decreasing the ratio t/D.

PROPOSED IMPROVEMENTS

Two features of the PSF have a negative impact when using MPO for imaging: the cross related to odd/even reflections on adjacent pore sides and the PSF spatial dependence. The cross might be transformed in a more isotropic feature by two different means. The first approach involves an array of square pores with a random orientation of the square cross section. This pore packing scheme was already proposed by R. Willingale et al. [START_REF] Willingale | Optimization of square pore optics for the x-ray spectrometer on bepi-columbo[END_REF] as well as other pore orientation patterns for the BEPI-COLUMBO spectrometer. As outlined in this reference the pore open fraction η has to be reduced in this case, the consequence being an overall reduction of the intensity of all the parts of the PSF. The second method consists in a rotation of a regular MPO, precise enough to achieve a negligible precession of the PSF. Figure 21 shows the PSF resulting from these two kind of modifications ( labeled b and c, respectively ) compared to the regular MPO PSF (a). In the case of the rotation the results depend on the position of the rotation axis, but changing this position gives similar results. Both new configurations are quite efficient in making the PSF more isotropic.

The efficiency of the two modified configurations are evaluated by imaging a Siemens star ( figure 22 ). It is interesting to see that both configurations have a similar effect at large distance ( 100 mm here), but that only the rotation provides a good result at short distance ( 5mm here). As a matter of fact the image obtained using the randomly oriented squares is not very different at short distance than the image obtained using a regular MPO. This can be explained by the limited number of channels involved for each point of the extended source, which is also the origin of the periodicity of the PSF discussed above in the case of a regular MPO.

CONCLUSION

Analytical expressions of the main PSF features of perfect planar MPOs were obtained using a semi-continuous model. This model is based on a two parameters approximation of the X-ray reflectivity curve and takes into account two contributions which were not considered in previous models. It was validated against results of ray tracing simulations by varying several parameters of a X-ray fluorescence imaging experiment: MPO parameters such as the thickness and ratio between the pore size and the thickness, and experimental parameters such as X-ray energy and working distance. The benefit of this analytical model is to evidence the influence of these parameters on the intensity and spatial resolution of an X-ray fluorescence experiment. It was also shown that, for short distances, it is necessary to use a discrete model which has however the disadvantage of clouding the influence of the different parameters.

Series of images of extended sources such as a Siemens star or a grid are interpreted in the light of the semi-continuous and the discrete models we have developed and solutions are also proposed to improve the quality of these images through an isotropization of the PSF.

Finally, it is worth noting that real MPOs show imperfections that might add another level of complexity to this analysis and bring important modifications to the behavior of MPOs [START_REF] Willingale | Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes[END_REF]. These imperfections were purposely not considered here because it appeared necessary to first establish precise models of a perfect MPO.

FUNDING INFORMATION

NASA MatISSE16_2-0005

Fig. 1 .

 1 Fig. 1. MPO geometrical parameters (a) Point source imaged in the 1:1 arrangement. (b) Detail showing the precise position (x 0 , y 0 ) of the optical axis with respect to the nearest pore center

Fig. 2 .

 2 Fig. 2. Example of the simplified version of the reflectivity curve used for analytic calculations (blue line) compared with the simulated reflectivity of an Ir layer on a silica substrate at 6400 eV ( red line).

Fig. 3 .

 3 Fig. 3. Ray tracing simulated point spread function (PSF). The reflecting material is a 25nm Ir layer on SiO 2 , with a surface and interface roughness both equal to 2nm. E = 15keV, l s = 0.05m, D = 20µm, T = 26µm, t = 4.8mm. Normalized intensity in log scale.

Fig. 4 .

 4 Fig. 4. Distribution of rays, at the entrance surface of the MPO, contributing to the PSF central spot. In the notation (n x , n y ), n x is the number of reflections in the xz plane, n y is the number of reflections in the yz plane. Because of the parameters used here n x and n y are either 1 or 0. Parameters : D = 20µm, T = 26µm , t = 1.2mm, l s = 10cm, E = 6400eV, reflective layer: Ir

Fig. 5 .

 5 Fig. 5. Plots of S 2 odd (α, ∆R) as a function of α for different values of ∆R showing the influence of absorption and roughness induced loss on Ω 2d

Fig. 6 .

 6 Fig. 6. External total reflection critical angle γ c and reflectivity loss ∆R as a function of energy in the case of Ir reflective material. ∆R is shown for two values of the root mean square surface roughness σ. The insert shows the 2-3 keV region where effective values γ ceff ( equation 4 ) replace in the case of the 2nm roughness ( dotted line) the critical angle value used in the case of a perfect flat surface ( continuous line). In this region and in the case of the 2nm rms roughness, ∆R = ∆R eff is constant and equal to -1.

Fig. 7 .

 7 Fig. 7. Ω 2d as a function of X-ray energy and t/D in the case of an Ir top layer with a 2nm root mean square roughness. Discontinuities induced by M and L absorption edges of Ir are clearly visible

Fig. 8 .

 8 Fig. 8. Ω eff /γ 2 c as a function of α = tγ c /D. Variations of α are obtained, for each energy, by changing the MPO thickness t, D being constant. Ray tracing simulations results (symbols) are compared with the model of equation 17 (continuous lines). Parameters : Ir layer with σ = 2nm, l s = 10cm, t = 1.2mm, D = 20µm, T = 26µm.

Fig. 11 .

 11 Fig. 11. Central spot x direction profile relative integral breadth W/D as a function of t represented as a function of α = tγ c /D for three different energies : ray tracing simulations results (symbols) and model of equation 20 ( continuous lines ). Parameters: Ir layer with σ = 2nm, l s = 0.1m, D = 20µm, T = 26µm.

Fig. 12 .

 12 Fig. 12. Ω arms /Ω 2d : (a) as a function of α for values of ∆R in the range 0 to -1, (b) as a function of energy for different values of t/D, (60,120,240,480), in the case of an Ir layer with a 2nm surface rms roughness. Continuous lines are obtained using equation 24, symbols are results from ray tracing simulations.

Fig. 13 .

 13 Fig. 13. Relative reach of the cross and pseudo background as a function of X-ray energy for t/D ratios ranging from 60 to 480

Fig. 14 .

 14 Fig. 14. Ray tracing simulation showing the enhancement of the periodic nature of the MPO point spread function at short distance (a) Source. (b) Image showing a periodic modulation at places where the source is uniform . Reflective material : Ir. E = 6400eV, l s = 1.2mm, D = 20µm, t = 1.2mm

Fig. 15 .

 15 Fig. 15. Side view of a channel with the distances and angles used in the discrete model

Fig. 16

 16 Fig.[START_REF] Chantler | Detailed tabulation of atomic form factors, photoelectric absorption and scattering cross section, and mass attenuation coefficients in the vicinity of absorption edges in the soft x-ray (z=30-36, z=60-89, e=0.1 kev-10 kev), addressing convergence issues of earlier work[END_REF]. Ω eff as a function of x for y = 0. The continuous line was calculated using equation 28. The first series of ray tracing simulations was carried out using the simplified reflectivity curve ( circles), the second series using the real reflectivity (squares). The x profile is quite sensitive to the exact reflectivity profile. Parameters : Ir, energy = 6400eV, σ = 2nm, l s = 5mm,t = 2.4mm, D = 20µm, T = 26µm.

  Fig.17. Ω eff as a function the X-ray energy in the case of a short l s distance. RT1 and RT2 are ray tracing simulations using two kinds of reflectivity curves ( see text). These ray tracing simulations were done using a point source at x = 0,y = 0 (square symbols), a point source at x = T/2,y = T/2 ( diamond symbols) or a uniform square source of side length T (circle symbols). The ray tracing simulations are compared with the discrete and the semi-continuous model (lines). Parameters : Ir, l s = 2.5mm, σ = 2nm, t = 2.4mm, D = 20µm, T = 26µm.

Fig. 18 .

 18 Fig. 18. Ray-tracing simulations of a Siemens star imaged at different energies using an MPO for a relatively large distance l s = l i = 100mm. MPO parameters : Ir with a 2nm rms roughness as reflecting material, pore size D = 20µm, period T = 26µm, thickness 1.2mm. The number of rays generated and the intensity scale are the same for the four simulations.

Fig. 19 .

 19 Fig. 19. Ray-tracing simulations of the MPO imaging of a Siemens star for different distances l s = l i . MPO parameters : Ir with a 2nm rms roughness is the reflecting material, X-ray energy is 6400eV, pore size D = 20µm, period T = 26µm, thickness 2.4mm. The number of rays generated and the intensity scale are the same for the four simulations.

Fig. 20 .

 20 Fig. 20. Ray tracing simulations of a grid imaged with MPOs of different thicknesses. For each MPO thickness two images with a different grid orientation ( 0 degree and 45 degree) are shown. The pitch of the grid is 10 microns, the holes are squares with a side length of 5 microns. MPO parameters : Ir with a 2nm rms roughness is the reflecting material, X-ray energy is 6400eV, pore size D = 20µm, period T = 26µm. The intensity is normalized for each image.

Fig. 21 .

 21 Fig. 21. Ray-tracing simulations of the MPO point spread function. (a) using a standard MPO with a fixed orientation. (b) using a modified static MPO with a random orientation of the pores square cross section. (c) a standard MPO with a continuous rotation. The rotation axis is parallel to the pore axis with a position at 150 microns from the PSF center. MPO parameters : l s = l i = 100mm,Ir with a 2nm rms roughness is the reflecting material, pore size D = 20µm, period T = 26µm, thickness 1.2mm. The number of rays generated and the intensity scale are the same for the three simulations.

Fig. 22 .

 22 Fig. 22. Effect of a random square orientation ( central row) or a MPO rotation (bottom row) on imaging for two working distances (100mm, left column and 5mm, right column), compared to the standard MPO (top row). MPO parameters : Ir with a 2nm rms roughness is the reflecting material, pore size D = 20µm, period T = 26µm, thickness 1.2mm. The intensity is normalized for each image.

  Ω eff , Ω 2d , 1d and Ω 0d dependence with l s . Ray tracing simulations results (symbols) are compared with the model of equations 17, 12,14, 16 ( continuous lines). Parameters : Ir layer with σ

		1e-04
				Ir σ=2nm t=1.2mm
				D=20µm T=26µm
				E=6.4keV	Ω eff
						Ω 2d
						Ω 1d
	Ω	5e-05				Ω 0d
						model
	0 0e+00	0.02	0.04	0.06	0.08	0.1
				l s (m)
	Fig. 9.


Thomas Gauthier

DYNAMICAL PAIRS WITH AN ABSOLUTELY CONTINUOUS BIFURCATION MEASURE by

In this article, we study algebraic dynamical pairs (f, a) parametrized by an irreducible quasi-projective curve Λ having an absolutely continuous bifurcation measure. We prove that, if f is non-isotrivial and (f, a) is unstable, this is equivalent to the fact that f is a family of Lattès maps. To do so, we prove the density of transversely prerepelling parameters in the bifucation locus of (f, a) and a similarity property, at any transversely prerepelling parameter λ0, between the measure µ f,a and the maximal entropy measure of f λ 0 . We also establish an equivalent result for dynamical pairs of P k , under an additional assumption.

Introduction

Let Λ be an irreducible quasi-projective complex curve. An algebraic dynamical pair (f, a) parametrized by Λ is an algebraic family f : Λ × P 1 → P 1 of rational maps of degree d ≥ 2, i.e. f is a morphism and f λ is a degree d rational map for all λ ∈ Λ, together with a marked point a, i.e. a morphism a : Λ → P 1 .

Recall that a dynamical pair (f, a) is stable if the sequence {λ → f n λ (a(λ))} n≥1 is a normal family on Λ. Otherwise, we say that the pair (f, a) is unstable. Recall also that f is isotrivial if there exists a branched cover X → Λ and an algebraic family of Möbius transformations M : X × P 1 → P 1 so that M λ • f λ • M -1 λ : P 1 → P 1 is independent of the parameter λ and that the pair (f, a) is isotrivial if, in addition, M λ (a(λ)) is also independent of the parameter λ. A result of DeMarco [De] states that any stable algebraic pair is either isotrivial or preperiodic, i.e. there exists n > m ≥ 0 such that f n λ (a(λ)) = f m λ (a(λ)) for all λ ∈ Λ.

When a dynamical pair (f, a) is unstable, the stability locus Stab(f, a) is the set of points λ 0 ∈ Λ admitting a neighborhood U on which the pair (f, a) the sequence {λ → f n λ (a(λ))} n≥1 is a normal family. The bifurcation locus Bif(f, a) of the pair (f, a) is its complement Bif(f, a) := Λ \ Stab(f, a). If a is the marking of a critical point, i.e. f λ (a(λ)) = 0 for all λ ∈ Λ, it is classical that the bifurcation locus Bif(f, a) has empty interior, [MSS].

The bifurcation locus of a pair (f, a) is the support of natural a positive (finite) measure: the bifurcation measure µ f,a of the pair (f, a), see Section 1 for a precise definition. The properties of this measure appear to be very important for studying arithmetic and dynamical properties of the pair (f, a), see e.g. [START_REF] Baker | Preperiodic points and unlikely intersections[END_REF][START_REF] Baker | Special curves and postcritically finite polynomials[END_REF][START_REF] Demarco | Bifurcations, intersections, and heights[END_REF][START_REF] Demarco | Variation of canonical height and equidistribution[END_REF][START_REF] De | Bifurcation measures and quadratic rational maps[END_REF][START_REF] Favre | Distribution of postcritically finite polynomials[END_REF][START_REF] Favre | Classification of special curves in the space of cubic polynomials[END_REF][START_REF] Favre | Continuity of the green function in meromorphic families of polynomials[END_REF]. Note also that the entropy theory of dynamical pairs has been recently developed in [DGV]. In the present article, we study algebraic dynamical pairs having an absolutely continuous bifurcation measure.

Assume that for some parameter λ 0 ∈ Λ, the marked point a eventually lands on a repelling periodic point x, that is f n λ 0 (a(λ 0 )) = x. Let x(λ) is the (local) natural continuation of x as a periodic point of f λ . We say that a is transversely prerepelling at λ 0 if the graphs of λ → f n λ (a(λ)) and λ → x(λ) are transverse at λ 0 . Finally, recall that a rational map f : P 1 → P 1 is a Lattès map if there exists an elliptic curve E, an endomorphism L : E → E and a finite branched cover p : E → P 1 such that p • L = f • p on E. Such a map has an absolutely continuous maximal entropy measure, see [Z]. On the other hand, when f is a family of Lattès maps and the pair (f, a) is unstable, then Bif(f, a) = Λ, see e.g. [START_REF] Demarco | Variation of canonical height and equidistribution[END_REF]§6].

Our main result is the following.

Theorem A. -Let (f, a) be a dynamical pair of P 1 of degree d ≥ 2 parametrized by an irreducible quasi-projective curve Λ. Assume that f is non-isotrivial and that (f, a) is unstable. The following assertions are equivalent:

1. The bifurcation locus of the dynamical pair (f, a) is Bif(f, a) = Λ, 2. Transversely prerepelling parameters are dense in Λ, 3. The bifurcation measure µ f,a of the pair (f, a) is absolutely continuous, 4. The family f is a family of Lattès maps.

Note that the hypothesis that f is not isotrivial is necessary to have the equivalence between 1. and 4. (see Proposition 4.3).

The first step of the proof consists in proving that transversely prerepelling parameters are dense in the support of µ f,a . Using properties of Polynomial-Like Maps in higher dimension and a transversality Theorem of Dujardin for laminar currents [Duj], under a mild assumption on Lyapunov exponents, we prove this property holds for the appropriate bifurcation current for any tuple (f, a 1 , . . . , a m ), where f : Λ×P k → P k is any holomorphic family of endomorphisms of P k and a 1 , . . . , a m : Λ → P k are any marked points.

In a second time, we adapt the similarity argument of Tan Lei to show that, if λ 0 is a transversely prerepelling parameter where the bifurcation measure is absolutely continuous, the maximal entropy measure µ f λ 0 of f λ 0 is also non-singular with respect to the Fubini-Study form on P 1 . As Zdunik [Z] has shown, this implies f λ 0 is a Lattès map.

We can see Theorem A as a partial parametric counterpart of Zdunik's result. However, the comparison with Zdunik's work ends there: Rational maps with P 1 as a Julia sets are, in general, not Lattès maps. Indeed, Lattès maps form a strict subvariety of the space of all degree d rational maps, and maps with J f = P 1 form a set of positive volume by [R]. In a way, Theorem A is a stronger rigidity statement that the dynamical one.

Recall that an endomorphism of P k of degree d ≥ 2 has a unique maximal entropy measure µ f which Lyapunov exponents χ 1 , . . . , χ k all satisfy χ j ≥ 1 2 log d, by [START_REF] Briend | Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CP k[END_REF]. We say that the Lyapunov exponents of f resonate if there exists 1 ≤ i < j ≤ k and an integer q ≥ 2 such that χ i = qχ j .

Recall also that, as in dimension 1, an endomorphism f of P k is a Lattès map if there exists an abelian variety A, a finite branched cover p : A → P k and an isogeny Berteloot and Dupont [BD3] generalized Zdunik's work to endomorphisms of P k : f is a Lattès map of P k if and only if the measure µ f is is not singular with respect to ω k P k , see also [Dup]. As an important part of our arguments applies in any dimension, we have the following.

I : A → A such that p • I = f • p on A.
Theorem B. -Fix integers d ≥ 2 and k ≥ 1 and let (f, a) be any holomorphic dynamical pair of degree d of P k parametrized by a Kähler manifold (M, ω) of dimension k. Assume that for all λ ∈ B, any J-repelling periodic point of f λ is linearizable and that the Lyapunov exponents of f λ do not resonate for all λ ∈ M . Assume in addition that µ f,a :

= T k f,a satisifies supp(µ f,a ) = M .
Then µ f,a is absolutely continuous with respect to ω k if and only if f is a family of Lattès maps of P k .

The paper is organised as follows. In section 1, we recall the construction of the biurcation currents of marked points and properties of Polynomial-Like Maps. Section 2 is dedicated to proving the density of transversely prerepelling parameters. In section 3, we establish the similarity proprety for the bifurcation and maximal entropy measures. Finally, in section 4 we prove Theorem A and B and list related questions.

Acknowledgements. -I would like to thank Charles Favre and Gabriel Vigny whose interesting discussions, remarks and questions led to an important part of this work. This research is partially supported by the ANR grant Fatou ANR-17-CE40-0002-01.

Dynamical preliminaries

The bifurcation current of a dynamical tuple

For this section, we follow the presentation of [DF, Duj]. Even though everything is presented in the case k = 1 and for marked critical points, the exact same arguments give what we present below.

Let Λ be a complex manifold and let f : Λ × P k → P k be a holomorphic family of endomorphisms of P k of algebraic degree d ≥ 2: f is holomorphic and f λ := f (λ, •) : P k → P k is an endomorphism of algebraic degree d.

Definition 1.1. -Fix integers m ≥ 1, d ≥ 2 and let Λ be a complex manifold. A dynamical (m + 1)-tuple (f, a 1 , . . . , a m ) of P k of degree d parametrized by Λ is a holomorphic family f of endomorphisms of P k of degree d parametrized by Λ, endowed with m holomorphic maps (marked points) a 1 , . . . , a m : Λ → P k .

Let ω P k be the standard Fubini-Study form on P k and π Λ : Λ × P k → Λ and π P k : Λ × P k → P k be the canonical projections. Finally, let ω := (π P k ) * ω P k . A family f : Λ × P k → P k naturally induces a fibered dynamical system f : Λ × P k → Λ × P k , given by f (λ, z) := (λ, f λ (z)). It is known that the sequence d -n ( f n ) * ω converges to a closed positive (1, 1)-current T on Λ×P k with continuous potential. Moreover, for any 1 ≤ j ≤ k,

f * T j = d j • T j and T k | {λ 0 }×P 1 = µ λ 0 is the unique measure of maximal entropy k log d of f λ 0 for all λ 0 ∈ Λ.
For any n ≥ 1, we have

T = d -n ( f n ) * ω + d -n dd c u n
, where ( u n ) n is a locally uniformly bounded sequence of continuous functions.

Pick now a dynamical (m + 1)-tuple (f, a 1 , . . . , a m ) of degree d of P k . Let Γ a j be the graph of the map a j and set a := (a 1 , . . . , a m ).

Definition 1.2. -For 1 ≤ i ≤ m, the bifurcation current T f,a i of the pair (f, a i ) is the closed positive (1, 1)-current on Λ defined by

T f,a i := (π Λ ) * T ∧ [Γ a j ]
and we define the bifurcation current T f,a of the (m + 1)-tuple (f, a 1 , . . . , a m ) as

T f,a := T f,a 1 + • • • + T f,a k .
For any ≥ 0, write

a (λ) := f λ (a 1 (λ)), . . . , f λ (a m (λ)) , λ ∈ Λ.
Let now K Λ be a compact subset of Λ and let Ω be some compact neighborhood of K, then (a ) * (ω P k ) is bounded in mass in Ω by Cd , where C depends on Ω but not on .

Applying verbatim the proof of [DF, Proposition-Definition 3.1], we have the following.

Lemma 1.3. -For any 1 ≤ i ≤ k, the support of T f,a i is the set of parameters λ 0 ∈ Λ such that the sequence {λ → f n λ (a i (λ))} is not a normal family at λ 0 . Moreover, writing a i, (λ) := f λ (a i (λ)), there exists a locally uniformly bounded family (u i, ) of continuous functions on Λ such that

(a i, ) * (ω P q ) = d T f,a i + dd c u i, on Λ.
As a consequence, for all j ≥ 1, we have

(a i, ) * (ω j P k ) = d j T j f,a i + dd c O(d (j-1)
) on compact subsets of Λ. In particular, one sees that

T k+1 f,a i = 0 on Λ. (1)
Let us still denote π Λ : Λ × (P k ) m → Λ be the projection onto the first coordinate and for 1 ≤ i ≤ k, let π i : Λ × (P k ) m → P q be the projection onto the i-th factor of the product (P k ) m . Finally, we denote by Γ a the graph of a:

Γ a := {(λ, z 1 , . . . , z m ), ∀j, z j = a j (λ)} ⊂ Λ × (P k ) m .
Following verbatim the proof of [START_REF] Astorg | Eckmann and the bifurcation measure[END_REF]Lemma 2.6], we get

1 (mk)! T mk f,a = m =1 T k f,a = (π Λ ) * m i=1 π * i T k ∧ [Γ a ] .
1.2. Hyperbolic sets supporting a PLB ergodic measure Definition 1.4. -Let W ⊂ C k be a bounded open set. We say that a positive measure ν compactly supported on W is PLB if the psh functions on W are integrable with respect to ν.

We aim here at proving the following lemma in the spirit of [START_REF] Dujardin | The supports of higher bifurcation currents[END_REF]Lemma 4.1]:

Proposition 1.5. -Pick an endomorphism f : P k → P k of degree d ≥ 2 which Lyapunov exponents don't resonate. There exists an integer m ≥ 1 a f m -invariant compact set K contained in a small ball B ⊂ P k and an integer N ≥ 2 such that f m | K is uniformly expanding and repelling periodic points of f m are dense in K,

there exists a unique probability measure ν supported on K such that (f m | K ) * ν = N ν which is PLB.

To prove Proposition 1.5, we need the notion of polynomial-like map. We refer to [DS]. Given an complex manifold M and an open set V ⊂ M , we say that V is S-convex if there exists a continuous strictly plurisubharmonic function on V . In fact, this implies that there exists a smooth strictly psh function ψ, whence there exists a Kähler form ω := dd c ψ on V .

Definition 1.6. -Given a connected S-convex open set and a relatively compact open set U V , a map f : U → V is polynomial-like if f is holomorphic and proper.

The filled-Julia set of f is the set

K f := n≥0 f -n (U ).
The set K f is full, compact, non-empty and it is the largest totally invariant compact subset of V , i.e. such that f

-1 (K f ) = K f .
The topological degree d t of f is the number of preimages of any z ∈ V by f , counted with multiplicity. Let k := dim V . We define

d * k-1 := sup ϕ d t • lim sup n→∞ Λ n dd c ϕ 1/n U ; ϕ is psh on V ,
where Λ := d -1 t f * . According to Theorem 3.2.1 and Theorem 3.9.5 of [DS], we have the following.

Theorem 1.7 (Dinh-Sibony). -Let f : U → V be a polynomial-like map of topological degree d t ≥ 2. There exists a unique probability measure µ supported by ∂K f which is ergodic and such that

1. for any volume form Ω of mass 1 in L 2 (V ), one has d -n t (f n ) * Ω → µ as n → ∞, 2. if d * k-1 < d t ,
≥ n 0 , f n has M (n) ≥ Cd nk inverse branches g 1 , . . . , g M (n) defined on B with -g i (B) B and g i is uniformly contracting on B for all i, -g i (B) ∩ g j (B) = ∅ for all i = j.
Fix m ≥ n 0 large enough so that Cd mk > d (k-1)m ≥ 2 and set

V := B, U := M (m) j=1 g j (B), N := M (m) and g := f m | U .
The map g : U → V is polynomial-like of topological degree N , whence its equilibrium measure ν is the unique probability measure which satisfies g * ν = N ν by the first part of Theorem 1.7. We let K := supp(ν). Since the g i 's are uniformly contracting, the compact set K is f m -hyperbolic.

To conclude, it is sufficient to verify that N > d * k-1 . Fix n ≥ 1 and ϕ psh on V . Let ω be the (normalized) restriction to V of the Fubini-Study form of P k . Then

Λ n (dd c ϕ) U = U (Λ n (dd c ϕ)) ∧ ω k-1 = U 1 N n ((g n ) * (dd c ϕ)) ∧ ω k-1 = 1 N n U dd c ϕ ∧ (g n ) * ω k-1 = 1 N n U dd c ϕ ∧ (d mn ω + dd c u nm ) k-1
where (u n ) n is a uniformly bounded sequence of continuous functions on P k . In particular, by the Chern-Levine-Niremberg inequality, if U W V , there exists a constant C > 0 depending only on W such that

Λ n (dd c ϕ) U = d (k-1)m N n U dd c ϕ ∧ (ω + d -nm dd c u nm ) k-1 ≤ d (k-1)m N n C dd c ϕ W .
Taking the n-th root and passing to the limit, we get

d * k-1 N ≤ d (k-1)m N < 1
by assumption. The second part of Theorem 1.7 allows us to conclude.

The support of bifurcation currents

Pick a complex manifold Λ and let m, k ≥ 1 be so that dim Λ ≥ km. Let (f, a 1 , . . . , a m ) be a dynamical (m + 1)-tuple of P k of degree d parametrized by Λ.

Definition 2.1. -We say that a 1 , . . . , a m are transversely J-prerepelling (resp. properly J-prerepelling) at a parameter λ 0 if there exists integers n 1 , . . . , n m ≥ 1 such that f

n j λ 0 (a j (λ 0 )) = z j is a repelling periodic point of f λ 0 and, if z j (λ) is the natural continua- tion of z j as a repelling periodic point of f λ in a neighborhood U of λ 0 , such that 1. z j (λ) ∈ J λ for all λ ∈ U and all 1 ≤ j ≤ m, 2. the graphs of A : λ → (f q 1 λ (a 1 (λ)), . . . , f qm λ (a m (λ))) and of Z : λ → (z 1 (λ), . . . , z m (λ)) intersect transversely (resp. properly) at λ 0 .
Recall that Lyapunov exponents of an endomorphism f resonate if there exists 1 ≤ i < j ≤ k and an integer q ≥ 2 such that χ i = qχ j , see 1.2. In this section, we prove Theorem 2.2. -Let (f, a 1 , . . . , a m ) be a dynamical (m + 1)-tuple of P k of degree d parametrized by Λ with km ≤ dim Λ. Assume that the Lyapunov exponent of f λ don't resonate for all λ ∈ Λ.

Then the support of

T k f,a 1 ∧ • • • ∧ T k f,am
coincides with the closure of the set of parameters at which a 1 , . . . , a m are transversely J-prerepelling.

Remark. -The hypothesis on the Lyapunov exponents is used only to prove the density of transversely prerepelling parameters and we think it is only a technical artefact.

Properly prerepelling marked points bifurcate

In a first time, we give a quick proof of the fact that properly J-prerepelling parameters belong to the support of

T k f,a 1 ∧ • • • ∧ T k f,am
, without any additional assumption.

Theorem 2.3. -Let (f, a 1 , . . . , a m ) be a dynamical (m + 1)-tuple of P k of degree d parametrized by Λ with km ≤ dim Λ. Pick any parameter λ 0 ∈ Λ such that a 1 , . . . , a m are properly J-prerepelling at λ 0 . Then

λ 0 ∈ supp T k f,a 1 ∧ • • • ∧ T k f,am .
The proof of this result is an adaptation of the strategy of Buff and Epstein [BE] and the strategy of Berteloot, Bianchi and Dupont [BBD], see also [G, AGMV]. Since it follows closely that of [AGMV, Theorem B], we shorten some parts of the proof.

Before giving the proof of Theorem 2.3, remark that our properness assumption is equivalent to saying that the local hypersurfaces

X j := {λ ∈ Λ ; f q j λ (a j (λ)) = z j (λ)} intersecting at λ 0 satisfy codim j X j = km.
Proof of Theorem 2.3. -According to [START_REF] Gauthier | Strong bifurcation loci of full Hausdorff dimension[END_REF]Lemma 6.3], we can reduce to the case when Λ is an open set of C km . Take a small ball B centered at λ 0 in Λ. Up to reducing B, we can assume z j (λ) can be followed as a repelling periodic point of f λ for all λ ∈ B. Up to reducing B, our assumption is equivalent to the fact that j X j = {λ 0 }.

We let µ := T k f,a 1 ∧ • • • ∧ T k f,am .
Our aim here is to exhibit a basis of neighborhood {Ω n } n of λ 0 in B with µ(Ω n ) > 0 for all n. For any m-tuple n := (n 1 , . . . , n m ) ∈ (N * ) m , we let

F n : Λ × (P k ) m -→ Λ × (P k ) m (λ, z 1 , . . . , z m ) -→ (λ, f n 1 λ (z 1 ), . . . , f nm λ (z m ))
. For a m-tuple n = (n 1 , . . . , n m ) of positie integers, we set

|n| := n 1 + • • • + n m .
We also denote

A n (λ) := f n 1 λ (a 1 (λ)), . . . , f nm λ (a m (λ)) , λ ∈ Λ.
As in [AGMV], we have the following.

Lemma 2.4. -For any m-tuple n = (n 1 , . . . , n m ) of positive integers, we let Γ n be the graph in Λ × (P k ) m of A n . Then, for any Borel set B ⊂ Λ, we have

µ(B) = d -k•|n| B×(P k ) m   m j=1 (π j ) * T k   ∧ Γ n .
Suppose that the point z j is r j -periodic. For the sake of simplicity, we let in the sequel A n := A q+nr , where q = (q 1 , . . . , q m ), r = (r 1 , . . . , r m ) are given as above and q + nr = (q 1 + nr 1 , . . . , q m + nr m ). Again as above, we let Γ n be the graph of A n .

Let z := (z 1 , . . . , z m ) and fix any small open neighborhood Ω of λ 0 in Λ. Set

I n := Ω×(P k ) m   m j=1 (π j ) * T k   ∧ [Γ n ] ,
and let δ be given by the expansion condition as above. Let S n be the connected component of Γ n ∩ Λ × B m δ (z) containing (λ 0 , z). Since z j (λ) is repelling and periodic for f λ for all λ ∈ B (if B has been choosen small enough), there exists a constant K > 1 such that

d P k (f r j λ (z), f r j λ (w)) ≥ K • d P k (z, w)
for all (z, w) ∈ B(z j (λ 0 ), ) and all λ ∈ B for some given > 0. In particular, the current [S n ] is vertical-like in in Λ × B m δ (z) and there exists n 0 ≥ 1 and a basis of neighborhood

Ω n of λ 0 in Λ such that supp([S n ]) = S n ⊂ Ω n × B m δ (z), for all n ≥ n 0 .
Let S be any weak limit of the sequence

[S n ]/ [S n ] . Then S is a closed positive (mk, mk)-current of mass 1 in B × B m δ (z) with supp(S) ⊂ {λ 0 } × B m δ (z). Hence S = M • [{λ 0 } × B m δ (z)]
, where M -1 > 0 is the volume of B m δ (z) for the volume form j (ω k j ), where ω j = (p j ) * ω P k and p j : (P k ) m → P k is the projection on the j-th coordinate.

As a consequence, [S n ]/ [S n ] converges weakly to S as n → ∞ and, since the (mk, mk)current m j=1 (π j ) * ( T k ) is the wedge product of (1, 1)-currents with continuous potentials, we have

m j=1 (π j ) * T k ∧ [S n ] [S n ] -→ m j=1 (π j ) * T k ∧ S as n → +∞. Whence lim inf n→∞ [S n ] -1 • I n ≥ lim inf n→∞ m j=1 (π j ) * T k ∧ [S n ] [S n ] ≥ m j=1 (π j ) * T k ∧ S
By the above, this gives

lim inf k→∞ [S n ] -1 • I n ≥ M • [{λ 0 } × B m δ (z)] ∧ m j=1 (π j ) * T k ,
In particular, there exists n 2 ≥ n 1 such that for all n ≥ n 2 ,

[S n ] -1 • I n ≥ M 2 • [{λ 0 } × B m δ (z)] ∧ m j=1 (π j ) * T k .
Finally, since [S n ] is a vertical current, up to reducing δ > 0, Fubini Theorem gives

lim inf n→∞ [S n ] ≥ m j=1 B(z j ,δ) ω k FS ≥ c • δ 2k m > 0 .
Up to increasing n 0 , we may assume [S n ] ≥ (cδ 2k ) m /2 for all n ≥ n 0 . Letting α = M (cδ 2k ) m /4 > 0, we find

Ω×(P k ) m   m j=1 (π j ) * T k   ∧ [Γ n ] ≥ α [{λ 0 } × B m δ (z)] ∧ m j=1 (π j ) * T k .
To conclude the proof of Theorem 2.3, we rely on the following purely dynamical result, which is an immediate adaptation of [START_REF] Astorg | Eckmann and the bifurcation measure[END_REF]Lemma 3.5].

Lemma 2.5. -For any δ > 0, and any x = (x 1 , . . . , x m ) ∈ (supp(µ λ 0 )) m , we have

[{λ 0 } × B m δ (x)] ∧ m j=1 (π j ) * T k = m j=1 µ λ 0 (B(x j , δ)) > 0.
We can now conclude the proof of Theorem 2.3. Pick any open neghborhood Ω of λ 0 in Λ. By the above and Lemma 2.5, we have an integer n 0 ≥ 1 and constants α, δ > 0 such that for all n ≥ n 0 ,

µ(Ω) ≥ α • d -k(|q|+n|r|) m j=1 µ f (D(z j , δ)) > 0 .
In particular, this yields µ(Ω) > 0. By assumption, this holds for a basis of neighborhoods of λ 0 in Λ, whence we have λ 0 ∈ supp(µ).

Density of transversely prerepelling parameters

To finish the proof fo Theorem 2.2, it is sufficient to prove that any point of the support of

T k f,a 1 ∧ • • • ∧ T k f,am
an be approximated by transversely J-prerepelling parameters. We follow the strategy of the proof of Theorem 0.1 of [Duj] to establish this approximation property. Precisely, we prove here the following.

Theorem 2.6. -Let (f, a 1 , . . . , a m ) be a dynamical (m + 1)-tuple of P k of degree d parametrized by Λ with km ≤ dim Λ. Assume that the Lyapunov exponent of f λ don't resonate for all λ ∈ Λ.

Then, any parameter λ ∈ Λ lying in the support of the current T k f,a 1 ∧ • • • ∧ T k f,am can be approximated by parameters at which a 1 , . . . , a m are transversely J-prerepelling.

We rely on the following property of PLB measures (see [DS]):

µ := T k f,a 1 ∧ • • • ∧ T k f,am
and pick λ 0 ∈ supp(Ω). According to Proposition 1.5, there exists an integer m ≥ 1 and a f m λ 0 -compact set K ⊂ P k contained in a ball and N ≥ 2 such that f m λ 0 | K is uniformly hyperbolic and repelling periodic points of f m λ 0 are dense in K, there exisys a unique probailibty measure ν supported on K such that (f m λ 0 | K ) * ν = N ν which is PLB. Since K is hyperbolic, there exists > 0 and a unique holomorphic motion h : B(λ 0 , ) × K → P k which conjugates the dynamics, i.e. h is continuous and such that for all λ ∈ B(λ 0 , ), the map h λ := h(λ, •) : K → P k is injective and h λ 0 = id K , for all z ∈ K, the map λ ∈ B(λ 0 , ) → h λ (z) ∈ P k is holomorphic, and [START_REF] De | One-dimensional dynamics[END_REF]Theorem 2.3 p. 255]. For all z := (z 1 , . . . , z m ) ∈ K m , we denote by Γ z the graph of the holomorphic map λ → (h λ (z 1 ), . . . , h λ (z m )).

-for all (λ, z) ∈ B(λ 0 , ) × K, we have h λ • f m λ 0 (z) = f m λ • h λ (z), see e.g.
We define a closed positive (km, km)-current on B(λ 0 , ) × (P k ) m by letting

ν := K m [Γ z ]dν ⊗m (z), where Γ z = {(λ, h λ (z 1 ), . . . , h λ (z m )) ; λ ∈ B(λ 0 , )} for all z = (z 1 , . . . , z m ) ∈ K m .
Claim. -There exists a (km -1, km -1)-current V on B(λ 0 , ) × (P k ) m which is locally bounded and such that ν = dd c V .

Recall that we have set a n (λ) := (f n λ (a 1 (λ)), . . . , f n λ (a m (λ))). We define a * n ν by a * n ν := (π 1 ) * (ν ∧ [Γ an ]) , where π 1 : B(t 0 , ) × (P k ) m → B(t 0 , ) is the canonical projection onto the first coordinate. According to the claim, locally we have ν = dd c V , for some bounded (km -1, km -1)current V . In particular, we get a * n ν = a * n (dd c V ), as expected. Let ω be the Fubini-Study form of P k and Ω :

= (π 2 ) * (ω k ⊗ • • • ⊗ ω k ), where π 2 : B(λ 0 , ) × (P k ) m → (P k ) m is the canonical projection onto the second coordinate. Then ν -Ω = dd c V where V is bounded on B(λ 0 , ) × (P k ) m , hence d -kmn a * n (ν) -d -kmn a * n ( Ω) = d -kmn a * n (dd c V ).
On the other hand, we have 1

d km f * ( Ω) = Ω + dd c W , where W is bounded on B(λ 0 , ) × (P k ) m , hence 1 d kmn ( f n ) * ( Ω) = ω + dd c W n , where W n -W n+1 = O(d -n ).
In particular,

1 d n ( f n ) * ( Ω)∧[Γ a ] = d -n Ω ∧ [Γ an ] +dd c O(d -n ), hence µ = lim n d -kmn a * n ( Ω). This yields lim n→∞ d -nkm (π 1 ) * (ν ∧ [Γ an ]) = µ.
We now use [START_REF] Dujardin | The supports of higher bifurcation currents[END_REF]Theorem 3.1]: as (2km, 2km)-currents on B(λ 0 , )

× (P k ) m , ν ∧ [Γ pn ] = K m [Γ z ] ∧ [Γ pn ]dν ⊗m (z)
and only the geometrically transverse intersections are taken into account. In particular, this means there exists a sequence of parameters λ n → λ 0 and z n ∈ K m such that the graph of a n and Γ zn intersect transversely at λ n . Now, since repelling periodic points of f m λ 0 are dense in K, there exists z n,j → z n as j → ∞, where z j,n ∈ K m and (f m λ 0 , . . . , f m λ 0 )periodic repelling. Since z j,n (λ) := (h λ , . . . , h λ )(z j,n ) remains in (h λ , . . . , h λ )(K m ) and remains periodic, it remains repelling for all λ ∈ B(λ 0 , ). By persistence of transverse intersections, for j large enough, there exists λ j,n where Γ an and Γ z j,n intersect transversely and λ j,n → λ n as j → ∞ and the proof is complete.

To finish this section, we prove the Claim.

Proof of the Claim. -Since the compact set K is contained in a ball, we can choose an affine chart C k such that K C k and, up to reducing > 0, we can assume K λ = h λ (K) C k for all λ ∈ B(λ 0 , ). Let (x 1 1 , . . . , x 1 k , . . . , x m 1 , . . . , x m k ) = (x 1 , . . . , x m ) be the coordinates of (C k ) m and let h λ,i be the i-th coordinate of the function h λ .

For all 1 ≤ i ≤ k and 1 ≤ j ≤ m, we define a psh function Ψ j i on B(λ 0 , ) × (C k ) m by letting

Ψ j i (t, w) := K m log |w j i -h t,i (z j )|dν ⊗m (z).
According to Lemma 2.7 and Proposition 1.5, we have

Ψ j i ∈ L ∞ loc B(λ 0 , ) × (C k ) m .
Moreover, according to [START_REF] Dujardin | The supports of higher bifurcation currents[END_REF]Theorem 3.1], we have

ν = i,j dd c Ψ j i = dd c   Ψ 1 1 • i,j>1 dd c Ψ j i   .
Since the functions Ψ j i are locally bounded, this ends the proof.

Local properties of bifurcation measures

A renormalization procedure

Pick k, m ≥ 1 and let B(0, ) be the open ball centered at 0 of radius in C km and let (f, a 1 , . . . , a m ) be a dynamical (m + 1)-tuple of degree d of P k parametrized by B(0, ). Assume there exists m holomorphically moving J-repelling periodic points z 1 , . . . , z m : B(0, ) → P k of respesctive period q j ≥ 1 with f n j 0 (a j (0)) = z j (0). We also assume that (a 1 , . . . , a m ) are transversely prerepelling at 0 and that z j (λ) is linearizable for all λ ∈ B(0, ) for all j. Let q := lcm(q 1 , . . . , q m ) and

Λ λ := (D z 1 (λ) (f q λ ), . . . , D zm(λ) (f q λ )) : m j=1 T z j (λ) P k -→ m j=1 T z j (λ) P k
and denote by φ λ = (φ λ,1 , . . . , φ λ,m ) : (C k , 0) → ((P k ) m , (z 1 (λ), . . . , z m (λ))), where φ λ,j is the linearizing coordinate of f q λ at z j (λ). Denote by π j : (P k ) m → P k the projection onto the j-th factor. Up to reducing > 0, we can also assume there exists r j > 0 independent of λ such that

f q λ • φ λ,j (z) = φ λ,j • D z j (λ) (f q j
λ )(z), z ∈ B(0, r j ), and D 0 φ λ,j : C k → T z j (λ) P k is an invertible linear map. Up to reducing again , we can also assume f n j λ (a j (λ)) always lies in the range of φ λ,j for all 1 ≤ j ≤ m. Recall that we denoted a n (λ) = (f n 1 λ (a 1 (λ)), . . . , f nm λ (a m (λ))), where n = (n 1 , . . . , n m ) and let

h(λ) := φ -1 λ • a n (t) = φ -1 λ,1 f n 1 λ (a 1 (λ)) , . . . , φ -1 λ,m f nm λ (a m (λ)) , λ ∈ B(0, ). Lemma 3.1. -The map h : B(0, ) → (C km , 0) is a local biholomorphism at 0. Proof. -Recall that f n j 0 (a j (0)) = z j (0). Write h = (h 1 , . . . , h m ) with h j : B(0, ) → (C k , 0) and let b j (λ) := f n j
λ (a j (λ)) for all λ ∈ B(0, ) so that b j (λ) = φ λ,j • h j (λ) for all λ ∈ B(0, ). Since φ λ,j (0) = z j (λ), differentiating and evaluating at λ = 0, we find

D 0 b j = D 0 z j + D 0 φ 0,j • D 0 h j .

Now our tranversality assumption implies that

L := ((D 0 b 1 -D 0 z 1 ), . . . , (D 0 b m -D 0 z m )) : C km → m j=1
T z j (0) P k is invertible. As a consequence, the linear map

D 0 h = (D 0 h 1 , . . . , D 0 h m ) = -(D 0 φ 0 ) -1 • L : C km → C km is invetible, ending the proof.
Up to reducing again , we assume h is a biholomorphism onto its image and let r :

= h -1 : h(B(0, )) → B(0, ). Fix δ 1 , . . . , δ m > 0 so that B C k (0, δ 1 ) × • • • × B C k (0, δ m ) ⊂ h(B(0, )). Finally, let Ω := B C k (0, δ 1 ) × • • • × B C k (0, δ m ) and, for any n ≥ 1, let r n (x) := r • Λ -n 0 (x), x ∈ B C k (0, δ 1 ) × • • • × B C k (0, δ m ).
The main goal of this paragraph is the following.

Proposition 3.2. -In the weak sense of measures on Ω, we have

m j=1 d k(n j +nq) • (r n ) * T k f,a 1 ∧ • • • ∧ T k f,am -→ n→∞ (φ 0 ) *   m j=1 (π j ) * µ f 0   .
To simplify notations, we let

a (n) := a n+nq , with n + nq = (n 1 + nq, . . . , n m + nq). Lemma 3.3. -The sequence (a (n) • r n ) n≥1 converges uniformly to φ 0 on Ω.
Proof. -Note first that

a (0) • r(x) = f n 1 r(x) (a 1 (r(x))), . . . , f nm r(x) (a m (r(x))) = φ r(x) (x), x ∈ Ω, by definition of r.
By definition, the sequence (r n ) n≥1 converges uniformly and exponentially fast to 0 on Ω, since we assumed z 1 (0), . . . , z m (0) are repelling periodic points and since r(0) = 0. Moreover, Λ rn → Λ 0 and φ rn(x) → φ 0 exponentially fast. In particular,

lim n→∞ Λ n rn(x) • Λ -n 0 (x) = x
and the convergence is uniform on Ω. Fix x ∈ Ω. Then

a (n) • r n (x) = (f qn rn(x) , . . . , f qn rn(x) ) a (0) • r • Λ -n 0 (x) = (f qn rn(x) , . . . , f qn rn(x) ) • φ rn(x) Λ -n 0 (x) = φ rn(x) Λ n rn(x) • Λ -n 0 (
x) and the conclusion follows.

Proof of Proposition 3.2. -Recall that we can assume there exists a holomorphic family of non-degenerate homogeneous polynomial maps

F λ : C k+1 → C k+1 of degree d such that, if π : C k+1 \ {0} → P k is the canonical projection, then π • F λ = f λ • π on C k+1 \ {0}. For 1 ≤ j ≤ m, let ãj : B(0, ) → C k+1 \ {0} be a lift of a j , i.e. a j = π • ãj . Recall that m j=1 T k a j = m j=1 (dd c G λ (ã j (λ))) k .
For 1 ≤ j ≤ m, pick a open set U j ⊂ P k such that φ 0,j (B C k (0, δ j )) U j and such that there exists a section σ j :

U j → C k+1 \ {0} of π on U j . Let U := U 1 × • • • × U k and σ := (σ 1 , . . . , σ k ) : U → (C k+1 \ {0}) m so that φ 0 (Ω) U . According to Lemma 3.3, there exists n 0 ≥ 1 such that a (n) • r n (Ω) U.
In other words, for any x ∈ Ω, any 1 ≤ j ≤ m and any n ≥ n 0 , a n,j (x) := f

n j +nq rn(x) (a j • r n (x)) ∈ U j . Moreover, for all x ∈ Ω, we have π • F n j +nq rn(x) (ã j • r n (x)) = f n j +nq rn(x) • π(ã j • r n (x)) = f n j +nq rn(x) (a j • r n (x)) = π • σ j a n,j (x) .
In particular, there exists a holomorphic function u n,j : Ω → C * such that

F n j +nq rn(x) (ã j • r n (x)) = u n,j (x) • σ j • a n,j (x) and d nq+n j G rn(x) (ã j • r n (x)) = G rn(x) F n j +nq rn(x) (ã j • r n (x)) = G rn(x) σ j • a n,j (x) + log |u n,j (x)|,
for all x ∈ Ω. Since log |u n,j | is pluriharmonic on Ω, the above gives

d nq+n j (r n ) * T f,a j = dd c G rn(x) σ j • a n,j (x) , so that µ n := m j=1 d k(n j +nq) • (r n ) * T k f,a 1 ∧ • • • ∧ T k f,am = m j=1 dd c G rn(x) σ j • a n,j (x) k .
Using again Lemma 3.3 gives

µ n -→ n→∞ m j=1 (dd c G 0 (σ j • φ 0,j (x))) k = m j=1 (φ 0,j ) * µ f 0 .
This ends the proof since φ 0,j = π j • φ 0 by definition of φ 0 .

Families with an absolutely continuous bifurcation measure

Fix integers k, m ≥ 1 and d ≥ 2. The following is a consequence of the above renormalization process.

Proposition 3.4. -Let (f, a 1 , . . . , a m ) be a dynamical (m + 1)-tuple of degree d of P k parametrized by the unit Ball B of C km . Assume that a 1 , . . . , a m are transeversely Jprerepelling at 0 to a J-repelling cycle of f 0 which moves holmorphically in B as a Jrepelling cycle of f λ which is linearizable for all λ ∈ B. Assume in addition that the measure µ :

= T k f,a 1 ∧ • • • ∧ T k f,am
is absolutely continuous with respect to the Lebesgue measure on B.

Then the measure µ f 0 is non-singular with respect to ω k P k .

Proof.

-By assumption, we can write µ = h • Leb where h :

B → R + is a measurable function. Let Ω := B C k (0, δ 1 ) × • • • × B C k (0, δ m ),
r n and φ 0 be given as in Section 3.1. We can apply Proposition 3.2:

m j=1 d k(n j +nq) h • r n • (r n ) * Leb = m j=1 d k(n j +nq) • (r n ) * µ -→ n→∞ (φ 0 ) *   m j=1 (π j ) * µ f 0   . Since φ 0 (0) = (z 1 (0), . . . , z m (0)) ∈ (supp(µ f 0 )) k , the measure (φ 0 ) *   m j=1 (π j ) * µ f 0  
has (finite) strictly positive mass in Ω. In particular, the measure

d knqm • (r n ) * (h • Leb) = d knqm • (h • r n ) • Λ -n 0 * (r * Leb)
converges to a non-zero finite mass positive measure on Ω. As r is a local holomorphic diffeomorphism, there exists a neighborhod of 0 in B such that we have r * Leb = v • Leb for some smooth function v > 0. Whence

d knqm • (r n ) * Leb = d knqm • (h • r n ) • (v • Λ -n 0 ) Λ -n 0 * (Leb) .
By the change of variable formula and Fubini,

Λ -n 0 * (Leb) = m j=1 | det D z j (0) (f q 0 )| -2nk • Leb.
For all n, define a measurable function α n : B → R + by letting

α n (x) := d knqm m j=1 | det D z j (0) (f q 0 )| -2nk • (h • r n (x)) • (v • Λ -n 0 (x)) ∈ R + .
By assumption, the measure α n • Leb converges weakly on Ω to a non-zero finite positive measure, whence α n → α ∞ , as n → ∞, where α ∞ : Ω → R + is not identically zero. As a consequence,

(φ 0 ) *   m j=1 (π j ) * µ f 0   = α ∞ • Leb.
Using again Fubini, on Ω, we find

(φ 0 ) *   m j=1 (π j ) * µ f 0   = α ∞ • Leb C k • • • Leb C k .
Finally, since as positive measures on φ 0 (Ω), we have

m j=1 (π j ) * µ f 0 = µ f 0 • • • µ f 0 ,
the measure µ f 0 is absolutely continuous with respect to Leb in an open set.

We now want to deduce Theorem B from the above, using [Z] when k = 1 and [BD3] when k > 1. In fact, they prove that f is a Lattès map if and only if the sum of its Lyapunov exponents L(f

) = P k log | det(Df )|µ f is equalt to k 2 log d.
We use this characterization to prove Theorem B.

Proof of Theorem B. -Assume firt that µ f,a is absolutely continuous with respect to ω k and let T be the set of parameters λ ∈ M such that a is transversely prerepelling at λ. The set T is dense in M by Theorem 2.2. Applying Proposition 3.4 at all λ ∈ T gives that µ f λ is non-singular with respect to ω k P k for all λ ∈ T . We then apply Zdunik or Berteloot-Dupont Theorem we have proven there exists a countable subset T which is dense in M such that the map f λ is a Lattès map for all λ ∈ T . In particular, L(f λ ) = k 2 log d for all λ ∈ T . As the function λ ∈ M → L(f λ ) is continuous, this implies L(f λ ) = k 2 log d for all λ ∈ M , i.e. f λ is a Lattès map for all λ ∈ M .

To conclude, we assume f is a family of Lattès maps and the pair (f, a) satisfies µ f,a > 0.

Let ω P k be the Fubini-Study form on P k . For all λ ∈ M , there exists a function u λ :

P k → R + such that µ f λ = u λ • ω k P k . Let u(λ, z) := u λ (z)
for all (λ, z) ∈ M × P k . The above can be expressed as

T = u • ωk ,
where ω = π * P k (ω P k ) and π P k : M × P k → P k is the canonical projection. Pick a local chart U ⊂ M and a local chart V ⊂ P k so that a(U ) ⊂ V and ω P k = dd c v on V where v is smooth. In U × V , the above gives

(π Λ ) * T k ∧ [Γ a ] = (π Λ ) * u • (dd c λ,z v(z)) k ∧ [Γ a ] = u(λ, a(λ)) (dd c λ (v • a(λ))) k .
Letting h(λ) := u(λ, a(λ)) and w(λ) := v • a(λ), we find

µ f,a = h • (dd c w) k on U.
Since w is smooth, the conclusion follows.

4. Proof of the main result and concluding remarks 4.1. J-stability and bifurcation of dynamical pairs on P 1

We say that a family f : Λ × P 1 -→ Λ × P 1 of degree d rational maps of P 1 is J-stable if all the repelling cycles can be followed holomorphically throughout the whole family Λ, i.e. if for all n ≥ 1, there exists N ≥ 0 and holomorphic maps z 1 , . . . , z N : Λ → P 1 such that {z 1 (λ), . . . , z N (λ)} is exactly the set of all repelling cycles of f λ of exact period n for all λ ∈ Λ.

Recall that an endomorphism of P 1 has a unique measure of maximal entropy µ f and let L(f ) := P 1 log |f |µ f be the Lyapunov exponents of f with respect to µ f . By a classical result of Mañé, Sad and Sullivan [MSS], it is also locally equivalent to the existence of a unique holomorphic motion of the Julia set which is compatible with the dynamics, i.e. for λ 0 ∈ Λ, there exists h : Λ × J f λ 0 -→ Λ × P 1 such that for any λ ∈ Λ, the map h λ := h(λ, •) : Proof. -Since Bif(f, a) = supp(µ f,a ) = ∅, the set D of parameters λ 0 ∈ D such that a is transversely prerepelling at λ 0 is a non-empty countable dense subset of Bif(f, a). As J-repelling points of f λ 0 are contained in J f λ 0 , this gives Bif(f, a) ⊂ {λ ∈ D ; a(λ) ∈ J f λ }.

J f λ 0 -→ P 1 is a homeomorphism which conjugates f λ 0 to f λ , i.e. h λ • f λ 0 = f λ • h λ on J f λ 0 , -for any z ∈ J f λ 0 , the map λ → h λ (z) is holomorphic on Λ, -h λ 0 is the identity on J f λ 0 .
Pick now λ 0 ∈ {λ ∈ D ; a(λ) ∈ J f λ } and assume λ 0 / ∈ Bif(f, a). Set a n (λ) := f n λ (a(λ)) for all n ≥ 0 and all λ ∈ D. Let h : D × J f 0 → P 1 be the unique holomorphic motion of J f 0 parametrized by D such that, if h λ := h(λ, •), then

h λ • f 0 = f λ • h λ on J f 0 .
Note that for all z ∈ J f 0 , the sequence {λ → h λ (f n 0 (z))} n is a normal family on D. Beware that for all periodic point z ∈ J f 0 of f 0 , the function z(λ) := h λ (z) is a marking of z as a periodic point of f λ . For all s ∈ D, if we let h s λ := h λ • h -1 s . The family (h s λ ) λ is a holomorphic motion of J fs which satisfies

h s λ • f s = f λ • h s λ on J fs , for all λ ∈ D(s, 1 -|s|).
Since we assumed λ 0 / ∈ Bif(f, a), there exists > 0 such that D(λ 0 , ) ∩ Bif(f, a) = ∅ and we can choose an affine chart of P 1 such that a n (λ) and h λ 0 λ (a n (λ 0 )) lie in this chart for all n ≥ 1 and all λ ∈ D(λ 0 , ) . For all n, set s n (λ) := a n (λ) -h λ 0 λ (a n (λ 0 )), λ ∈ D(λ 0 , ). Assume first s m ≡ 0 on D(λ 0 , ) for some m ≥ 0. This implies a m (λ) = h λ (a m (0)) for all λ ∈ D(λ 0 , ). By the Isolated Zero Theorem, we thus have a m (λ) = h λ (a m (0)) for all λ ∈ D.

As h λ • f 0 = f λ • h λ , this yields a n (λ) ≡ h λ (a n (0)) for all n ≥ m, and (a n ) is a normal family on D. This is a contradiction, since we assumed Bif(f, a) = ∅. We thus may assume s m ≡ 0 on D(λ 0 , ). In particular, up to reducing , we may assume s m (λ) = 0 for all λ ∈ D(λ 0 , ) \ {λ 0 }. Let z 0 := a m (λ 0 ). By Rouché Theorem, there exists η > 0 such that for any z ∈ D(z 0 , η) ∩ J f λ 0 , the function s m,z (λ) := a m (λ) -h λ 0 λ (z) has finitely many isolated zeros in D(λ 0 , ). As repelling periodic points are dense in J f λ 0 , there exists z 1 ∈ D(z 0 , η) ∩ J f λ 0 which is f λ 0 -periodic and repelling. The implies there exists λ 1 ∈ D(λ 0 , ) such that a is properly prerepelling at λ 1 . Finally, Theorem 2.3 (or simply Montel Theorem in this case) gives λ 1 ∈ Bif(f, a) ending the proof.

Using Montel theorem, one can deduce the following. Proposition 4.2. -Let (f, a) be a dynamical pair of degree d of P 1 parametrized by a one-dimensional complex manifold Λ. Assume that Bif(f, a) = Λ. Then f is J-stable and if f has a persistent attracting cycle, then it has period at most 2.

Proof. -Assume first f has a persistent attracting cycle of period at p ≥ 3. Pick a topological disk D ⊂ Λ. Then there exists holomorphic functions z 1 , . . . , z p : D → P 1 which paramerize this attracting cycle. In particular, z i (λ) = z j (λ) for all i = j and all λ ∈ D. Since we assumed Bif(f, a) = Λ, the sequence {λ -→ f n λ (a(λ))} n≥1 is not a normal family on D. By Montel Theorem, there exists n ≥ 1, 1 ≤ i ≤ p and λ 0 ∈ D such that

f n λ 0 (a(λ 0 )) = z i (λ 0 ).
By Lemma 4.1, since λ 0 ∈ Bif(f, a) this implies z i (λ 0 ) ∈ J f λ 0 . This is a contradiction with the fact that z i is attracting. If f is not J-stable, by Montel Theorem, there exists a non-empty open set U of Λ such that (f λ ) λ∈U is J-stable with an attracting periodic z 1 , . . . , z p of period p ≥ 3, and we reduce to the previous case (see e.g. [START_REF] Dujardin | Distribution of rational maps with a preperiodic critical point[END_REF]Proposition 2.4]).

Proof of the main result

Proof of Theorem A. -Firs, remark that points 1. and 2. are equivalent by Theorem 2.2. Assume Bif(f, a) = Λ. By Proposition 4.2 the family f is J-stable. By [START_REF] Curtis | The Mandelbrot set is universal[END_REF]Theorem 2.4], since f is not isotrivial, f is a family of Lattès maps. Assume now f is a non-isotrivial family of Lattès and that µ f,a is non-zero. Recall that, since f is a family of Lattès maps, it is stable. We want to prove that supp(µ f,a ) = Λ. Assume it is not the case, then there exists a non-empty open set U ⊂ Λ such that U ⊂ Λ \ supp(µ f,a ). The pair (f, a) being stable in U , a(λ) cannot be a repelling periodic point of f λ for any λ ∈ U . From the uniqueness of the holomorphic motion, it follows that there exists z 0 ∈ P 1 such that a(λ) = h λ (z 0 ) for all λ ∈ U . By analytic contuinuation, this gives a(λ) = h λ (z 0 ) for all λ ∈ Λ. This contradicts the fact that µ f,a is non-zero.

As in the proof of Theorem 4.5, let T be the set of parameters λ ∈ Λ such that a is transversely prerepelling at λ. The set T is dense in supp(µ f,a ) by Theorem 2.2. Applying Proposition 3.4 at all λ ∈ T gives that µ f λ is non-singular with respect to ω P 1 for all λ ∈ T . By [Z], the map f λ is a Lattès map for all λ ∈ T . Consider now the morphism ϕ : λ ∈ Λ → f λ ∈ Rat d and let X ⊂ Rat d be the set of Lattès maps of degree d. X is a strict subvariety of Rat d . By the above, the algebraic curve ϕ(Λ) has an intersection with the subvariety X which admits accumulation points. By the Isolated Zeros Theorem, we have ϕ(Λ) ⊂ X, i.e. f is a family of Lattès maps.

The converse implication follows immediately from Theorem B.

Recall that when f is isotrivial, either J f λ = P 1 for all λ, or J f λ = P 1 for all λ. We conclude this section with the following easy proposition, which clarifies the case when f is isotrivial.

Proposition 4.3. -Let f be an isotrivial algebraic family parametrized by an irreducible quasiprojective curve Λ and let a : Λ → P 1 be such that the pair (f, a) is unstable.

1. If J f λ = P 1 for all λ ∈ Λ, we have Bif(f, a) = Λ. 2. If µ f,a is absolutely continuous, f is an isotrivial family of Lattès maps.

Proof. -Assume first J f λ = P 1 for all λ ∈ Λ. Applying Lemma 4.1 in local charts gives Bif(f, a) = {λ ∈ Λ ; a(λ) ∈ J f λ } and the conclusion follows. When µ f,a is absolutely continuous, the conclusion follows as in the proof of Theorem A.

Concluding remarks and questions

Dynamical pairs with a non-singular bifurcation measure. -First, when k > 1, the statement of Theorem B holds only if the Lyapunov exponents of f don't resonate and if all repelling J-cycles are linearizable.

This results raises several questions:

Question 4.4. -Can we generalize Theorem B to the cases when 1. At some parameter, Lyapunov exponents do resonate? 2. There exists J-repelling cycles that are non-linearizable? 3. T k a is just non-singular with respect to a smooth volume form?

In fact, Zdunik [Z] completely classifies rational maps with a maximal entropy measure which is not singular with respect to a Hausdorff measure H α : either α = 1 and the rational map is conjugated to a monomial map z ±d or to a Chebichev polynomial T d , i.e. the only polynomial satisfying T d z + 1 z = z d + 1 z d for all z ∈ C, or α = 2 and the rational map is a Lattès map.

We expect the following complete parametric counterpart to [Z] to be true:

Question 4.5. -Let (f, a) be any holomorphic dynamical pair of P 1 of degree d ≥ 2 parametrized by the unit disk D of C. Assume that (f, a) is unstable. Assume also there exists α > 0 and a function h : D → R + such that µ f,a = h • H α on D. Ca we prove that either α = 2 and f is a family of Lattès maps, or α = 1, f is isotrivial and all f λ 's are conjugated to z ±d or a Chebichev polynomial?

As in the case of families of Lattès maps, we can expect the proof to generalize to the case when k > 1. This raises the following question.

Question 4.6. -Classify endomorphisms of P k which maximal entropy measure is not singular with respect to some Hausdorff measure H α on P k (and possible values of α).

As seen above, the case α = 2k has been treated by Berteloot and Dupont [BD3]. Of course, there are also easy examples where α = k: take f : P 1 → P 1 which maximal entropy measure has dimension 1, then the endomorphism F : P k -→ P k making the following diagram commute (P 1 ) k (f,...,f ) / / η k (P 1 ) k η k P k F / / P k where η k is te quotient map of the action by permutation of coordinates of the symmetric group S k , satisfies dim(µ F ) = k (see [GHK] for a study of symmetric products).

J-stability and dynamical pairs, when k ≥ 2. -

We say that a family f : Λ × P k -→ P k of degree d ≥ 2 endomorphisms of P k is weakly J-stable if all the Jrepelling cycles can be followed holomorphically throughout the whole family Λ, i.e. if for all n ≥ 1, there exists N ≥ 0 and holomorphic maps z 1 , . . . , z N : Λ → P k such that {z 1 (λ), . . . , z N (λ)} is exactly the set of all repelling J-cycles of f λ of exact period n for all λ ∈ Λ.

For any endomorphism f of P k , let L(f ) := P k log | det Df |µ f be the sum of the Lyapunov exponents of f with respect to its Green measure µ f . By a beautiful result of Berteloot, Bianchi and Dupont [BBD], f is J-stable if and only if λ -→ L(f λ ) is pluriharmonic on Λ.

A natural question is then the following:

Question 4.7. -Given any dynamical pair (f, a) of degree d of P k parametrized by the unit disk D such that f is a weakly J-stable family, do we still have Supp(T k a ) = {λ ∈ D ; a(λ) ∈ J f λ } ?

One of the difficulties is that the weak J-stability is equivalent to the existence of a branched holomorphic motion.

  the measure µ is PLB and repelling periodic points are dense in supp(µ).Proof of Proposition 1.5. -First, we fix an open set O ⊂ P k with µ f (O) > 0. According to[START_REF] Berteloot | Perturbations d'exemples de Latt\'es et dimension de Hausdorff du lieu de bifurcation[END_REF] Proposition 3.2], there exists a ball B ⊂ O, a constant C > 0 depending only on O and n 0 ≥ 1 such that for all n

  Lemma 2.7. -Let ν be PLB with compact support in a bounded open set W ⊂ C k and let ψ be a psh function on C k . The function G ψ defined by G ψ (z) := ψ(z -w)dν(w), z ∈ C k , is psh and locally bounded on C k . Proof of Theorem 2.6. -We follow the strategy of the proof of [Duj, Theorem 0.1]. Write

  Lemma 4.1. -Let (f, a) be any dynamical pair of P 1 of degree d ≥ 2 parametrized by the unit disk D. If f is J-stable and supp(µ f,a ) = ∅, we have supp(µ f,a ) = {λ ∈ D ; a(λ) ∈ J f λ }.