arXiv subjects

Matthieu Astorg

Publications and source records attributed to Matthieu Astorg.

13 recordsLinked to original sources

arXiv2026

Parabolic implosion in dimension 2

In this paper, we extend the theory of parabolic implosion in complex dimension 2 to the case of holomorphic maps tangent to the identity at order 2. We investigate the bifurcation phenomena that occur when a fully parabolic fixed point is perturbed. Under the assumption of a non-degenerate characteristic direction with a formal invariant curve and director $\alpha$ satisfying $\re\alpha> 2$, we establish the existence of Lavaurs maps as limits of iterates $f_{\epsilon_n}^n$ for specific sequences of the perturbation parameter $\epsilon_n$. Finally, we apply these results to prove the discontinuity of the Julia sets $J_1$ and $J_2$ for holomorphic endomorphisms of $\mathbb{P}^2$, generalizing classical one-dimensional results to this higher-dimensional setting.

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arXiv2026

Non-autonomous parabolic implosion

We study parabolic implosion in a general non-autonomous setting. Let $f(w)=w+w^2+O(w^3)$ be a holomorphic germ tangent to the identity. We consider the iteration of non-autonomous perturbations of the form \[ w_{j+1}=f(w_j)+\varepsilon_{j,n}^2. \] We show that, when the $\varepsilon_{j,n}^2$'s satisfy a Lavaurs-type condition, the element $w_n$ can be described by means of a suitable Lavaurs map $L_{u_n}$, whose phase $u_n$ is an explicit function of the perturbation parameters. In particular, whenever $u_n\to u\in \mathbb C$, the non-autonomous dynamics converges locally uniformly on compact subsets of the parabolic basin to the corresponding Lavaurs map $L_u$. Our study provides a general description of additive non-autonomous parabolic implosion and yields several deterministic and random convergence results as corollaries, as well as a unified proof of several previous results. As an application, we also obtain strong discontinuity results for the Julia sets of fibered holomorphic endomorphisms of $\mathbb P^2(\mathbb C)$.

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arXiv2024

Cubic Siegel polynomials and the bifurcation measure

We prove that cubic polynomial maps with a fixed Siegel disk and a critical orbit eventually landing inside that Siegel disk lie in the support of the bifurcation measure. This answers a question of Dujardin in positive. Our result implies the existence of holomorphic disks in the support of the bifurcation measure, and also implies that the set of rigid parameters is not closed in the moduli space of cubic polynomials.

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arXiv2024

Bifurcation for families of Ahlfors island maps

We extend Ma\~n\'e-Sad-Sullivan and Lyubich's equivalent characterization of stability to the setting of Ahlfors island maps, which include notably all meromorphic maps. As a consequence we also obtain the density of $J$-stability for finite type maps in the sense of Epstein.

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arXiv2022

Dynamics of skew-products tangent to the identity

We study the local dynamics of generic skew-products tangent to the identity, i.e. maps of the form $P(z,w)=(p(z), q(z,w))$ with $dP_0=\mathrm{id}$. More precisely, we focus on maps with non-degenerate second differential at the origin; such maps have local normal form $P(z,w)=(z-z^2+O(z^3),w+w^2+bz^2+O(\|(z,w)\|^3))$. We prove the existence of parabolic domains, and prove that inside these parabolic domains the orbits converge non-tangentially if and only if $b \in (\frac{1}{4},+\infty)$. Furthermore, we prove the existence of a type of parabolic implosion, in which the renormalization limits are different from previously known cases. This has a number of consequences: under a diophantine condition on coefficients of $P$, we prove the existence of wandering domains with rank 1 limit maps. We also give explicit examples of quadratic skew-products with countably many grand orbits of wandering domains, and we give an explicit example of a skew-product map with a Fatou component exhibiting historic behaviour. Finally, we construct various topological invariants, which allow us to answer a question of Abate.

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arXiv2021

Bifurcation loci of families of finite type meromorphic maps

We show that $J-$ stability is open and dense in natural families of meromorphic maps of one complex variable with a finite number of singular values, and even more generally, to finite type maps. This extends the results of Ma\~{n}\'e-Sad-Sullivan for rational maps of the Riemann sphere and those of Eremenko and Lyubich for entire maps of finite type of the complex plane, and essentially closes the problem of density of structural stability for holomorphic dynamical systems in one complex variable with finitely many singular values. This result is obtained as a consequence of a detailed study of a new type of bifurcation that arises with the presence of both poles and essential singularities (namely periodic orbits exiting the domain of definition of the map along a parameter curve), and in particular its relation with the bifurcations in the dynamics of singular values. The presence of these new bifurcation parameters require essentially different methods to those used in previous work for rational or entire maps.

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arXiv2020

Higher bifurcations for polynomial skew-products

We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any bifurcation parameter, one can find parameters where $k$ critical points bifurcate \emph{independently}, with $k$ up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an analytical criterion for the non-vanishing of the self-intersections of the bifurcation current, we deduce the equality of the supports of the bifurcation current and the bifurcation measure for such families. Combined with results by Dujardin and Taflin, this also implies that the support of the bifurcation measure in these families has non-empty interior.As part of our proof we construct, in these families, subfamilies of codimension 1 where the bifurcation locus has non empty interior. This provides a new independent proof of the existence of holomorphic families of arbitrarily large dimension whose bifurcation locus has non empty interior. Finally, it shows that the Hausdorff dimension of the support of the bifurcation measure is maximal at any point of its support.

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arXiv2018

Hyperbolicity and bifurcations in holomorphic families of polynomial skew products

We initiate a parametric study of holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of $\mathbb{C}^2$ of the form $F(z,w)= (p(z), q(z,w))$ that extend to holomorphic endomorphisms of $\mathbb{P}^2(\mathbb{C})$. We prove that dynamical stability in the sense of arXiv:1403.7603 preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family $z^2 +c$. We also precisely describe the geometry of the bifurcation locus and current near the boundary of the parameter space. One of our tools is an asymptotic equidistribution property for the bifurcation current. This is established in the general setting of families of endomorphisms of $\mathbb{P}^k$ and is the first equidistribution result of this kind for holomorphic dynamical systems in dimension larger than one.

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arXiv2017

Collet, Eckmann and the bifurcation measure

The moduli space $\mathcal{M}_d$ of degree $d\geq2$ rational maps can naturally be endowed with a measure $\mu_\mathrm{bif}$ detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure $\mu_\mathrm{bif}$ has positive Lebesgue measure. To do so, we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of $\mu_\mathrm{bif}$ and we exhibit a large set of Collet-Eckmann rational maps which satisfy this condition. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps.

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arXiv2016

Dynamics of post-critically finite maps in higher dimension

We study the dynamics of post-critically finite endomorphisms of P^k(C). We prove that post-critically finite endomorphisms are always post-critically finite all the way down under a mild regularity condition on the post-critical set. We study the eigenvalues of periodic points of post-critically finite endomorphisms. Then, under a weak transversality condition and assuming Kobayashi hyperbolicity of the complement of the post-critical set, we prove that the only possible Fatou components are super-attracting basins, thus partially extending to any dimension a result of Fornaess-Sibony and Rong holding in the case k = 2.

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arXiv2016

Summability Condition and Rigidity for Finite Type Maps

We extend a series of results due to Makienko, Dominguez and Sienra on the rigidity of some holomorphic dynamical systems with summable critical values to the setting of finite type maps. We also recover a shorter proof of a transversality theorem of Levin. Our methods are based on the deformation theory introduced by Epstein.

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arXiv2014

A two-dimensional polynomial mapping with a wandering Fatou component

We show that there exist polynomial endomorphisms of C^2, possessing a wandering Fatou component. These mappings are polynomial skew-products, and can be chosen to extend holomorphically of P^2(C). We also find real examples with wandering domains in R^2. The proof is based on parabolic implosion techniques, and is based on an original idea of M. Lyubich.

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arXiv2014

On the dynamical Teichm\"uller space

We prove that the dynamical Teichm\"uller space of a rational map immerses into the space of rational maps of the same degree, answering a question of McMullen and Sullivan. This is achieved through a new description of the tangent and cotangent space to the dynamical Teichm\"uller space.

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