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Arnaud Chéritat

Publications and source records attributed to Arnaud Chéritat.

At least 19 recordsLinked to original sources
arXiv2026

Biggest bounded type Siegel disks of monic polynomials include those that stick to all critical points

We prove that for all degree $d\geq 2$ and all bounded type irrational $θ$, in the space of monic polynomials having a period $1$ Siegel disk $Δ$ of rotation number $θ$, the maximum locus of the conformal radius of $Δ$ with respect to its fixed point contains polynomials having all critical points on the boundary of $Δ$. We apply this to reduce a conjecture of Douady (optimality of the Bruno condition) to a weaker statement.

math.DS↗
arXiv2026

Hybrid connections on Hessian manifolds

We introduce a new class of affine connections on Hessian manifolds, called hybrid connections, characterized by the compatibility between their projective geometry with the underlying affine structure on the one hand and their infinitesimal holonomy with the Hessian metric on the other hand. In this paper, we investigate the properties of hybrid connections and prove that, on a given Hessian manifold, they are completely determined by the choice of a Hessian potential for the metric. In the special case of pseudo Euclidean manifolds, we identify canonical models and construct, in particular, a natural connection on the open unit ball that combines features of the Cayley Klein and Poincaré models of hyperbolic geometry. We also prove the existence and uniqueness (up to scaling) of a pseudo-Riemannian metric $h$ such that the geodesics of $\nabla$ admit parameterizations of constant speed with respect to $h$, which we call the isochrone metric.

math.DG↗
arXiv2026

Dynamics of generic 1-parameter perturbations of a vector field with a singular point of codimension k

The paper describes the bifurcation diagram of germs of generic $1$-parameter families of complex vector fields $\dot z = ω_ε(z)$ on $\mathbb{C}$, unfolding a singular point of multiplicity $k+1$: $ω_0= z^{k+1} +o(z^{k+1})$. As a preparatory step, the bifurcation diagram of the family of vector fields $\dot z = z^{k+1}-ε$ over $\mathbb{CP}^1$ is given, through a description of its associated translation surface as a star shaped domain (a straightening coordinate, a.k.a.\ time coordinate, of a vector field defines a translation surface structure on the complement of the singularities). In this article, we define a notion of generic families of vector fields $\dot z = ω_ε(z)$, and classify them up to conjugacy by a holomorphic change of coordinate and parameter. A description of the modulus space and several (almost) unique normal forms are provided. Then, we describe an analogue of the star shaped domain for these generic vector fields, that we call the geometric model and which allows to describe the local bifurcation diagram for generic families of vector fields in fine details and compare it with that of the simple vector field $\dot z = z^{k+1}-ε$ restricted to $B(0,r)$.

math.DS↗
arXiv2025

Some invariant classes for parabolic renormalization in the multicritical case

Parabolic renormalization associates to a holomorphic map f with a parabolic fixed point another holomorphic map with a parabolic fixed point. This procedure is essential for understanding the phenomenon of parabolic enrichment, which occurs when one perturbs f appropriately. Shishikura defined in [Shi98] (see also [LY14]) a class of maps that is stable under this parabolic renormalization operator. These maps have only one critical point in their immediate basins. We extend here this result to the more general classes of maps conjugated on their immediate basins to finite Blaschke products. For these classes, we introduce an analogue of Milnor's mapping schemes [Mil12] and describe the action of parabolic renormalization on the scheme. This shall be a starting point to the fine study of perturbation of these bigger classes of maps.

math.DS↗
arXiv2025

The affine geometry of meromorphic connections with irregular singularities

A meromorphic connection on the tangent bundle of a Riemann surface induces a complex affine structure on the complement of the poles. Local models for Fuchsian singularities are already known. In this paper, we introduce a complete set of local invariants for a meromorphic connection and provide local models for a complex affine structure in a punctured neighborhood of an irregular singularity. Generalizing a construction attributed to Veech, we introduce the Delaunay decomposition of a compact Riemann surface endowed with a meromorphic connection with irregular singularities. In particular, we give upper bounds on the complexity of the decomposition.

math.CV↗
arXiv2025

Similarity surfaces, connections, and the measurable Riemann mapping theorem

This article studies a particular process that approximates solutions of the Beltrami equation (straightening of ellipse fields, a.k.a. measurable Riemann mapping theorem) on $\mathbb{C}$. It passes through the introduction of a sequence of similarity surfaces constructed by gluing polygons, and we explain the relation between their conformal uniformization and the Schwarz-Christoffel formula. Numerical aspects, in particular the efficiency of the process, are not studied, but we draw interesting theoretical consequences. First, we give an independent proof of the analytic dependence, on the data (the Beltrami form), of the solution of the Beltrami equation (Ahlfors-Bers theorem). For this we prove, without using the Ahlfors-Bers theorem, the holomorphic dependence, with respect to the polygons, of the Christoffel symbol appearing in the Schwarz-Christoffel formula. Second, these Christoffel symbols define a sequence of parallel transports on the range, and in the case of a data that is $C^2$ with compact support, we prove that it converges to the parallel transport associated to a particular affine connection, which we identify.

math.CV↗
arXiv2025

Horn maps of holomorphic functions locally pseudo-conjugate on their parabolic basins

The lifted horn map of a holomorphic function with a simple parabolic point is well known to be a complete local conjugacy invariant; this is a classical result proved independently by Écalle, Voronin, Martinet and Ramis. Lanford and Yampolski have shown that, if two functions $f_1, f_2$ with simple parabolic points at $z_1, z_2$ are globally conjugate on their immediate parabolic basins, with the conjugacy and its inverse continuous at $z_1$, resp. $z_2$, then their horn maps must be cover-equivalent: there are isomorphisms $ψ^+ : \mathcal{D}_1^+\to \mathcal{D}_2^+$ and $ψ^- : \mathcal{D}_1^-\to \mathcal{D}_2^-$ between the top and bottom connected components of their domains, and a translation $T$ on the cylinder, such that $\mathbb{h}_2\circψ^+ = T\circ \mathbb{h}_1$ and $\mathbb{h}_2\circψ^- = T\circ \mathbb{h}_1$ holds on these domains. In this article, we introduce a notion of (semi) local conjugacy on immediate parabolic basins, which we call local pseudo-conjugacy and which in particular does not make any continuity assumption, and show that the horn maps $\mathbb{h}_1$ and $\mathbb{h}_2$ satisfy the condition above if and only if the two functions $f_1, f_2$ are locally pseudo-conjugate. This result is a first step to better understand invariant classes by parabolic renormalization.

math.DS↗
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.

math.DS↗
arXiv2024

Observations on the hex clusters of the Spectre tilings

Decorating the Spectre tile with hexagons reveals triangular hexagonal clusters whose structure we study. In the process we reprove that the Spectre tilings exist and are uniquely hierarchical. The proof is not computer-assisted.

math.CO↗
arXiv2020

Near parabolic renormalization for unicritical holomorphic maps

Inou and Shishikura provided a class of maps that is invariant by near-parabolic renormalization, and that has proved extremely useful in the study of the dynamics of quadratic polynomials. We provide here another construction, using more general arguments. This will allow to extend the range of applications to unicritical polynomials of all degrees.

math.DS↗
arXiv2020

On the size of Siegel disks with fixed multiplier for cubic polynomials

We study the slices of the parameter space of cubic polynomials where we fix the multiplier of a fixed point to some value $λ$. The main object of interest here is the radius of convergence of the linearizing parametrization. The opposite of its logarithm turns out to be a sub-harmonic function of the parameter whose Laplacian $μ_λ$ is of particular interest. We relate its support to the Zakeri curve in the case the multiplier is neutral with a bounded type irrational rotation number. In the attracting case, we define and study an analogue of the Zakeri curve, using work of Petersen and Tan. In the parabolic case, we define an analogue using the notion of asymptotic size. We prove a convergence theorem of $μ_{λ_n}$ to $μ_λ$ for $λ_n= \exp(2πi p_n/q_nn)$ and $λ= \exp(2πiθ)$ where $θ$ is a bounded type irrational and $p_n/q_n$ are its convergents.

math.DS↗
arXiv2019

Smooth Siegel disks everywhere

We prove the existence of Siegel disks with smooth boundaries in most families of holomorphic maps fixing the origin. The method can also yield other types of regularity conditions for the boundary. The family is required to have an indifferent fixed point at $0$, to be parameterized by the rotation number $α$, to depend on $α$ in a Lipschitz-continuous way, and to be non-degenerate. A degenerate family is one for which the set of non-linearizable maps is not dense. We give a characterization of degenerate families, which proves that they are quite exceptional.

math.DS↗
arXiv2015

A Proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type

Marmi Moussa and Yoccoz conjectured that some error function Upsilon, related to the approximation of the size of Siegel disk by some arithmetic function of the rotation number theta, is a Holder continuous function of theta with exponent 1/2. Using the renormalization invariant class of Inou and Shishikura, we prove this conjecture for the restriction of Upsilon to a class of high type numbers.

math.DS↗
arXiv2014

Yet another sphere eversion

We present a (possibly) new sphere eversion based on the contractibility* of a certain subset of the space of immersions of the circle in the plane. (*: by strong deformation retraction)

math.GT↗
arXiv2013

Proof of a convergence of affine Riemann surfaces to a richer one

We give a proof of a phenomenon conjectured in our former article: "Beltrami forms, affine surfaces and the Schwarz-Christoffel formula: a worked out example of straightening". We also start an abstract discussion of the notion of limits of Riemann surfaces.

math.CV↗
arXiv2012

Tan Lei and Shishikura's example of non-mateable degree 3 polynomials without a Levy cycle

After giving an introduction to the procedure dubbed slow polynomial mating and stating a conjecture relating this to other notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree 3 post critically finite polynomials introduced by Shishikura and Tan Lei as an example of a non matable pair of polynomials without a Levy cycle. The pictures show a limit for the Julia sets, which seems to be related to the Julia set of a degree 6 rational map. We give a conjectural interpretation of this in terms of pinched spheres and show further conformal representations.

math.DS↗