arXiv subjects

Valentin Huguin

Publications and source records attributed to Valentin Huguin.

8 recordsLinked to original sources

arXiv2026

An arithmetic approach to parabolic multiplicity in complex dynamics

When $\omega$ is a primitive $n$-th root of unity, the quadratic polynomial $F(z) = \omega z (1 -z)$ and the entire map $F(z) = \omega z \mathrm{e}^{-z}$ both have a parabolic fixed point at $0$. Their parabolic multiplicity is equal to $1$, that is, $F^{\circ n}(z) = z \bigl( 1 +c z^n +\mathcal{O}(z^{n+1}) \bigr)$ with $c \neq 0$. The classical proof of this fact is transcendental. We present an arithmetic proof which may be extracted from [Towards global models near homoclinic tangencies of dissipative diffeomorphisms; H. Broer, C. Sim\'{o}, J.C. Tatjer] in the transcendental case and requires working in $\mathbb{Z}/(n -1) \mathbb{Z}$, and which is new in the polynomial case and requires working in the $p$-adic field $\mathbb{Q}_p$ for a suitable prime $p$ such that the order of $2$ in $(\mathbb{Z}/p \mathbb{Z})^\times$ is exactly $n$.

math.DS↗

arXiv2025

Entire maps with rational preperiodic points and multipliers

Given a number field $\mathbb{K} \subset \mathbb{C}$ that is not contained in $\mathbb{R}$, we prove the existence of a dense set of entire maps $f \colon \mathbb{C} \rightarrow \mathbb{C}$ whose preperiodic points and multipliers all lie in $\mathbb{K}$. This contrasts with the case of rational maps. In addition, we show that there exists an escaping quadratic-like map that is not conjugate to an affine escaping quadratic-like map and whose multipliers all lie in $\mathbb{Q}$.

math.DS↗

arXiv2024

Moduli spaces of polynomial maps and multipliers at small cycles

Fix an integer $d \geq 2$. The space $\mathcal{P}_{d}$ of polynomial maps of degree $d$ modulo conjugation by affine transformations is naturally an affine variety over $\mathbb{Q}$ of dimension $d -1$. For each integer $P \geq 1$, the elementary symmetric functions of the multipliers at all the cycles with period $p \in \lbrace 1, \dotsc, P \rbrace$ induce a natural morphism $\operatorname{Mult}_{d}^{(P)}$ defined on $\mathcal{P}_{d}$. In this article, we show that the morphism $\operatorname{Mult}_{d}^{(2)}$ induced by the multipliers at the cycles with periods $1$ and $2$ is both finite and birational onto its image. In the case of polynomial maps, this strengthens results by McMullen and by Ji and Xie stating that $\operatorname{Mult}_{d}^{(P)}$ is quasifinite and birational onto its image for all sufficiently large integers $P$. Our result arises as the combination of the following two statements: $\mathord{\bullet}$ A sequence of polynomials over $\mathbb{C}$ of degree $d$ with bounded multipliers at its cycles with periods $1$ and $2$ is necessarily bounded in $\mathcal{P}_{d}(\mathbb{C})$. $\mathord{\bullet}$ A generic conjugacy class of polynomials over $\mathbb{C}$ of degree $d$ is uniquely determined by its multipliers at its cycles with periods $1$ and $2$.

math.DS↗

arXiv2023

Entire or rational maps with integer multipliers

Let $\mathcal{O}_{K}$ be the ring of integers of an imaginary quadratic field $K$. Recently, Ji and Xie proved that every rational map $f \colon \widehat{\mathbb{C}} \rightarrow \widehat{\mathbb{C}}$ of degree $d \geq 2$ whose multipliers all lie in $\mathcal{O}_{K}$ is a power map, a Chebyshev map or a Lattès map. Their proof relies on a result from non-Archimedean dynamics obtained by Rivera-Letelier. In the present note, we show that one can avoid using this result by considering a differential equation instead. Our proof of Ji and Xie's result also applies to the case of entire maps. Thus, we also show that every nonaffine entire map $f \colon \mathbb{C} \rightarrow \mathbb{C}$ whose multipliers all lie in $\mathcal{O}_{K}$ is a power map or a Chebyshev map.

math.DS↗

arXiv2022

Rational maps with rational multipliers

In this article, we show that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Lattès map. This strengthens a conjecture by Milnor concerning rational maps with integer multipliers, which was recently proved by Ji and Xie.

math.DS↗

arXiv2021

Quadratic rational maps with integer multipliers

In this article, we prove that every quadratic rational map whose multipliers all lie in the ring of integers of a given imaginary quadratic field is a power map, a Chebyshev map or a Lattès map. In particular, this provides some evidence in support of a conjecture by Milnor concerning rational maps whose multipliers are all integers.

math.DS↗

arXiv2020

Unicritical polynomial maps with rational multipliers

In this article, we prove that every unicritical polynomial map that has only rational multipliers is either a power map or a Chebyshev map. This provides some evidence in support of a conjecture by Milnor concerning rational maps whose multipliers are all integers.

math.DS↗

arXiv2019

Simultaneously preperiodic integers for quadratic polynomials

In this article, we study the set of parameters $c \in \mathbb{C}$ for which two given complex numbers $a$ and $b$ are simultaneously preperiodic for the quadratic polynomial $f_{c}(z) = z^{2} +c$. Combining complex-analytic and arithmetic arguments, Baker and DeMarco showed that this set of parameters is infinite if and only if $a^{2} = b^{2}$. Recently, Buff answered a question of theirs, proving that the set of parameters $c \in \mathbb{C}$ for which both $0$ and $1$ are preperiodic for $f_{c}$ is equal to $\lbrace -2, -1, 0 \rbrace$. Following his approach, we complete the description of these sets when $a$ and $b$ are two given integers with $\lvert a \rvert \neq \lvert b \rvert$.

math.DS↗