arXiv subjects

Virgile Tapiero

Publications and source records attributed to Virgile Tapiero.

4 recordsLinked to original sources

arXiv2026

Normal forms, Lyapunov exponents, and pluripotential theory on $\mathbb{P}^k(\mathbb{C})$

We study the dynamical properties of endomorphisms $f$ of $\mathbb{P}^k$ of algebraic degree $d \geq 2$. We investigate the relationships between the Green current $T$ of $f$, the equilibrium measure $μ= T^k$, and the Lyapunov exponents $λ\_1 \geq \cdots \geq λ\_k$ of $μ$. The latter are bounded below by $\frac{1}{2} \mathrm{Log} \ d$. Dujardin proved in \cite{Duj12} that if $μ\ll T^r \wedge ω\_{\mathbb{P}^k}^{k-r}$ for some $1 \leq r \leq k-1$, then $λ\_{r+1} = \cdots = λ\_k = \frac{1}{2} \mathrm{Log} \ d$. In this article we prove that, conversely, if $λ\_r>λ\_{r+1} = \cdots = λ\_k = \frac{1}{2} \mathrm{Log} \ d$, then $μ\ll T^r\wedgeω\_{\mathbb{P}^k}^{k-r}$, answering a question asked by Dujardin. Our arguments rely on pluripotential theory, ergodic theory, and normal forms for the inverse branches of the endomorphism. We also use normal forms to provide another proof of Dujardin's result.

math.CV↗

arXiv2025

Hyperbolic components and iterated monodromy of polynomial skew-products of $\mathbb{C}^2$

We study the hyperbolic components of the family $\mathrm{Sk}(p,d)$ of regular polynomial skew-products of $\mathbb{C}^2$ of degree $d\geq2$, with a fixed base $p\in\mathbb{C}[z]$. Using a homogeneous parametrization of the family, we compute the accumulation set $E$ of the bifurcation locus on the boundary of the parameter space. Then in the case $p(z)=z^d$, we construct a map $π_0(\mathcal{D}')\to AB_d$ from the set of unbounded hyperbolic components that do not fully accumulate on $E$, to the set of algebraic braids of degree $d$. This map induces a second surjective map $π_0(\mathcal{D}')\to\mathrm{Conj}(\mathfrak{S}_d)$ towards the set of conjugacy classes of permutations on $d$ letters. This article is a continuation in higher degrees of the work of Astorg-Bianchi in the quadratic case $d=2$, for which they provided a complete classification of the hyperbolic components belonging to $π_0(\mathcal{D}')$.

math.DS↗

arXiv2024

Invariant foliations for endomorphims of $\mathbb{P}^2$ with a pluripotentialist product structure

Let $f$ be a holomorphic endomorphism of $\mathbb{P}^2$, let $T$ be its Green current and $μ=T\wedge T$ be its equilibrium measure. We prove that if $μ$ has a local product structure with respect to $T$ then (an iterate of) $f$ preserves a local foliation $\mathcal{F}$ on a neighborhood of $\mathrm{Supp}(T )\backslash\mathcal{E}$,where $\mathcal{E}$ denotes the exceptional set of f . If the local foliation $\mathcal{F}$ extends through $\mathcal{E}$,then it extends to $\mathbb{P}^2$ and is an invariant pencil of lines.

math.CV↗

arXiv2024

On slice measures of Green currents on CP(2)

Let $f$ be a holomorphic map of $\mathbb{C}\mathbb{P}^2$ of degree $d\geq 2$, let $T$ be its Green current and $μ=T\wedge T$ be its equilibrium measure. We give a new proof of a theorem due to Dujardin asserting that $μ\ll T\wedgeω_{\mathbb{P}^2}$ implies $λ_2=\frac{1}{2} \log\ d$, where $λ_1 \geq λ_2$ are the Lyapunov exponents of $μ$. Then, assuming $μ\ll T\wedgeω_{\mathbb{P}^2}$, we study slice measures $ν:=T\wedge dd^c|W|^2$, where $W$ is a holomorphic local submersion. We give sufficient conditions on the Radon-Nikodym derivative of $μ$ with respect to the trace measure $T\wedgeω_{\mathbb{P}^2}$ ensuring $μ=ν$. The involved submersion $W$ comes from normal coordinates for the inverse branches of the iterates of $f$.

math.CV↗