arXiv subjects
Publications and source records attributed to Gabriel Vigny.
Let $f$ and $g$ be two H\'enon-Sibony maps of $\mathbb{C}^k$. We show that if they have the same forward Julia set, then they share a common iterate, thereby extending Lamy's results from dimension 2.
We show that periodic points of period $n$ of a complex rational map of degree $d$ equidistribute towards the equilibrium measure $\mu_f$ of the rational map with a rate of convergence of $(nd^{-n})^{1/2}$ for $\mathscr{C}^1$-observables. This is a consequence of a quantitative equidistribution of Galois invariant finite subsets of preperiodic points \`a la Favre and Rivera-Letelier. Our proof relies on the H\"older regularity of the quasi-psh Green function of a rational map, an estimate of Baker concerning Hsia kernel, as well as on the product formula and its generalization by Moriwaki for finitely generated fields over $\mathbb{Q}$.
We prove that several dynamically defined fractals in $\mathbb{C}$ and $\mathbb{C}^2$ which arise from different type of polynomial dynamical systems can not be the same objects. One of our main results is that the closure of Misiurewicz PCF cubic polynomials (the strong bifurcation locus) cannot be the Julia set of a regular polynomial endomorphism of $\mathbb{C}^2$. We also show that the Julia set of a H\'enon map and a polynomial endomorphism cannot coincide.
For H\'enon maps, Bianchi and Dinh recently proved the exponential mixing of all orders for the measure of maximal entropy and, as a consequence of the recent work of Bj\"orklund and Gorodnik, the CLT for H\"older observables. We extend their results to generic birational maps of $\mathbb{P}^k$. Because of the indeterminacy set, H\"older maps are not stable under iteration, so we need to work with a suitable space of test functions.
We propose a set of questions on the dynamics of H\'enon maps from the real, complex, algebraic and arithmetic points of view.
When an endomorphism $f:X\to X$ of a projective variety which is polarized by an ample line bundle $L$, i.e. such that $f^*L\simeq L^{\otimes d}$ with $d\geq2$, is defined over a number field, Call and Silverman defined a canonical height $\widehat{h}_f$ for $f$. In a family $(\mathcal{X},\mathcal{f},\mathcal{L})$ parametrized by a curve $S$ together with a section $P:S\to \mathcal{X}$, they show that $\widehat{h}_{f_t}(P(t))/h(t)$ converges to the height $\widehat{h}_{f_\eta}(P_\eta)$ on the generic fiber. In the present paper, we prove the equivalent statement when studying the variation of canonical heights of subvarieties $Y_t$ varying in a family $\mathcal{Y}$ of any relative dimension.
The aim of this note is to give a proof of Theorem A from our work on the geometric Northcott property in the simpler case of the quadratic family; being in dimension $1$ in both the dynamical space and the parameter space, and having a simple and explicit parametrization of the family allow to simplify the proof and, we hope, make the ideas more apparent.
Let $\varphi$ be a function in the complex Sobolev space $W^*(U)$, where $U$ is an open subset in $\mathbb{C}^k$. We show that the complement of the set of Lebesgue points of $\varphi$ is pluripolar. The key ingredient in our approach is to show that $|\varphi|^\alpha $ for $\alpha \in [1,2)$ is locally bounded from above by a plurisubharmonic function.
An endomorphism $f:\mathbb{P}^k\to\mathbb{P}^k$ of degree $d\geq2$ is said to be postcritically finite (or PCF) if its critical set $\mathrm{Crit}(f)$ is preperiodic, i.e. if there are integers $m>n\geq0$ such that $f^m(\mathrm{Crit}(f))\subseteq f^n(\mathrm{Crit}(f))$. When $k\geq2$, it was conjectured by Ingram, Ramadas and Silverman that, in the space $\mathrm{End}_d^k$ of all endomorphisms of degree $d$ of $\mathbb{P}^k$, such endomorphisms are not Zariski dense. We prove this conjecture. Further, in the space $\mathrm{Poly}_d^2$ of all regular polynomial endomorphisms of degree $d\geq2$ of the affine plane $\mathbb{A}^2$, we construct a dense and Zariski open subset where we have a uniform bound on the number of preperiodic points lying in the critical set. The proofs are a combination of the theory of heights in arithmetic dynamics and methods from real dynamics to produce open subsets with maximal bifurcation.
Jonsson and Reschke showed that birational selfmaps on projective surface defined over a number field satisfy the energy condition of Bedford and Diller so their ergodic properties are very well understood. Under suitable hypotheses on the indeterminacy loci, we extend that result to birational maps $\mathbb{P}^k\dashrightarrow\mathbb{P}^k$, $k\geq2$, defined over a number field, showing that they satisfy a similar energy condition introduced by De Th\'elin and the second author. As a consequence, we can construct for such maps their Green measure and deduce several important ergodic consequences. Under a mild additional hypothesis, we show that generic sequences of Galois invariant subset of periodic points equidistribute toward the Green measure.
We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $\mathbf{K}$ of characteristic zero, improving results of Ingram. For that, we show that when $\mathbf{K}$ is the field of rational functions of a smooth complex projective curve, the canonical height of a subvariety is the mass of an appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a similarity argument.
In an algebraic family of rational maps of $\mathbb{P}^1$, we show that, for almost every parameter for the trace of the bifurcation current of a marked critical value, the critical value is Collet-Eckmann. This extends previous results of Graczyk and Świcatek in the unicritical family, using Makarov theorem. Our methods are based instead on ideas of laminar currents theory.
For a Kahler manifold X, we study a space of test functions W* which is a complex version of H1. We prove for W* the classical results of the theory of Dirichlet spaces: the functions in W* are defined up to a pluripolar set and the functional capacity associated to W* tests the pluripolar sets. This functional capacity is a Choquet capacity. The space W* is not reflexive and the smooth functions are not dense in it for the strong topology. So the classical tools of potential theory do not apply here. We use instead pluripotential theory and Dirichlet spaces associated to a current.
Let $X$ be a compact Kaehler manifold of dimension $k$ and $T$ be a positive closed current on $X$ of bidimension $(p,p)$ ($1\leq p < k-1$). We construct a continuous linear transform $\mathcal{L}_p(T)$ of $T$ which is a positive closed current on $X$ of bidimension $(k-1,k-1)$ which has the same Lelong numbers as $T$. We deduce from that construction self-intersection inequalities for positive closed currents of any bidegree.
We establish the dynamical Northcott property for polarized endomorphisms of a projective variety over a function field $\mathbf{K}$ of characteristic zero, and we relate this property to the notion of stability in complex dynamics. This extends previous results of Benedetto, Baker and DeMarco in dimension $1$, and of Chatzidakis-Hrushovski in higher dimension. Our proof uses complex dynamics arguments and does not rely on the previous ones. We first show that, when $\mathbf{K}$ is the field of rational functions of a normal complex projective variety, the canonical height of a subvariety is the mass of an appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the geometric dynamical Northcott property characterizing points of height zero in this setting, using a similarity argument. Moving from points to subvarieties, we propose, for polarized endomorphisms, a dynamical version of the geometric Bogomolov conjecture, recently proved by Cantat, Gao, Habegger, and Xie in the original setting of abelian varieties.
We study the dynamics near infinity of polynomial mappings $f$ in $\mathbb{C}^2$. We assume that $f$ has indeterminacy points and is non constant on the line at infinity $L_\infty$. If $L_\infty$ is $f$-attracting, we decompose the Green current along itineraries defined by the indeterminacy points and their preimages. The symbolic dynamics that arises is a subshift on an infinite alphabet.
For every $m\in\mathbb{N}$, we establish the equidistribution of the sequence of the averaged pull-backs of a Dirac measure at any given value in $\mathbb{C}\setminus\{0\}$ under the $m$-th order derivatives of the iterates of a polynomials $f\in \mathbb{C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of $f$ with pole at $\infty$. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field $k$ for a sequence of effective divisors on $\mathbb{P}^1(\overline{k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of $\mathbb{C}^2$ has a given eigenvalue.
Let $f : X\to X$ be a dominating meromorphic map on a compact Kähler manifold $X$ of dimension $k$. We extend the notion of topological entropy $h^l_{\mathrm{top}}(f)$ for the action of $f$ on (local) analytic sets of dimension $0\leq l \leq k$. For an ergodic probability measure $ν$, we extend similarly the notion of measure-theoretic entropy $h_ν^l(f)$. Under mild hypothesis, we compute $h^l_{\mathrm{top}}(f)$ in term of the dynamical degrees of $f$. In the particular case of endomorphisms of $\mathbb{P}^2$ of degree $d$, we show that $h^1_{\mathrm{top}}(f)= \log d$ for a large class of maps but we give examples where $h^1_{\mathrm{top}}(f)\neq \log d$.