arXiv subjects

Gabriel Vigny

Publications and source records attributed to Gabriel Vigny.

29 records · Page 2Linked to original sources

arXiv2018

The bifurcation measure has maximal entropy

Let $Λ$ be a complex manifold and let $(f_λ)_{λ\in Λ}$ be a holomorphic family of rational maps of degree $d\geq 2$ of $\mathbb{P}^1$. We define a natural notion of entropy of bifurcation, mimicking the classical definition of entropy, by the parametric growth rate of critical orbits. We also define a notion a measure-theoretic bifurcation entropy for which we prove a variational principle: the measure of bifurcation is a measure of maximal entropy. We rely crucially on a generalization of Yomdin's bound of the volume of the image of a dynamical ball. Applying our technics to complex dynamics in several variables, we notably define and compute the entropy of the trace measure of the Green currents of a holomorphic endomorphism of $\mathbb{P}^k$.

math.DS↗

arXiv2018

Approximation of non-archimedean Lyapunov exponents and applications over global fields

Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of $\mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of $n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and $K$. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over $K$. Combined with our former archimedean version, this non-archimedean quantitative approximation allows us to show: - a quantified version of Silverman's and Ingram's recent comparison between the critical height and any ample height on the moduli space $\mathcal{M}_d(\bar{\mathbb{Q}})$, - two improvements of McMullen's finiteness of the multiplier maps: reduction to multipliers of cycles of exact given period and an effective bound from below on the period, - a characterization of non-affine isotrivial rational maps defined over the function field $\mathbb{C}(X)$ of a normal projective variety $X$ in terms of the growth of the degree of the multipliers.

math.DS↗

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 $μ_\mathrm{bif}$ detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure $μ_\mathrm{bif}$ 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 $μ_\mathrm{bif}$ 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.

math.DS↗

arXiv2017

Hyperbolic components of rational maps: Quantitative equidistribution and counting

Let $Λ$ be a quasi-projective variety and assume that, either $Λ$ is a subvariety of the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, or $Λ$ parametrizes an algebraic family $(f_λ)_{λ\inΛ}$ of degree $d$ rational maps on $\mathbb{P}^1$. We prove the equidistribution of parameters having $p$ distinct neutral cycles towards the $p$-th bifurcation current letting the periods of the cycles go to $\infty$, with an exponential speed of convergence. We deduce several fundamental consequences of this result on equidistribution and counting of hyperbolic components. A key step of the proof is a locally uniform version of the quantitative approximation of the Lyapunov exponent of a rational map by the $\log^+$ of the modulus of the multipliers of periodic points.

math.DS↗

arXiv2016

Distribution of postcritically finite polynomials II: Speed of convergence

In the moduli space of degree d polynomials, we prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, we prove the exponential speed of convergence for C 2-observables. This improves results obtained with arithmetic methods by Favre and Rivera-Letellier in the unicritical family and Favre and the first author in the space of degree d polynomials. We deduce from that the equidistribution of hyperbolic parameters with (d -- 1) distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for C 1-observables. As an application, we prove the equidistribution (up to an explicit extraction) of parameters with (d -- 1) distinct cycles with prescribed multiplier toward the bifurcation measure for any (d -- 1) multipliers outside a pluripolar set.

math.DS↗

arXiv2016

Distribution of points with prescribed derivative in polynomial dynamics

In analogy to the equidistribution of preimages of a prescribed point by the iterates of a polynomial map in the complex plane towards the equilibrium measure, we show here the equidistribution of points for which the derivative of the n-th iterate of the polynomial takes a suitable prescribed value towards the equilibrium measure. We then give a similar statement in the space of degree d polynomials for the equidistribution of parameters for which the n-derivative at a given critical value has a prescribed derivative towards the activity current of the corresponding critical point.

math.DS↗

arXiv2016

Distribution of postcritically finite polynomials iii: Combinatorial continuity

In the first part of the present paper, we continue our study of distribution of postcritically finite parameters in the moduli space of polynomials: we show the equidistribution of Misiurewicz parameters with prescribed combinatorics toward the bifurcation measure. Our results essentially rely on a combinatorial description of the escape locus and of the bifurcation measure developped by Kiwi and Dujardin-Favre. In the second part of the paper, we construct a bifurcation measure for the connectedness locus of the quadratic anti-holomorphic family which is supported by a strict subset of the boundary of the Tricorn. We also establish an approximation property by Misiurewicz parameters in the spirit of the previous one. Finally, we answer a question of Kiwi, exhibiting in the moduli space of degree 4 polynomials, non-trivial Impression of specific combinatorics.

math.DS↗

arXiv2014

Hyperbolic measure of maximal entropy for generic rational maps of P^k

Let f be a dominant rational map of P^k such that there exists s lambda_l(f) for all l. Under mild hypotheses, we show that, for A outside a pluripolar set of the group of automorphisms of P^k, the map f o A admits a hyperbolic measure of maximal entropy log(lambda_s(f)) with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of f to P^{k+1}. This provides many examples where non uniform hyperbolic dynamics is established. One of the key tools is to approximate the graph of a meromorphic function by a smooth positive closed current. This allows us to do all the computations in a smooth setting, using super-potentials theory to pass to the limit.

math.CV↗

arXiv2014

On the measures of large entropy on a positive closed current

Let $f:X\to X$ be a dominating meromorphic map of a compact Kähler surface of large topological degree. Let $S$ be a positive closed current on $X$ of bidegree $(1,1)$. We consider an ergodic measure $ν$ of large entropy supported by $\mathrm{supp}(S)$. Defining dimensions for $ν$ and $S$, we give inequalities à la Mañé involving the Lyapunov exponents of $ν$ and those dimensions. We give dynamical applications of those inequalities.

math.DS↗

arXiv2013

Exponential decay of correlations for generic birational maps of P^k

We prove the exponential decay of correlations for C^α-observables (0<α=<2) for generic birational maps of P^k à la Bedford-Diller. In the particular case of regular birational maps, we give a better estimate of the speed of the decay, getting results as sharp as Dinh's results for Hénon maps.

math.DS↗

arXiv2008

Entropy of meromorphic maps and dynamics of birational maps

We study the dynamics of meromorphic maps for a compact Kaehler manifold X. More precisely, we give a simple criterion that allows us to produce a measure of maximal entropy. We can apply this result to bound the Lyapunov exponents. Then, we study the particular case of a family of generic birational maps of P^k for which we construct the Green currents and the equilibrium measure. We use for that the theory of super-potentials. We show that the measure is mixing and gives no mass to pluripolar sets. Using the criterion we get that the measure is of maximal entropy. It implies finally that the measure is hyperbolic.

math.DS↗