arXiv subjects
Publications and source records attributed to Romain Dujardin.
We study the asymptotic behavior of the Lyapunov exponent in a meromorphic family of random products of matrices in SL(2, C), as the parameter converges to a pole. We show that the blow-up of the Lyapunov exponent is governed by a quantity which can be interpreted as the non-Archimedean Lyapunov exponent of the family. We also describe the limit of the corresponding family of stationary measures on P 1 (C).
We prove that for a polynomial diffeomorphism of C^2 , the support of any invariant measure, apart from a few obvious cases, is contained in the closure of the set of saddle periodic points.
A well-known theorem due to Mañé-Sad-Sullivan and Lyubich asserts that J-stable maps are dense in any holomorphic family of rational maps in dimension 1. In this paper we show that the corresponding result fails in higher dimension. More precisely, we construct open subsets in the bifurcation locus in the space of holomorphic mappings of degree d of P^k (C) for every d $\ge$ 2 and k $\ge$ 2.
Let $f$ be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve $C$. We conjecture that this happens if and only if $f$ admits a time-reversal symmetry; in particular the Jacobian $\mathrm{Jac}(f)$ must be a root of unity. As a step towards this conjecture, we prove that the Jacobian of $f$ and all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed $\mathrm{Jac}(f)$ is a root of unity. We use these results to show in various cases that any two automorphisms sharing an infinite set of periodic points must have a common iterate, in the spirit of recent results by Baker-DeMarco and Yuan-Zhang.
We show that there exist polynomial endomorphisms of C^2, possessing a wandering Fatou component. These mappings are polynomial skew-products, and can be chosen to extend holomorphically of P^2(C). We also find real examples with wandering domains in R^2. The proof is based on parabolic implosion techniques, and is based on an original idea of M. Lyubich.
Let $(f_λ)_{λ\in Λ}$ be a holomorphic family of polynomial automorphisms of $\mathbb{C}^2$. Following previous work of Dujardin and Lyubich, we say that such a family is weakly stable if saddle periodic orbits do not bifurcate. It is an open question whether this property is equivalent to structural stability on the Julia set $J^*$ (that is, the closure of the set of saddle periodic points). In this paper we introduce a notion of regular point for a polynomial automorphism, inspired by Pesin theory, and prove that in a weakly stable family, the set of regular points moves holomorphically. It follows that a weakly stable family is probabilistically structurally stable, in a very strong sense. Another consequence of these techniques is that weak stability preserves uniform hyperbolicity on $J^*$.
We study stability and bifurcations in holomorphic families of polynomial automorphisms of C^2. We say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there. We first show that this defines a meaningful notion of stability, which parallels in many ways the classical notion of J-stability in one-dimensional dynamics. In the second part of the paper, we prove that under an assumption of moderate dissipativity, the parameters displaying homoclinic tangencies are dense in the bifurcation locus. This confirms one of Palis' Conjectures in the complex setting. The proof relies on the formalism of semi-parabolic bifurcation and the construction of "critical points" in semi-parabolic basins (which makes use of the classical Denjoy-Carleman-Ahlfors and Wiman Theorems).
A holomorphic endomorphism f of CP^2 admits a Julia set J_1, defined as usual to be the locus of non-normality of its iterates, and a (typically) smaller Julia set J_2, which is essentially the closure of the set of repelling periodic orbits. The question has been raised whether J_1\J_2 is filled (possibly in a measure-theoretic sense) with "Fatou subvarieties" along which the dynamics is locally equicontinuous. In this article we construct examples showing that this is not the case in general
Let (ρ_\la)_{\la\in \La} be a holomorphic family of representations of a surface group π_1(S) into PSL(2,C), where S is a topological (possibly punctured) surface with negative Euler characteristic. Given a structure of Riemann surface of finite type on S we construct a bifurcation current on the parameter space \La, that is a (1,1) positive closed current attached to the bifurcations of the family. It is defined as the $dd^c$ of the Lyapunov exponent of the representation with respect to the Brownian motion on the Riemann surface S, endowed with its Poincare metric. We show that this bifurcation current describes the asymptotic distribution of various codimension 1 phenomena in \La. For instance, the random hypersurfaces of \La defined by the condition that a random closed geodesic on S is mapped under ρ_\la to a parabolic element or the identity are asymptotically equidistributed with respect to the bifurcation current. The proofs are based on our previous work "Random walks, Kleinian groups, and bifurcation currents", and on a careful control of a discretization procedure of the Brownian motion.
We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developing map; and a family of harmonic measures on the Riemann sphere, previously introduced by Hussenot. We show that the degree and the Lyapunov exponent are related by a simple formula and give estimates for the Hausdorff dimension of the harmonic measures in terms of the Lyapunov exponent. In accordance with the famous "Sullivan dictionary", this leads to a description of the space of such projective structures that is reminiscent of that of the space of polynomials in holomorphic dynamics.
Not much is known about the dynamics outside the support of the maximal entropy measure $μ$ for holomorphic endomorphisms of $\mathbb{CP}^k$. In this article we study the structure of the dynamics on the Julia set, which is typically larger than $Supp(μ)$. The Julia set is the support of the so-called Green current $T$, so it admits a natural filtration by the supports of the exterior powers of $T$. For $1\leq q \leq k$, let $J_q= Supp(T^q)$. We show that for a generic point of $J_q\setminus J_{q+1}$ there are at least $(k-q)$ "Fatou directions" in the tangent space. We also give estimates for the rate of expansion in directions transverse to the Fatou directions.
In this short note we prove an estimate on the rank a.e. of the tangent (p,p) vector to a a positive closed current of bidimension (p,p) in CP^k, in terms of the dimension of its trace measure.
Let (f_λ) be a holomorphic family of rational mappings of degree d on the Riemann sphere, with k marked critical points c_1,..., c_k, parameterized by a complex manifold Λ. To this data is associated a closed positive current T_1\wedge ... \wedge T_k of bidegree (k,k) on Λ, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which c_1,..., c_k eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of Supp(T_1\wedge ... \wedge T_k).
In this paper we review the use of techniques of positive currents for the study of parameter spaces of one-dimensional holomorphic dynamical systems (rational mappings on P^1 or subgroups of the Moebius group PSL(2,C)). The topics covered include: the construction of bifurcation currents and the characterization of their supports, the equidistribution properties of dynamically defined subvarieties on parameter space.
Let (ρ_λ)_{λ\in Λ} be a holomorphic family of representations of a finitely generated group G into PSL(2,C), parameterized by a complex manifold Λ. We define a notion of bifurcation current in this context, that is, a positive closed current on Λdescribing the bifurcations of this family of representations in a quantitative sense. It is the analogue of the bifurcation current introduced by DeMarco for holomorphic families of rational mappings on the Riemann sphere. Our definition relies on the theory of random products of matrices, so it depends on the choice of a probability measure μon G. We show that under natural assumptions on μ, the support of the bifurcation current coincides with the bifurcation locus of the family. We also prove that the bifurcation current describes the asymptotic distribution of several codimension 1 phenomena in parameter space, like accidental parabolics or new relations, or accidental collisions between fixed points.
We continue our study of the dynamics of meromorphic mappings with small topological degree on a compact Kähler surface $X$. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and admits a natural geometric description. Our hypotheses are always satisfied when $X$ has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of $\mathbf{C}^2$. They are new even in the birational case. We also exhibit families of mappings where our assumptions are generically satisfied and show that if counterexamples exist, the corresponding measure must give mass to a pluripolar set.
We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalize results that were known in the invertible case and is, to our knowledge, one among not very many instances in which a natural invariant measure for a non-invertible dynamical system is well-understood.
In this paper we give the first examples of positive closed currents in $\mathbb{C}^2$ with continuous potentials, vanishing self-intersection, and which are not laminar. More precisely, they are supported on sets "without analytic structure". The result is mostly interesting when the potential has regularity close to $C^2$, because laminarity is expected to hold in that case. We actually construct examples which are $C^{1,α}$ for all $α<1$.