arXiv subjects
Publications and source records attributed to Christophe Dupont.
Let $G$ be a compact Lie group. Let ${\mathsf{F}}_n$ be the free group of rank $n$. We describe the orbits of ${\mathsf{Aut}}(\mathsf{F}_n)$ on ${\mathsf{Hom}}(\mathsf{F}_n;G)$ when $n$ is sufficiently large. The dynamics stabilizes: orbit closures and invariant probability measures are algebraic, as in Ratner's theorems.
Let $\mathcal{F}$ be a singular holomorphic foliation on an algebraic complex surface $S$, with hyperbolic singularities and no foliated cycle. We prove a formula for the transverse Hausdorff dimension of the unique harmonic current, involving the Furstenberg entropy and the Lyapunov exponent. In particular, we extend Brunella's inequality to every holomorphic foliation $\mathcal{F}$ on $\mathbb P^2$: if $\mathcal{F}$ has degree $d \geq 2$, then the Hausdorff dimension of its harmonic current is smaller than or equal to ${d-1 \over d+2}$, in particular the harmonic current is singular with respect to the Lebesgue measure. We also show that the Hausdorff dimension of the harmonic current of the Jouanolou foliation of degree $2$ is equal to $1/4$, and that the same property holds for topologically conjugate foliations on $\mathbb P^2$.
Consider the four punctured sphere ${\mathbb{S}}_4^2$. Each choice of four traces, one for each puncture, determines a relative character variety for the representations of the fundamental group of ${\mathbb{S}}_4^2$ in ${\sf{SL}}_2(\mathbb{C})$. We classify the stationary probability measures for the action of the mapping class group ${\sf{Mod}}({\mathbb{S}}_4^2)$ on these character varieties.
Let $f$ be a holomorphic map of $\mathbb{C}\mathbb{P}^2$ of degree $d\geq 2$, let $T$ be its Green current and $\mu=T\wedge T$ be its equilibrium measure. We give a new proof of a theorem due to Dujardin asserting that $\mu \ll T\wedge\omega_{\mathbb{P}^2}$ implies $\lambda_2=\frac{1}{2} \log\ d$, where $\lambda_1 \geq \lambda_2$ are the Lyapunov exponents of $\mu$. Then, assuming $\mu\ll T\wedge\omega_{\mathbb{P}^2}$, we study slice measures $\nu :=T\wedge dd^c|W|^2$, where $W$ is a holomorphic local submersion. We give sufficient conditions on the Radon-Nikodym derivative of $\mu$ with respect to the trace measure $T\wedge\omega_{\mathbb{P}^2}$ ensuring $\mu=\nu$. The involved submersion $W$ comes from normal coordinates for the inverse branches of the iterates of $f$.
Let $\mathcal F$ be a holomorphic foliation on a compact Káhler surface with hyperbolic singularities and no foliation cycle. We prove that if the limit set of $\mathcal F$ has zero Lebesgue measure, then its complement is a modification of a Stein domain. The proof consists in building, in several steps, a metric of positive curvature for the normal bundle of $\mathcal F$ near the limit set. Then we construct a proper strictly plurisubharmonic exhaustion function for the complement of the limit set, by adapting Brunella's ideas to our singular context. The arguments hold more generally when the limit set is thin, a property relying on Brownian motion.
We study the structure and the Lyapunov exponents of the equilibrium measure of endomorphisms of $\mathbb P^k$ preserving a fibration. We extend the decomposition of the equilibrium measure obtained by Jonsson for polynomial skew products of $\mathbb C^2$. We also show that the sum of the sectional exponents satisfies a Bedford-Jonsson formula when the fibration is linear, and that this function is plurisubharmonic on families of fibered endomorphisms. In particular, the sectional part of the bifurcation current is a closed positive current on the parameter space.
We linearize the inverse branches of the iterates of holomorphic endomorphisms of CP(k) and thus overcome the lack of Koebe distortion theorem in this setting when k $\ge$ 2. We review several applications of this result in holomorphic dynamics.
LLet $f$ be a holomorphic endomorphism of $\mathbb P^ 2$ of degree $d \geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $ν$. We infer several applications. The first one shows that the currents $S$ containing a measure of entropy $h\_ν> \log d$ have a directional dimension $>2$, which answers a question by de Thélin-Vigny. The second application asserts that the Dujardin's semi-extremal endomorphisms are close to suspensions of one-dimensional Lattès maps. Finally, we obtain an upper bound for the dimension of the equilibrium measure, towards the formula conjectured by Binder-DeMarco.
We introduce a notion of stability for equilibrium measures in holomorphic families of endomorphisms of CP(k) and prove that it is equivalent to the stability of repelling cycles and equivalent to the existence of some measurable holomorphic motion of Julia sets which we call equilibrium lamination. We characterize the corresponding bifurcations by the strict subharmonicity of the sum of Lyapunov exponents or the instability of critical dynamics and analyze how repelling cycles may bifurcate. Our methods deeply exploit the properties of Lyapunov exponents and are based on ergodic theory and on pluripotential theory.
We classify complex projective surfaces with an automorphism of positive entropy for which the unique invariant measure of maximal entropy is absolutely continuous with respect to Lebesgue measure.
We focus on the topology and dynamics of minimal sets and Levi-flats in surfaces of general type. Our method relies on the ergodic theory of Riemann surfaces laminations: we use harmonic measures and Lyapunov exponents. Our first result establishes that minimal sets have large Hausdorff dimension when a leaf is simply connected. Our second result shows that the class of Anosov Levi-flats does not occur in surfaces of general type. In particular, by using rigidity results, we obtain that Levi-flats are not virtually diffeomorphic to unitary tangent bundles of hyperbolic compact surfaces, nor to hyperbolic torus bundles.
Let $f$ be an endomorphism of $\mathbb{CP}^k$ and $ν$ be an $f$-invariant measure with positive Lyapunov exponents $(λ_1,\...,λ_k)$. We prove a lower bound for the pointwise dimension of $ν$ in terms of the degree of $f$, the exponents of $ν$ and the entropy of $ν$. In particular our result can be applied for the maximal entropy measure $μ$. When $k=2$, it implies that the Hausdorff dimension of $μ$ is estimated by $\dim_{\cal H} μ\geq {\log d \over λ_1} + {\log d \over λ_2}$, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the $ν$-generic inverse branches of $f^n$ in $\mathbb{CP}^k$. Our tools are a volume growth estimate for the bounded holomorphic polydiscs in $\mathbb{CP}^k$ and a normalization theorem for the $ν$-generic inverse branches of $f^n$.
Let $f$ be an holomorphic endomorphism of $\mathbb{C}\mathbb{P}^k$. We construct by using coding techniques a class of ergodic measures as limits of non-uniform probability measures on preimages of points. We show that they have large metric entropy, close to $\log d^k$. We establish for them strong stochastic properties and prove the positivity of their Lyapunov exponents. Since they have large entropy, those measures are supported in the support of the maximal entropy measure of $f$. They in particular provide lower bounds for the Hausdorff dimension of the Julia set.
Let $f$ be an holomorphic endomorphism of $\mathbb{P}^k$ and $μ$ be its measure of maximal entropy. We prove an Almost Sure Invariance Principle for the systems $(\mathbb{P}^k,f,μ)$. Our class $\cal{U}$ of observables includes the Hölder functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map $ω: (Σ, s, ν) \to (\mathbb{P}^k,f,μ)$. We obtain the invariance principle for an observable $ψ$ on $(\mathbb{P}^k,f,μ)$ by applying Philipp-Stout's theorem for $χ= ψ\circ ω$ on $(Σ, s, ν)$. The invariance principle implies the Central Limit Theorem as well as several statistical properties for the class $\cal{U}$. As an application, we give a \emph{direct} proof of the absolute continuity of the measure $μ$ when it satisfies Pesin's formula. This approach relies on the Central Limit Theorem for the unbounded observable $\log \textsf{Jac} f \in \cal{U}$.
Let $f$ be a dominating meromorphic self-map of a compact Kähler manifold. Assume that the topological degree of $f$ is larger than the other dynamical degrees. We give estimates of the dimension of the equilibrium measure of $f$, which involve the Lyapounov exponents.
We show that the Lattes endomorphisms are the only holomorphic endomorphisms of the complex k-dimensional projective space whose measure of maximal entropy is absolutely continuous with respect to the Lebesgue measure. As a consequence, Lattes endomorphisms are also characterized by other extremal properties as the maximality of the Hausdorff dimension of their measure of maximal entropy or the minimality of their Liapounov exponents. Our proof uses a linearization method which is of independant interest and a previous characterization by the regularity of the Green current.
Let f an holomorphic endomorphism of CP(k) with degree larger than 2. We show that if the Green measure of f is not singular, then f is rigid : it is a Lattes example. The proof relies on a linearization property of the iterates of f, along typical orbits. This property allow us to regularize the Green current, and to prove the rigidity. -- -- -- Soit f un endomorphisme holomorphe de CP(k) de degre plus grand que 2. Nous montrons que si la mesure de Green de f n'est pas singuliere, alors f est tres rigide : c'est un exemple de Lattes. La demonstration repose sur une propriete de linearisation des iteres de f le long d'orbites typiques. Cette propriete nous permet de "regulariser" le courant de Green, et d'en deduire la rigidite.