arXiv subjects

Lucas Kaufmann

Publications and source records attributed to Lucas Kaufmann.

14 recordsLinked to original sources

arXiv2024

Exponential equidistribution of periodic points for endomorphisms of $\mathbb P^k$

Let $f$ be a holomorphic endomorphism of $\mathbb P^k$ of algebraic degree $d\geq 2$. We show that the periodic points of $f$ of period $n$ equidistribute towards the equilibrium measure of $f$ exponentially fast as $n$ tends to infinity. This quantifies a theorem of Lyubich for $k=1$ and of Briend-Duval for $k\geq 2$. A byproduct of our proof is the existence of a large number of periodic cycles in the small Julia set with large multipliers.

math.DS↗

arXiv2024

Global Chern currents of coherent sheaves and Baum Bott currents

We provide global extensions of previous results about representations of characteristic classes of coherent analytic sheaves and of Baum-Bott residues of holomorphic foliations. We show in the first case that they can be represented by currents with support on the support of the given coherent analytic sheaf, and in the second case, by currents with support on the singular set of the foliation. In previous works, we have constructed such representatives provided global resolutions of the appropriate sheaves existed. In this article, we show that the definition of Chern classes of Green and the associated techniques, which work on arbitrary complex manifolds without any assumption on the existence of global resolutions, may be combined with our previous constructions to yield the desired representatives. We also prove a transgression formula for such representatives, which is new even in the case when global resolutions exist. More precisely, the representatives depend on local resolutions of the sheaf, and on choices of metrics and connections on these bundles, i.e., the currents for two different choices differ by a current of the form $dN$, where $N$ is an explicit current, which in the first case above has support on the support of the given coherent analytic sheaf, and in the second case above has support on the singular set of the foliation.

math.CV↗

arXiv2024

Baum-Bott residue currents

Let $\mathscr{F}$ be a holomorphic foliation of rank $κ$ on a complex manifold $M$ of dimension $n$, let $Z$ be a compact connected component of the singular set of $\mathscr{F}$, and let $Φ\in \mathbb C[z_1,\ldots,z_n]$ be a homogeneous symmetric polynomial of degree $\ell$ with $n-κ< \ell \leq n$. Given a locally free resolution of the normal sheaf of $\mathscr{F}$, equipped with Hermitian metrics and certain smooth connections, we construct an explicit current $R^Φ_Z$ with support on $Z$ that represents the Baum-Bott residue $\text{res}^Φ(\mathscr{F}; Z)\in H_{2n-2\ell}(Z, \mathbb C)$ and is obtained as the limit of certain smooth representatives of $\text{res}^Φ(\mathscr{F}; Z)$. If the connections are $(1,0)$-connections and $\text{codim} Z\geq \ell$, then $R^Φ_Z$ is independent of the choice of metrics and connections. When $\mathscr{F}$ has rank one we give a more precise description of $R^Φ_Z$ in terms of so-called residue currents of Bochner-Martinelli type. In particular, when the singularities are isolated, we recover the classical expression of Baum-Bott residues in terms of Grothendieck residues.

math.CV↗

arXiv2023

Random walks on SL_2(C): spectral gap and limit theorems

We obtain various new limit theorems for random walks on SL_2(C) under low moment conditions. For non-elementary measures with a finite second moment, we prove a Local Limit Theorem for the norm cocycle, yielding the optimal version of a theorem of E. Le Page. For measures with a finite third moment, we obtain the Local Limit Theorem for the matrix coefficients, improving a recent result of Grama-Quint-Xiao and the authors, and Berry-Esseen bounds with optimal rate $O(1 / \sqrt n)$ for the norm cocycle and the matrix coefficients. The main tool is a detailed study of the spectral properties of the Markov operator and its purely imaginary perturbations acting on different function spaces. We introduce, in particular, a new function space derived from the Sobolev space $W^{1,2}$ that provides uniform estimates.

math.PR↗

arXiv2022

Berry-Esseen bounds with targets and Local Limit Theorems for products of random matrices

Let $μ$ be a probability measure on $\text{GL}_d(\mathbb R)$ and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$'s are i.i.d.'s with law $μ$. We study statistical properties of random variables of the form $$σ(S_n,x) + u(S_n x),$$ where $x \in \mathbb P^{d-1}$, $σ$ is the norm cocycle and $u$ belongs to a class of admissible functions on $\mathbb P^{d-1}$ with values in $\mathbb R \cup \{\pm \infty\}$. Assuming that $μ$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we obtain optimal Berry-Esseen bounds and the Local Limit Theorem for such variables using a large class of observables on $\mathbb R$ and Hölder continuous target functions on $\mathbb P^{d-1}$. As particular cases, we obtain new limit theorems for $σ(S_n,x)$ and for the coefficients of $S_n$.

math.PR↗

arXiv2021

Berry-Esseen bound and Local Limit Theorem for the coefficients of products of random matrices

Let $μ$ be a probability measure on $\text{GL}_d(\mathbb{R})$ and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$ are i.i.d. with law $μ$. Under the assumptions that $μ$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we prove a Berry-Esseen bound with the optimal rate $O(1/\sqrt n)$ and a general Local Limit Theorem for the coefficients of $S_n$.

math.PR↗

arXiv2021

Decay of Fourier coefficients for Furstenberg measures

Let $ν$ be the Furstenberg measure associated with a non-elementary probability measure $μ$ on SL_2(R). We show that, when $μ$ has a finite second moment, the Fourier coefficients of $ν$ tend to zero at infinity. In other words, $ν$ is a Rajchman measure. This improves a recent result of Jialun Li.

math.PR↗

arXiv2021

Intersection of (1,1)-currents and the domain of definition of the Monge-Ampere operator

We study the Monge-Amp\` ere operator within the framework of Dinh-Sibony's intersection theory defined via density currents. We show that if $u$ is a plurisubharmonic function belonging to the Blocki-Cegrell class, then the Dinh-Sibony $n$-fold self-product of $\text{dd}^c u$ exists and coincides with the classically defined Monge-Ampère measure $(\text{dd}^c u)^n$.

math.CV↗

arXiv2019

Random products of matrices: a dynamical point of view

We study random products of matrices in SL_2(C) from the point of view of holomorphic dynamics. For non-elementary measures with finite first moment we obtain the exponential convergence towards the stationary measure in Sobolev norm. As a consequence we obtain the exponentially fast equidistribution of forward images of points towards the stationary measure. We also give a new proof of the Central Limit Theorem for the norm cocycle under a second moment condition, originally due to Benoist-Quint, and obtain some general regularity results for stationary measures.

math.CV↗

arXiv2018

Dynamics of holomorphic correspondences on Riemann Surfaces

We study the dynamics of holomorphic correspondences $f$ on a compact Riemann surface $X$ in the case, so far not well understood, where $f$ and $f^{-1}$ have the same topological degree. Under a mild and necessary condition that we call non weak modularity, $f$ admits two canonical probability measures $μ^+$ and $μ^-$ which are invariant by $f^*$ and $f_*$ respectively. If the critical values of $f$ (resp. $f^{-1}$) are not periodic, the backward (resp. forward) orbit of any point $a \in X$ equidistributes towards $μ^+$ (resp. $μ^-$), uniformly in $a$ and exponentially fast.

math.DS↗

arXiv2018

Density and intersection of (1,1)-currents

We study density currents associated with a collection of positive closed (1,1)-currents. We prove that the density current is unique and determined by the usual wedge product in some classical situations including the case where the currents have bounded potentials. As an application, we compare density currents with the non-pluripolar product and the Andersson-Wulcan product. We also analyse some situations where the wedge product is not well-defined but the density can be explicitly computed.

math.CV↗

arXiv2016

Self-intersection of foliated cycles on complex manifolds

Let X be a compact Kahler manifold and let T be a foliated cycle directed by a transversally Lipschitz lamination on X . We prove that the self-intersection of the cohomology class of T vanishes as long as T does not contain currents of integration along compact manifolds. As a consequence we prove that transversally Lipschitz laminations of low codimension in certain manifolds, e.g. projective spaces, do not carry any foliated cycles except those given by integration along compact leaves.

math.CV↗

arXiv2015

Commuting pairs of endomorphisms of P^2

We consider commuting pairs of holomorphic endomorphisms of P^2 with disjoint sequence of iterates. The remaining case to be studied is when their degrees coincide after some number of iterations. We show in this case that they are either commuting Lattès maps or commuting homogeneous polynomial maps of C^2 inducing a Lattès map on the hyperplane at infinity.

math.CV↗

arXiv2014

Some properties of plurisubharmonic functions

Two properties of plurisubharmonic functions are proven. The first result is a Skoda type integrability theorem with respect to a Monge-Ampère mass with Hölder continuous potential. The second one says that locally, a p.s.h. function is $k$-Lipschitz outside a set of Lebesgue measure smaller that $c/k^2$.

math.CV↗