arXiv subjects

Thomas Gauthier

Publications and source records attributed to Thomas Gauthier.

At least 19 recordsLinked to original sources

arXiv2025

Quantitative equidistribution of periodic points for rational maps

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 $\mathbb{Q}$.

math.DS↗

arXiv2024

Julia sets and bifurcation loci

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.

math.DS↗

arXiv2024

H\"older estimates and uniformity in arithmetic dynamics

In this note we study common preperiodic points of rational maps of the Riemann Sphere. We show that given any degrees $d_1,d_2\geq2$, outside a Zariski closed subset of the space of pairs of rational maps $(f,g)$ of degree $d_1$ and $d_2$ respectively, the maps $f$ and $g$ share at most a uniformly bounded number of common preperiodic points. This generalizes a result of DeMarco and Mavraki to maps of possibly different degrees. Our main contribution is the use of H\"older properties of the Green function of a rational map to obtain height estimates.

math.DS↗

arXiv2024

Domain Adaptation of Llama3-70B-Instruct through Continual Pre-Training and Model Merging: A Comprehensive Evaluation

We conducted extensive experiments on domain adaptation of the Meta-Llama-3-70B-Instruct model on SEC data, exploring its performance on both general and domain-specific benchmarks. Our focus included continual pre-training (CPT) and model merging, aiming to enhance the model's domain-specific capabilities while mitigating catastrophic forgetting. Through this study, we evaluated the impact of integrating financial regulatory data into a robust language model and examined the effectiveness of our model merging techniques in preserving and improving the model's instructive abilities. The model is accessible at hugging face: https://huggingface.co/arcee-ai/Llama-3-SEC-Base, arcee-ai/Llama-3-SEC-Base. This is an intermediate checkpoint of our final model, which has seen 20B tokens so far. The full model is still in the process of training. This is a preprint technical report with thorough evaluations to understand the entire process.

cs.CL↗

arXiv2023

Variation of canonical heights of subvarieties for polarized endomorphisms

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.

math.NT↗

arXiv2023

The Geometric Dynamical Northcott Property in the quadratic family

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.

math.DS↗

arXiv2023

Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach

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.

math.DS↗

arXiv2023

Complex Dynamics of birational maps of $\mathbb{P}^k$ defined over a number field

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.

math.DS↗

arXiv2022

Entire or rational maps with integer multipliers

Let $\mathcal{O}_{K}$ be the ring of integers of an imaginary quadratic field $K$. Recently, Ji and Xie proved that every rational map $f \colon \widehat{\mathbb{C}} \rightarrow \widehat{\mathbb{C}}$ of degree $d \geq 2$ whose multipliers all lie in $\mathcal{O}_{K}$ is a power map, a Chebyshev map or a Latt\`{e}s map. Their proof relies on a result from non-Archimedean dynamics obtained by Rivera-Letelier. In the present note, we show that one can avoid using this result by considering a differential equation instead. Our proof of Ji and Xie's result also applies to the case of entire maps. Thus, we also show that every nonaffine entire map $f \colon \mathbb{C} \rightarrow \mathbb{C}$ whose multipliers all lie in $\mathcal{O}_{K}$ is a power map or a Chebyshev map.

math.DS↗

arXiv2022

Self-Supervised 3D Monocular Object Detection by Recycling Bounding Boxes

Modern object detection architectures are moving towards employing self-supervised learning (SSL) to improve performance detection with related pretext tasks. Pretext tasks for monocular 3D object detection have not yet been explored yet in literature. The paper studies the application of established self-supervised bounding box recycling by labeling random windows as the pretext task. The classifier head of the 3D detector is trained to classify random windows containing different proportions of the ground truth objects, thus handling the foreground-background imbalance. We evaluate the pretext task using the RTM3D detection model as baseline, with and without the application of data augmentation. We demonstrate improvements of between 2-3 % in mAP 3D and 0.9-1.5 % BEV scores using SSL over the baseline scores. We propose the inverse class frequency re-weighted (ICFW) mAP score that highlights improvements in detection for low frequency classes in a class imbalanced dataset with long tails. We demonstrate improvements in ICFW both mAP 3D and BEV scores to take into account the class imbalance in the KITTI validation dataset. We see 4-5 % increase in ICFW metric with the pretext task.

cs.CV↗

arXiv2022

Simulation-to-Reality domain adaptation for offline 3D object annotation on pointclouds with correlation alignment

Annotating objects with 3D bounding boxes in LiDAR pointclouds is a costly human driven process in an autonomous driving perception system. In this paper, we present a method to semi-automatically annotate real-world pointclouds collected by deployment vehicles using simulated data. We train a 3D object detector model on labeled simulated data from CARLA jointly with real world pointclouds from our target vehicle. The supervised object detection loss is augmented with a CORAL loss term to reduce the distance between labeled simulated and unlabeled real pointcloud feature representations. The goal here is to learn representations that are invariant to simulated (labeled) and real-world (unlabeled) target domains. We also provide an updated survey on domain adaptation methods for pointclouds.

cs.CV↗

arXiv2022

When do two rational functions have locally biholomorphic Julia sets?

In this article we address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics: under which conditions does there exist a local biholomorphism between the Julia sets of two given one-dimensional rational maps? In particular we find criteria ensuring that such a local isomorphism is induced by an algebraic correspondence. This extends and unifies classical results due to Baker, Beardon, Eremenko, Levin, Przytycki and others. The proof involves entire curves and positive currents.

math.DS↗

arXiv2021

Good height functions on quasi-projective varieties: equidistribution and applications in dynamics

In the present article, we define a notion of good height functions on quasi-projective varieties $V$ defined over number fields and prove an equidistribution theorem of small points for such height functions. Those good height functions are defined as limits of height functions associated with semi-positive adelic metrization on big and nef $\mathbb{Q}$-line bundles on projective models of $V$ satisfying mild assumptions. Building on a recent work of the author and Vigny as well as on a classical estimate of Call and Silverman, and inspiring from recent works of K\"uhne and Yuan and Zhang, we deduce the equidistribution of generic sequence of preperiodic parameters for families of polarized endomorphisms with marked points.

math.NT↗

arXiv2021

Exploring 2D Data Augmentation for 3D Monocular Object Detection

Data augmentation is a key component of CNN based image recognition tasks like object detection. However, it is relatively less explored for 3D object detection. Many standard 2D object detection data augmentation techniques do not extend to 3D box. Extension of these data augmentations for 3D object detection requires adaptation of the 3D geometry of the input scene and synthesis of new viewpoints. This requires accurate depth information of the scene which may not be always available. In this paper, we evaluate existing 2D data augmentations and propose two novel augmentations for monocular 3D detection without a requirement for novel view synthesis. We evaluate these augmentations on the RTM3D detection model firstly due to the shorter training times . We obtain a consistent improvement by 4% in the 3D AP (@IoU=0.7) for cars, ~1.8% scores 3D AP (@IoU=0.25) for pedestrians & cyclists, over the baseline on KITTI car detection dataset. We also demonstrate a rigorous evaluation of the mAP scores by re-weighting them to take into account the class imbalance in the KITTI validation dataset.

cs.CV↗

arXiv2020

The Geometric Dynamical Northcott Property For Regular Polynomial Automorphisms of the Affine Plane

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.

math.DS↗

arXiv2020

The arithmetic of polynomial dynamical pairs

We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an "unlikely intersection" statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this result the dynamical Andr\'e-Oort conjecture for curves in the moduli space of polynomials, by describing one-dimensional families in this parameter space containing infinitely many post-critically finite parameters.

math.DS↗

arXiv2020

Parametric Lyapunov exponents

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 \'{S}wi\c{a}tek in the unicritical family, using Makarov theorem. Our methods are based instead on ideas of laminar currents theory.

math.DS↗

arXiv2019

The Geometric Dynamical Northcott and Bogomolov Properties

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.

math.DS↗