Search

In the context of holomorphic families of ${\mathbb P}^k$ endomorphisms, we show that various notions of stability are equivalent. This allows us to both extend and simplify the architecture of the proof of certain results of [BBD], Comment: 7 pages

We present a renormalization lemma for certain maps defined on the unit disc of C and taking values in some metric space. We show that the classical renormalization lemmas of Zalcman and Miniowitz can be deduced from our lemma. We also use it to establish a general normality statement for the Pinchuk's scaling method in C^2 and, incidentally, reprove the Catlin's estimates for the Kobayashi metric in finite type domains.

Ferroelectric nematic liquid crystals exhibit unique non-linear optical properties, with the potential to become transformative materials for photonic applications. A promising direction relies on the fabrication of tailored polar orientational patterns via photoalignment, thus shaping the non-linear optical susceptibility through thin slabs of the ferroelectric fluid. Here, we explore the fabrication of 2D periodic SHG active arrays in ferroelectric nematic fluids, for different materials, cell thicknesses and motifs. Based on polarizing optical microscopy observations in combination with optical simulations, second harmonic generation microscopy and interferometry, the 3D structure of the motifs are revealed. Two different 2D periodic patterns are explored, showing that the balance between flexoelectric and electrostatic energy can lead to different domain structures, an effect which is rooted in the difference between the flexoelectric properties of the materials. It is shown that by combining the surface-inscribed alignment with different spontaneous degrees of twist, 2D SHG active arrays can be obtained in the micrometre scale, in which adjacent areas exhibit maximum SHG signals at opposite angles., Comment: 24 pages, 5 Images main, 15 supplementary images

Association rule mining is one of the most studied research fields of data mining, with applications ranging from grocery basket problems to explainable classification systems. Classical association rule mining algorithms have several limitations, especially with regards to their high execution times and number of rules produced. Over the past decade, neural network solutions have been used to solve various optimization problems, such as classification, regression or clustering. However there are still no efficient way association rules using neural networks. In this paper, we present an auto-encoder solution to mine association rule called ARM-AE. We compare our algorithm to FP-Growth and NSGAII on three categorical datasets, and show that our algorithm discovers high support and confidence rule set and has a better execution time than classical methods while preserving the quality of the rule set produced.

The recently discovered ferroelectric nematic liquids incorporate to the functional combination of fluidity, processability and anisotropic optical properties of nematic liquids, an astonishing range of physical properties derived from the phase polarity. Among them, the remarkably large values of second order optical susceptibility encourage to exploit these new materials for non-linear photonic applications. Here we show that photopatterning of the alignment layer can be used to structure polarization patterns. To do so, we take advantage of the flexoelectric effect and design splay structures that geometrically define the polarization direction. We demonstrate the creation of periodic polarization structures and the possibility of guiding polarization by embedding splay structures in uniform backgrounds. The demonstrated capabilities of polarization patterning, open a promising new route for the design of ferroelectric nematic based photonic structures and their exploitation., Comment: 10 pages manuscript, 7 Figures, 36 pages Supplementary Information, 25 Figures

In the context of holomorphic families of endomorphisms of $\mathbb P^k$, we prove that stability in the sense of [BBD18] is equivalent to a summability condition for the post-critical mass and to the convergence of a suitably defined ramification current. This allows us to both simplify the approach of [BBD18] and better relate stability to post-critical normality., Comment: 15 pages

Association rule mining is one of the most studied research fields of data mining, with applications ranging from grocery basket problems to highly explainable classification systems. Classical association rule mining algorithms have several flaws especially with regards to their execution times, memory usage and number of rules produced. An alternative is the use of meta-heuristics, which have been used on several optimisation problems. This paper has two objectives. First, we provide a comparison of the performances of state-of-the-art meta-heuristics on the association rule mining problem. We use the multi-objective versions of those algorithms using support, confidence and cosine. Second, we propose a new algorithm designed to mine rules efficiently from massive datasets by exploring a large variety of solutions, akin to the explosion of species diversity of the Cambrian Explosion. We compare our algorithm to 20 benchmark algorithms on 22 real-world data-sets, and show that our algorithm present good results and outperform several state-of-the-art algorithms.

We construct transcendental automorphims of $\mathbb{C}^{2}$ having an unbounded and regular Siegel domain., Comment: 10 pages, no figures

Working within the polynomial quadratic family, we introduce a new point of view on bifurcations which naturally allows to see the seat of bifurcations as the projection of a Julia set of a complex dynamical system in dimension three. We expect our approach to be extendable to other holomorphic families of dynamical systems.

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.

We give an estimate for the Hausdorff dimension of the bifurcation locus of a family of endomorphisms of $\mathbb{P}^k (\mathbb{C})$. This dimension is maximal near isolated Latt\`es examples., Comment: in French

We build measurable holomorphic motions for Julia sets of holomorphic families of endomorphisms of CP(k) under various equivalent notions of stability., Comment: 15 pages

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., Comment: 43 pages, to appear in Ann. Sci. Ecole Norm. Sup

We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive (1, 1)-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.

The aim of these lectures is the study of bifurcations within holomorphic families of polynomials or rational maps by mean of ergodic and pluripotential theoretic tools., Comment: 109 pages

In the family of degree $d$ polynomials the hypersurfaces defined by the existence of a cycle of period $n$ and multiplier $e^{i\theta}$ are shown to equiditribute the bifurcation current.

In this paper, the existence of parabolic boundary points of certain convex domains in $\mathbb C^2$ is given. On the other hand, the nonexistence of parabolic boundary points of infinite type of certain domains in $\mathbb C^2$ is also shown., Comment: 19 pages

Bifurcation loci in the moduli space of degree $d$ rational maps are shaped by the hypersurfaces defined by the existence of a cycle of period $n$ and multiplier 0 or $e^{i\theta}$. Using potential-theoretic arguments, we establish two equidistribution properties for these hypersurfaces with respect to the bifurcation current. To this purpose we first establish approximation formulas for the Lyapunov function. In degree $d=2$, this allows us to build holomorphic motions and show that the bifurcation locus has a lamination structure in the regions where an attracting basin of fixed period exists.

We establish a Poincar\'e-Dulac theorem for sequences (G_n)_n of holomorphic contractions whose differentials d_0 G_n split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of contraction. Our results are actually stated in the framework of bundle maps. Such sequences of holomorphic contractions appear naturally as iterated inverse branches of endomorphisms of CP(k). In this context, our normalization result allows to precisely estimate the distortions of ellipsoids along typical orbits. As an application, we show how the Lyapunov exponents of the equilibrium measure are approximated in terms of the multipliers of the repulsive cycles., Comment: 29 pages, references added, to appear in Ann. Inst. Fourier

We establish a formula for the sum of the Lyapounov exponents of an holomorphic endomorphism of ${\bf P}^k$. For an holomorphic family of such endomorphisms we define the {\em bifurcation current} as $dd^cL$ and show that it vanishes when the repulsive cycles move holomorphically. We then prove a formula which relates this current with the interaction between the Green current and the current of integration on the critical set. In the 1-dimensional case (i.e. for ${\bf P}^1$) we find a geometrical description of the support of this current and its powers. Finally we introduce the {\em bifurcation measure} giving some applications. This last part may be interpreted as a generalization of Mane-Sad-Sullivan theory based on pluri-potentialist methods., Comment: 32 pages

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., Comment: a paraitre in Comment. Math. Helv

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., Comment: 19 pages, in french