Pluripotential theory on the support of closed positive currents and applications to dynamics in Cn\mathbb{C}^n

We extend certain classical theorems in pluripotential theory to a class of functions defined on the support of a $(1,1)$-closed positive current $T$, analogous to plurisubharmonic functions, called $T$-plurisubharmonic functions. These functions are defined as limits, on the support of $T$, of sequences of plurisubharmonic functions decreasing on this support. In particular, we show that the poles of such functions are pluripolar sets. We also show that the maximum principle and the Hartogs's theorem remain valid in a weak sense. We study these functions by means of a class of measures, so-called "pluri-Jensen measures", about which we prove that they are numerous on the support of $(1,1)$-closed positive currents. We also obtain, for any fat compact set, an expression of its relative Green's function in terms of an infimum of an integral over a set of pluri-Jensen measures. We then deduce, by means of these measures, a characterization of the polynomially convex fat compact sets, as well as a characterization of pluripolar sets, and the fact that the support of a closed positive $(1,1)$-current is nowhere pluri-thin. In the second part of this article, these tools are used to study dynamics of a certain class of automorphisms of $\mathbb{C}^n$ which naturally generalize H\'enon's automorphisms of $\mathbb{C}^2$. First we study the geometry of the support of canonical invariant currents. Then we obtain an equidistribution result for the convergence of pull-back of certain measures towards an ergodic invariant measure, with compact support

{\L}S condition for filled Julia sets in C\mathbb{C}

In this article, we derive an inequality of {\L}ojasiewicz-Siciak type for certain sets arising in the context of the complex dynamics in dimension 1. More precisely, if we denote by $dist$ the euclidian distance in $\mathbb{C}$, we show that the Green function $G_K$ of the filled Julia set $K$ of a polynomial such that $\mathring{K}\neq \emptyset$ satisfies the so-called {\L}S condition $\displaystyle G_A\geq c\cdot dist(\cdot, K)^{c'}$ in a neighborhood of $K$, for some constants $c,c'>0$. Relatively few examples of compact sets satisfying the {\L}S condition are known. Our result highlights an interesting class of compact sets fulfilling this condition. The fact that filled Julia sets satisfy the {\L}S condition may seem surprising, since they are in general very irregular. In order to prove our main result, we define and study the set of obstruction points to the {\L}S condition. We also prove, in dimension $n\geq 1$, that for a polynomially convex and L-regular compact set of non empty interior, these obstruction points are rare, in a sense which will be specified

Dynamics of non cohomologically hyperbolic automorphisms of C 3

We study the dynamics of a family of non cohomologically hyperbolic automorphisms f of C^3. We construct a compactification X of C^3 where their extensions are algebraically stable. We finally construct canonical invariant closed positive (1, 1)-currents for f^* , f_* and we study several of their properties. Moreover, we study the well defined current T_f ∧ T_{f^{−1}} and the dynamics of f on its support. Then we construct an invariant positive measure T_f ∧T_{f^{−1}} ∧φ_∞ , where φ_∞ is a function defined on the support of T_f ∧ T_{f^{−1}}. We prove that the support of this measure is compact and pluripolar. We prove also that this measure is canonical, in some sense that will be precised

A new algorithm for graph center computation and graph partitioning according to the distance to the center

We propose a new algorithm for finding the center of a graph, as well as the rank of each node in the hierarchy of distances to the center. In other words, our algorithm allows to partition the graph according to nodes distance to the center. Moreover, the algorithm is parallelizable. We compare the performances of our algorithm with the ones of Floyd-Warshall algorithm, which is traditionally used for these purposes. We show that, for a large variety of graphs, our algorithm outperforms the Floyd-Warshall algorithm

Epidemic Dynamics via Wavelet Theory and Machine Learning with Applications to Covid-19

We introduce the concept of epidemic-fitted wavelets which comprise, in particular, as special cases the number I(t) of infectious individuals at time t in classical SIR models and their derivatives. We present a novel method for modelling epidemic dynamics by a model selection method using wavelet theory and, for its applications, machine learning-based curve fitting techniques. Our universal models are functions that are finite linear combinations of epidemic-fitted wavelets. We apply our method by modelling and forecasting, based on the Johns Hopkins University dataset, the spread of the current Covid-19 (SARS-CoV-2) epidemic in France, Germany, Italy and the Czech Republic, as well as in the US federal states New York and Florid

LS condition for filled Julia sets in C

International audienceIn this article we derive an inequality of Lojasiewicz-Siciak type for certain sets arising in the context of the complex dynamics in dimension 1. More precisely, if we denote by dist the euclidian distance in C, we show that the Green function G_K of the filled Julia set K of a polynomial such that K is of non-empty interior satisfies the so-called LS condition G_A ≥ c · dist(·, K)^c' in a neighborhood of K, for some constants c, c' > 0. Relatively few examples of compact sets satisfying the LS condition are known. Our result highlights an interesting class of compact sets fulfilling this condition. For instance, this is the case for the filled Julia sets of quadratic polynomials of the form z → z^2 + a, provided that the parameter a is parabolic, hyperbolic or Siegel. The fact that filled Julia sets satisfy the LS condition may seem surprising, since they are in general very irregular and sometimes they have cusps. However, we provide an explicit example of a curve which has a cusp and satisfies the LS condition. In order to prove our main result, we define and study the set of obstruction points to the LS condition. We also prove, in dimension n ≥ 1, that for a polynomially convex and L-regular compact set of non empty interior, these obstruction points are rare, in a sense which will be specified

Dynamics of non cohomologically hyperbolic automorphisms of C 3

We study the dynamics of a family of non cohomologically hyperbolic automorphisms f of C^3. We construct a compactification X of C^3 where their extensions are algebraically stable. We finally construct canonical invariant closed positive (1, 1)-currents for f^* , f_* and we study several of their properties. Moreover, we study the well defined current T_f ∧ T_{f^{−1}} and the dynamics of f on its support. Then we construct an invariant positive measure T_f ∧T_{f^{−1}} ∧φ_∞ , where φ_∞ is a function defined on the support of T_f ∧ T_{f^{−1}}. We prove that the support of this measure is compact and pluripolar. We prove also that this measure is canonical, in some sense that will be precised