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

Quantales of open groupoids

It is well known that inverse semigroups are closely related to \'etale groupoids. In particular, it has recently been shown that there is a (non-functorial) equivalence between localic \'etale groupoids, on one hand, and complete and infinitely distributive inverse semigroups (abstract complete pseudogroups), on the other. This correspondence is mediated by a class of quantales, known as inverse quantal frames, that are obtained from the inverse semigroups by a simple join completion that yields an equivalence of categories. Hence, we can regard abstract complete pseudogroups as being essentially ``the same'' as inverse quantal frames, and in this paper we exploit this fact in order to find a suitable replacement for inverse semigroups in the context of open groupoids that are not necessarily \'etale. The interest of such a generalization lies in the importance and ubiquity of open groupoids in areas such as operator algebras, differential geometry and topos theory, and we achieve it by means of a class of quantales, called open quantal frames, which generalize inverse quantal frames and whose properties we study in detail. The resulting correspondence between quantales and open groupoids is not a straightforward generalization of the previous results concerning \'etale groupoids, and it depends heavily on the existence of inverse semigroups of local bisections of the quantales involved.Comment: 55 page

Introduction to Pylog

PyLog is a minimal experimental proof assistant based on linearised natural deduction for intuitionistic and classical first-order logic extended with a comprehension operator. PyLog is interesting as a tool to be used in conjunction with other more complex proof assistants and formal mathematics projects (such as Coq and Coq-based projects). Proof assistants based on dependent type theory are at once very different and profoundly connected to the one employed by Pylog via the Curry-Howard correspondence. The Tactic system of Coq presents us with a top-down approach to proofs (finding a term inhabiting a given type via backtracking the rules, typability and type-inference being automated) whilst the classical approach of Pylog follows how mathematical proofs are usually written. Pylog should be further developed along the lines of Coq in particular through the introduction of many "micro-automatisations" and a nice IDE

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

From Heterogeneous expectations to exchange rate dynamic:

The purpose of this paper is to analyze how heterogeneous behaviors of agents influence the exchange rates dynamic in the short and long terms. We examine how agents use the information and which kind of information, in order to take theirs decisions to form an expectation of the exchange rate. We investigate a methodology based on interactive agents simulations, following the Santa Fe Artificial Stock Market. Each trader is modeled as an autonomous, interactive agent and the aggregation of their behavior results in foreign exchange market dynamic. Genetic algorithm is the tool used to compute agents, and the simulated market tends to replicate the real EUR/USD exchange rate market. We consider six kinds of agents with pure behavior: fundamentalists, positive feedback traders and negative ones, naive traders, news traders (positive and negative). To reproduce stylized facts of the exchange rates dynamic, we conclude that the key factor is the correct proportion of each agents type, whiteout any need of mimetic behaviors, adaptive agents or pure noisy agentsexchange rates dynamic, heterogeneous interactive agents behaviour, genetic algorithm, learning process

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