The Lefschetz zeta function associated to a continuous self-map f of a compact manifold is a rational function P/Q. According to the parity of the degrees of the polynomials P and Q, we analyze when the set of periodic points of f is infinite and when the topological entropy is positive.

Les algèbres différentielles sont apparues comme des outils commodes ou même inévitables pour exprimer les symétries continues, exactes ou brisées, suivant la situation physique envisagée, dans le cadre de l’algorithme de Feynman de la théorie quantique des champs perturbative. Les algèbres de courants, les théories de Yang-Mills, la première quantification de la corde, sont proposées comme exemples classiques.

We provide a detailed treatment of the Camassa-Holm (CH) hierarchy with special emphasis on its algebro-geometric solutions. In analogy to other completely integrable hierarchies of soliton equations such as the KdV or AKNS hierarchies, the CH hierarchy is recursively constructed by means of a basic polynomial formalism invoking a spectral parameter. Moreover, we study Dubrovin-type equations for auxiliary divisors and associated trace formulas, consider the corresponding algebro-geometric initial...

We show that piecewise smooth maps with a finite number of pieces of monotonicity and nowhere vanishing Lipschitz continuous derivative can have only period doubling solenoids. The proof is based on the fact that if $p_1<...<p_n$ is a periodic orbit of a continuous map f then there is a union set $q_1,...,q_{n-1}$ of some periodic orbits of f such that $p_i<q_i<p_{i+1}$ for any i.

Let (X, μ) be a σ-finite measure space and let τ be an ergodic invertible measure preserving transformation. We study the a.e. convergence of the Cesàro-α ergodic averages associated with τ and the boundedness of the corresponding maximal operator in the setting of Lp,q(wdμ) spaces.

It is well-known that a probability measure $\mu$ on the circle $𝕋$ satisfies $\parallel {\mu }^n{*f-\int f\mathrm{d}m\parallel }_p\to 0$ for every $f\in L_p$, every (some) $p\in [1,\infty )$, if and only if $|\widehat{\mu }\left(n\right)|lt;1$ for every non-zero $n\in \mathbb{Z}$ ($\mu$ is strictly aperiodic). In this paper we study the a.e. convergence of ${\mu }^n*f$ for every $f\in L_p$ whenever $pgt;1$. We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of $\mu$, for the strong sweeping out property (existence of a Borel set $B$ with $lim\; sup{\mu }^n*1_B=1$ a.e. and $lim\; inf{\mu }^n*1_B=0$ a.e.). The results are extended to general compact Abelian groups $G$ with Haar...

We show that some results of Gaposhkin about a.e. convergence of series associated to a unitary operator U acting on L²(X,Σ,μ) (μ is a σ-finite measure) may be extended to the case where U is an invertible power-bounded operator acting on $L^p(X,\Sigma ,\mu )$, p > 1. The proofs make use of the spectral integration initiated by Berkson-Gillespie and, more specifically, of recent results of the author.

We extend the convergence method introduced in our works [8–10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in ${ℝ}^d$ to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in ${ℝ}^3$. The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated to the equation,...