In LaSalle's book "The Stability of Dynamical Systems", the author gives four conditions which imply that the origin of a discrete dynamical system defined on ℝ is a global attractor, and proposes to study the natural extensions of these conditions in ℝⁿ. Although some partial results are obtained in previous papers, as far as we know, the problem is not completely settled. In this work we first study the four conditions and prove that just one of them implies that the origin is a global attractor...

Let $X$ be a connected closed manifold and $f$ a self-map on $X$. We say that $f$ is almost quasi-unipotent if every eigenvalue $\lambda$ of the map $f_{*k}$ (the induced map on the $k$-th homology group of $X$) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of $\lambda$ as eigenvalue of all the maps $f_{*k}$ with $k$ odd is equal to the sum of the multiplicities of $\lambda$ as eigenvalue of all the maps $f_{*k}$ with $k$ even. We prove that if $f$ is $C^1$ having finitely many periodic points all of them hyperbolic,...

Soit $f$ un difféomorphisme lisse de ${ℝ}_{+}$ fixant seulement l’origine, et ${𝒵}^r$ son centralisateur dans le groupe des difféomorphismes ${𝒞}^r$. Des résultat classiques de Kopell et Szekeres montrent que ${𝒵}^1$ est toujours un groupe à un paramètre. En revanche, Sergeraert a construit un $f$ dont le centralisateur ${𝒵}^{\infty }$ est réduit au groupe des itérés de $f$. On présente ici le résultat principal de [3] : ${𝒵}^{\infty }$ peut en fait être un sous-groupe propre et non-dénombrable (donc dense) de ${𝒵}^1$.

Consider a bifurcation problem, namely, its bifurcation equation. There is a diffeomorphism $\Phi$ linking the actual solution set with an unfolded normal form of the bifurcation equation. The differential $D\Phi \left(0\right)$ of this diffeomorphism is a valuable information for a numerical analysis of the imperfect bifurcation. The aim of this paper is to construct algorithms for a computation of $D\Phi \left(0\right)$. Singularity classes containing bifurcation points with $\mathrm{c}odim\le 3$, $\mathrm{c}orank=1$ are considered.

We prove that if the Cantor set K, dynamically defined by a function $S\in C^{1+\alpha }$, satisfies the conditions of McDuff’s conjecture then it cannot be C¹-minimal.

We present a new method to compute normal forms, applied to the germs of reversible mappings. We translate the classification problem of these germs to the theory of ideals in the space of the coefficients of their jets. Integral factorization coupled with Gröbner basis constructionjs the key factor that makes the process efficient. We also show that a language with typed objects like AXIOM is very convenient to solve these kinds of problems.

Let F be a C ∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney Wr topology. Then contains a subset consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = . In this paper we present...

We study the jumps of topological entropy for $C^r$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f\in C^r\left([0,1]\right)$ with $h_{top}\left(f\right)>\left(log⁺\right||f^{\text{'}}{\left|\right|}_{\infty })/r$. To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^r$ interval maps.

We prove that simple transformations are disjoint from those which are infinitely divisible and embeddable in a flow. This is a reinforcement of a previous result of A. del Junco and M. Lemańczyk [1] who showed that simple transformations are disjoint from Gaussian processes.