Let Ω be a bounded domain in ${ℝ}^N$ with smooth boundary. Consider the following elliptic system: $-\Delta u={\partial }_{v}H(u,v,x)$ in Ω, $-\Delta v={\partial }_{u}H(u,v,x)$ in Ω, u = 0, v = 0 in ∂Ω. (ES) We assume that H is an even "-"-type Hamiltonian function whose first order partial derivatives satisfy appropriate growth conditions. We show that if (0,0) is a hyperbolic solution of (ES), then (ES) has at least 2|μ| nontrivial solutions, where μ = μ(0,0) is the renormalized Morse index of (0,0). This proves a conjecture by Angenent and van der Vorst.

Let F be a homeomorphism of 𝕋² = ℝ²/ℤ² isotopic to the identity and f a lift to the universal covering space ℝ². We suppose that κ ∈ H¹(𝕋²,ℝ) is a cohomology class which is positive on the rotation set of f. We prove the existence of a smooth Lyapunov function of f whose derivative lifts a non-vanishing smooth closed form on 𝕋² whose cohomology class is κ.

For a topological group G we introduce the algebra SUC(G) of strongly uniformly continuous functions. We show that SUC(G) contains the algebra WAP(G) of weakly almost periodic functions as well as the algebras LE(G) and Asp(G) of locally equicontinuous and Asplund functions respectively. For the Polish groups of order preserving homeomorphisms of the unit interval and of isometries of the Urysohn space of diameter 1, we show that SUC(G) is trivial. We introduce the notion of fixed point on a class...

En topologie dynamique, une famille classique de systèmes est celle formée par les rotations minimales. La classe des nilsystèmes et de leurs limites projectives en est une extension naturelle. L’étude de ces systèmes est ancienne mais connaît actuellement un renouveau à cause de ses applications, à la fois à la théorie ergodique et en théorie additive des nombres. Les rotations minimales sont caractérisées par le fait que la relation de proximalité régionale est l’égalité. Nous introduisons une...

The dynamics of singular Lagrangean systems is described by a distribution the rank of which is greater than one and may be non-constant. Consequently, these systems possess two kinds of conserved functions, namely, functions which are constant along extremals (constants of the motion), and functions which are constant on integral manifolds of the corresponding distribution (first integrals). It is known that with the help of the (First) Noether theorem one gets constants of the motion. In this...

Soit $\beta$ un nombre de Pisot ; nous montrons que pour tout entier $n$ assez grand il existe une matrice carrée à coefficients positifs ou nuls dont l’ordre est égal au degré de $\beta$ et dont ${\beta }^n$ est valeur propre.Soit $\beta =a_1/\beta +a_2/{\beta }^2+\cdots +a_n/{\beta }^n+\cdots$ le $\beta$-développement de $\beta$ ; si $\beta$ est un nombre de Pisot, alors la suite ${\left(a_n\right)}_{n\ge 1}$ est périodique après un certain rang $n_0$ (pour $n\ge n_0$, $a_{n+k}=a_n$) et le polynôme$$X^{n_0+k}-(a_1{X}^{n_0+k-1}+\cdots +a_{n_0+k})-(X^{n_0}-(a_1{X}^{n_0}+\cdots +a_{n_0}))$$est appelé polynôme de Parry. Nous montrons qu’il existe un ensemble relativement dense d’entiers $n$ tels que le polynôme minimal de ${\beta }^n$ est égal à son polynôme...

We consider continuous extensions of minimal rotations on a locally connected compact group X by cocycles taking values in locally compact Lie groups and prove regularity (i.e. the existence of orbit closures which project onto the whole basis X) in certain special situations beyond the nilpotent case. We further discuss an open question on cocycles acting on homogeneous spaces which seems to be the missing key for a general regularity theorem.

A conjecture of [swTAMS] states that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from symbolic dynamics.

We examine the structure of countable closed invariant sets under a dynamical system on a compact metric space. We are motivated by a desire to understand the possible structures of inhomogeneities in one-dimensional nonhyperbolic sets (inverse limits of finite graphs), particularly when those inhomogeneities form a countable set. Using tools from descriptive set theory we prove a surprising restriction on the topological structure of these invariant sets if the map satisfies a weak repelling or...

For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if f is minimal and A ⊆ X then both f(A) and $f^{-1}\left(A\right)$ share with A those topological properties which describe how large a set is. Using these results...

We establish that for a fairly general class of topologically transitive dynamical systems, the set of non-transitive points is very small when the rate of transitivity is very high. The notion of smallness that we consider here is that of σ-porosity, and in particular we show that the set of non-transitive points is σ-porous for any subshift that is a factor of a transitive subshift of finite type, and for the tent map of [0,1]. The result extends to some finite-to-one factor systems. We also show...

We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space ${\ell }^1\left(\mathbb{N}\right)$, any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for $c_0\left(\mathbb{N}\right)$ or ${\ell }^p\left(\mathbb{N}\right)$, $1\<p\<\infty$. Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.

Although Sarnak's conjecture holds for compact group rotations (irrational rotations, odometers), it is not even known whether it holds for all Jewett-Krieger models of such rotations. In this paper we show that it does, as long as the model is at the same a topological extension, via the same map that establishes the isomorphism, of an equicontinuous model. In particular, we recover (after [AKL]) that regular Toeplitz systems satisfy Sarnak's conjecture, and, as another consequence, so do...