A proof of the C⁰-closing lemma for noninvertible discrete dynamical systems and its extension to the noncompact case are presented.

Under very mild assumptions, any Lipschitz continuous conjugacy between the closures of the postcritical sets of two C¹-unimodal maps has a derivative at the critical point, and also on a dense set of its preimages. In a more restrictive situation of infinitely renormalizable maps of bounded combinatorial type the Lipschitz condition automatically implies the C¹-smoothness of the conjugacy. Here the critical degree can be any real number α > 1.

Hydrogen atoms placed in external fields serve as a paradigm of a strongly coupled multidimensional Hamiltonian system. This system has been already very extensively studied, using experimental measurements and a wealth of theoretical methods. In this work, we apply the Morales-Ramis theory of non-integrability of Hamiltonian systems to the case of the hydrogen atom in perpendicular (crossed) static electric and magnetic uniform fields.

Let $f_s$ and $f_t$ be tent maps on the unit interval. In this paper we give a new proof of the fact that if the critical points of $f_s$ and $f_t$ are periodic and the inverse limit spaces $(I,f_s)$ and $(I,f_t)$ are homeomorphic, then s = t. This theorem was first proved by Kailhofer. The new proof in this paper simplifies the proof of Kailhofer. Using the techniques of the paper we are also able to identify certain isotopies between homeomorphisms on the inverse limit space.

We introduce the cohomological Conley type index theory for multivalued flows generated by vector fields which are compact and convex-valued perturbations of some linear operators.

Consider the ordinary differential equation (1) ẋ = Lx + K(x) on an infinite-dimensional Hilbert space E, where L is a bounded linear operator on E which is assumed to be strongly indefinite and K: E → E is a completely continuous but not necessarily locally Lipschitzian map. Given any isolating neighborhood N relative to equation (1) we define a Conley-type index of N. This index is based on Galerkin approximation of equation (1) by finite-dimensional ODEs and extends...

Given d ≥ 2 consider the family of polynomials $P_c\left(z\right)=z^d+c$ for c ∈ ℂ. Denote by $J_c$ the Julia set of $P_c$ and let ${ℳ}_d=c|J_{c}isconnected$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c₀\in \partial {ℳ}_d$: those for which the critical point 0 is not recurrent by $P_{c₀}$ and without parabolic cycles. The Hausdorff dimension of $J_c$, denoted by $HD\left(J_c\right)$, does not depend continuously on c at such $c₀\in \partial {ℳ}_d$; on the other hand the function $c\mapsto HD\left(J_c\right)$ is analytic in $ℂ-{ℳ}_d$. Our first result asserts that there is still some continuity...

Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map $g_{\sigma }$. We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set $J_{0,\sigma }$ is continuous at σ₀ as the function of the parameter $\sigma \in \overline{₀}$ if and only if $HD\left(J_{0,\sigma ₀}\right)\ge 4/3$. Since $HD\left(J_{0,\sigma }\right)>4/3$ on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of $HD\left(J_{0,\sigma }\right)$ on an open and dense subset of ∂₀.

If $f:[0,1]\to ℝ$ is strictly increasing and continuous define $T_{f}x=f\left(x\right)(mod1)$. A transformation $\tilde{T}:[0,1]\to [0,1]$ is called $\epsilon$-close to $T_f$, if $\tilde{T}x=\tilde{f}\left(x\right)(mod1)$ for a strictly increasing and continuous function $\tilde{f}:[0,1]\to ℝ$ with $\parallel \tilde{f}{-f\parallel }_{\infty }<\epsilon$. It is proved that the topological pressure $p(T_f,g)$ is lower semi-continuous, and an upper bound for the jumps up is given. Furthermore the continuity of the maximal measure is shown, if a certain condition is satisfied. Then it is proved that the topological pressure is upper semi-continuous for every continuous function $g:[0,1]\to ℝ$, if and only if $0$ is...

In this work we are interested in the study of controllability and stabilization of the linearized Benjamin-Ono equation with periodic boundary conditions, which is a generic model for the study of weakly nonlinear waves with nonlocal dispersion. It is well known that the Benjamin-Ono equation has infinite number of conserved quantities, thus we consider only controls acting in the equation such that the volume of the solution is conserved. We study also the stabilization with a feedback law which...

In this work we are interested in the study of controllability and stabilization of the linearized Benjamin-Ono equation with periodic boundary conditions, which is a generic model for the study of weakly nonlinear waves with nonlocal dispersion. It is well known that the Benjamin-Ono equation has infinite number of conserved quantities, thus we consider only controls acting in the equation such that the volume of the solution is conserved. We study also the stabilization with a feedback law...

We show that for any irrational number α and a sequence ${m_l}_{l\in \mathbb{N}}$ of integers such that $li{m}_{l\to \infty }\left|\right||m_l\alpha \left|\right||=0$, there exists a continuous measure μ on the circle such that $li{m}_{l\to \infty }{\int }_{}\left|\right||m_l\theta \left|\right||d\mu \left(\theta \right)=0$. This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any α ∈ ℝ - ℚ, there exists a sequence ${m_l}_{l\in \mathbb{N}}$ of integers such that $\left|\right||m_l\alpha \left|\right||\to 0$ and such that $m_l\theta \left[1\right]$ is dense on the circle if and only if θ ∉ ℚα + ℚ.