We prove the ultimate boundedness of solutions of some third order nonlinear ordinary differential equations using the Lyapunov method. The results obtained generalize earlier results of Ezeilo, Tejumola, Reissig, Tunç and others. The Lyapunov function used does not involve the use of signum functions as used by others.
We give two examples of tent maps with uncountable (as it happens, post-critical) ω-limit sets, which have isolated points, with interesting structures. Such ω-limit sets must be of the form C ∪ R, where C is a Cantor set and R is a scattered set. Firstly, it is known that there is a restriction on the topological structure of countable ω-limit sets for finite-to-one maps satisfying at least some weak form of expansivity. We show that this restriction does not hold if the ω-limit set is uncountable....
Un mot sturmien est la discrétisation d’une droite de pente irrationnelle. Un nombre de Sturm est la pente d’un mot sturmien qui est invariant par une substitution non triviale. Ces nombres sont certains irrationnels quadratiques caractérisés par la forme de leur développement en fraction continue. Nous donnons une caractérisation très simple des nombres de Sturm : un nombre irrationnel positif est de Sturm (de première espèce) si et seulement s’il est quadratique et à conjugué négatif.
We investigate the structure of kneading sequences that belong to unimodal maps for which the omega-limit set of the turning point is a minimal Cantor set. We define a scheme that can be used to generate uniformly recurrent and regularly recurrent infinite sequences over a finite alphabet. It is then shown that if the kneading sequence of a unimodal map can be generated from one of these schemes, then the omega-limit set of the turning point must be a minimal Cantor set.
Let be a unimodular Pisot substitution over a letter alphabet and let be the associated Rauzy fractals. In the present paper we want to investigate the boundaries () of these fractals. To this matter we define a certain graph, the so-called contact graph of . If satisfies a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries . From this graph...
We prove that by considering a finitary (almost continuous) symbolic extension of a topological dynamical system instead of a continuous extension, one cannot achieve any drop of the entropy of the extension.