A quasiperiodically forced skew-product on the cylinder without fixed-curves

In [FJJK] the Sharkovski\u{i} Theorem was extended to periodic orbits of strips of quasiperiodic skew products in the cylinder. In this paper we deal with the following natural question that arises in this setting: Does Sharkovski\u{i} Theorem holds when restricted to curves instead of general strips?mWe answer this question in the negative by constructing a counterexample: We construct a map having a periodic orbit of period 2 of curves (which is, in fact, the upper and lower circles of the cylinder) and without any invariant curve. In particular this shows that there exist quasiperiodic skew products in the cylinder without invariant curves

Sistemes dinámics : les matemàtiques del professor W. Szlenk

L'objectiu d'aquesta xerrada és donar a conèixer a nivell humà i matemàtic la figura d'en Wiesław Szlenk, mort el 26 de juliol de 1995 a l'Hospital Clínic de Barcelona, a aquells membres de la comunitat matemàtica de la Universitat Autònoma de Barcelona que no hi varen tenir un contacte intens

Semiconjugacy to a map of a constant slope

It is well known that a continuous piecewise monotone interval map with positive topological entropy is semiconjugate to a map of a constant slope and the same entropy, and if it is additionally transitive then this semiconjugacy is actually a conjugacy. We generalize this result to piecewise continuous piecewise monotone interval maps, and as a consequence, get it also for piecewise monotone graph maps. We show that assigning to a continuous transitive piecewise monotone map of positive entropy a map of constant slope conjugate to it defines an operator, and show that this operator is not continuous