Non-integrability of measure preserving maps via Lie symmetries

We consider the problem of characterizing, for certain natural number $m$, the local $\mathcal{C}^m$-non-integrability near elliptic fixed points of smooth planar measure preserving maps. Our criterion relates this non-integrability with the existence of some Lie Symmetries associated to the maps, together with the study of the finiteness of its periodic points. One of the steps in the proof uses the regularity of the period function on the whole period annulus for non-degenerate centers, question that we believe that is interesting by itself. The obtained criterion can be applied to prove the local non-integrability of the Cohen map and of several rational maps coming from second order difference equations.Comment: 25 page

Basin of attraction of triangular maps with applications

We consider some planar triangular maps. These maps preserve certain fibration of the plane. We assume that there exists an invariant attracting fiber and we study the limit dynamics of those points in the basin of attraction of this invariant fiber, assuming that either it contains a global attractor, or it is filled by fixed or 2-periodic points. Finally, we apply our results to a variety of examples, from particular cases of triangular systems to some planar quasi-homogeneous maps, and some multiplicative and additive difference equations, as well.Comment: 1 figur

Bótes i barrils

En aquest treball ens interessem per la validesa de distintes fórmules usades a la pràctica per a calcular la capacitat de les bótes de vi o de sidra. Hi trobareu, per exemple, la fórmula de Simpson per a calcular integrals definides, un llibre escrit pel mateix Kepler sobre el tema i el mètode dels mínims quadrats, introduït per Gauss i Legendre per a trobar la millor solució per a sistemes sobredeterminats i incompatibles.In this paper we consider the validity of several practical formulas used to calculate the volume of wine or cider barrels. In our study we will find, for instance, Simpsons formula for calculating definite integrals; a book written by Kepler on the subject; and the method of least squares established by Gauss and Legendre, for calculating the best solution for overdetermined and incompatible systems

Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points

We show that for periodic non-autonomous discrete dynamical systems, even when a common fixed point for each of the autonomous associated dynamical systems is repeller, this fixed point can became a local attractor for the whole system, giving rise to a Parrondo's dynamic type paradox.Comment: 21 page