# Congress detail

**Authors**: Muro, Santiago; Cardeccia, Rodrigo.

**Resumen**: Let X be a Banach space. A function F : X ¡æ X is said to be hypercyclic if there exists x ¡ô X whose orbit OrbF(x) = {Fn(x) : n ¡ô N0} is dense in X. It is a known fact that there exist hypercyclic linear operators in arbitrary separable inﬁnite dimensional Banach spaces. The dynamical system induced by a (non linear) homogeneous polynomial is quite diﬀerent. Associated to each homogeneous polynomial there is a ball, centered at zero, with the following property: orbits that meet this limit ball tend to zero. Therefore homogeneous polynomials on Banach spaces cannot be hypercyclic. However, the behavior of the orbits that never enter the limit ball can be non trivial. Indeed, in [1] Bernardes showed the existence of orbits oscillating between inﬁnity and the limit ball. He also proved that there are supercyclic homogeneous polynomials in arbitrary separable inﬁnite dimensional Banach spaces. In this talk we will exhibit a simple and natural 2-homogeneous polynomial that is at the same time d-hypercyclic (the orbit meets every ball of radius d), weakly hypercyclic (the orbit is dense with to respect the weak topology) and ¥Ã-supercyclic (¥Ã¡¤OrbP(x) is dense) for each subset ¥Ã ¡ö C unbounded or not bounded away from zero. To prove this, the properties of its Julia set are studied. We will also generalize the construction to arbitrary inﬁnite dimensional Fr¢¥echet spaces.References[1] N.C. Bernardes. On orbits of polynomial maps in Banach spaces. Quaest. Math., 21(34):318,1998.

**Meeting type**: Workshop.

**Production**: Dynamics of homogeneous polynomials on Banach spaces.

**Scientific meeting**: International Workshop on Nonlinear Dynamical Systems and Functional Analysis.

**Meeting place**: Brasilia.

**It's published?**: No

**Meeting month**: 8