# Non-hypercyclicity for certain classes of linear dynamical systems

### Laboratório de Computação 3 do DMAT (Edifício 6… | 2020-01-29 | 14:30

Alexandre Baraviera
Instituto de Matemática e Estatística - Universidade Federal do Rio Grande do Sul (UFRGS)

The investigation of the properties of bounded linear maps on certain vector spaces (Hilbert or Banach spaces, for example) is a very rich andactive area. In particular, the existence of dense orbits (that in this context is known as hypercyclicity) attracts a lot of attention, as well as the extension of classical results to this setting, like hyperbolicity and shadowing, amongmany others.
A source of examples is the weighted shift, defined as $B_w(x_1, x_2, x_3, \ldots) = (w_2 x_2, w_3 x_3, \ldots)$ where $w_i$ are positive and bounded real numbers and $x = (x_1, x_2, \ldots)$ is a point of the space $\ell_p(N)$. Another map,with a less rich dynamics, is the diagonal map defined on the same space by $D_{\lambda}(x_1, x_2, \ldots)= (\lambda_1 x_1, \lambda_2 x_2, \ldots)$, where $\lambda_i$ is a complex number with norm $1$. Is is also usefull to consider the map $T_{w, \lambda} = D_{\lambda} + B_w$, where hypercyclicity is known to hold for some parameters.
Our goal in this talk is to exhibit some conditions for $\lambda$ and $w$  where the map $T_{w, \lambda}$ is NOT hypercyclic; we also show how to extend the methodfor anohter class of linear maps. This is a joint work with G. Pessil (UFRGS).