This is an announcement for the paper "Linear operators with wild dynamics" by Jean-Matthieu Auge. Abstract: If $X$ is a separable infinite dimensional Banach space, we construct a bounded and linear operator $R$ on $X$ such that $$ A_R=\{x \in X, \|R^tx\| \rightarrow \infty\} $$ is not dense and has non empty interior with the additional property that $R$ can be written $I+K$, where $I$ is the identity and $K$ is a compact operator. This answers two recent questions of H\'ajek and Smith. Archive classification: math.FA Mathematics Subject Classification: Primary 47A05, Secondary 47A15, 47A16 Remarks: 14 pages Submitted from: jean-matthieu.auge@math.u-bordeaux1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2044 or http://arXiv.org/abs/1204.2044