This is an announcement for the paper "Orbits of linear operators and Banach space geometry" by Jean-Matthieu Auge. Abstract: Let $T$ be a bounded linear operator on a (real or complex) Banach space $X$. If $(a_n)$ is a sequence of non-negative numbers tending to 0. Then, the set of $x \in X$ such that $\|T^nx\| \geqslant a_n \|T^n\|$ for infinitely many $n$'s has a complement which is both $\sigma$-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents $q>0$, such that for every non nilpotent operator $T$, there exists $x \in X$ such that $(\|T^nx\|/\|T^n\|) \notin \ell^{q}(\mathbb{N})$, using techniques which involve the modulus of asymptotic uniform smoothness of $X$. Archive classification: math.FA Mathematics Subject Classification: Primary 47A05, 47A16, Secondary 28A05 Remarks: 16 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/1204.2046 or http://arXiv.org/abs/1204.2046