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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 $n$'shas 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 everynon nilpotent operator $T$, there exists $x \in X$ such that$(\|T^nx\|/\|T^n\|) \notin \ell^{q}(\mathbb{N})$, using techniques whichinvolve the modulus of asymptotic uniform smoothness of $X$.

Orbits of linear operators and Banach space geometry