In this paper we study dynamical properties of linear actions by free groups via the induced action on projective space. This point of view allows us to introduce techniques from Thermodynamic Formalism. In particular, we obtain estimates on the growth of orbits and their limiting distribution on projective space.

Let $H\subset GL\left(V\right)$ be a connected complex reductive group where $V$ is a finite-dimensional complex vector space. Let $v\in V$ and let $G=\{g\in \mathrm{GL}(V)\mid gHv=Hv\}$. Following Raïs we say that the orbit $Hv$ is characteristic for $H$ if the identity component of $G$ is $H$. If $H$ is semisimple, we say that $Hv$ is semi-characteristic for $H$ if the identity component of $G$ is an extension of $H$ by a torus. We classify the $H$-orbits which are not (semi)-characteristic in many cases.