Linear actions of free groups.(English)Zbl 0967.37016

Given a discrete subgroup $$\Gamma$$ of $$SL(d,{\mathbb R})$$ and a fixed non zero vector, it is interesting to consider the orbits $$\{Av, A \in \Gamma\}$$. For groups which are lattices, the set $$\{A \in \Gamma$$, $$\|Av \|\leq T\}$$ is infinite. For other groups, this set may be finite for certain choices of $$v$$. The question is how this counting function behaves as $$T$$ tends to infinity.
The authors consider the case where $$\Gamma$$ is a free group which satisfies two generic conditions. They get an estimate of the cardinality of $$\{A^l \in \Gamma$$, $$\|Av\|\leq T\}$$ for all sufficiently large $$l$$: the counting function is equivalent with $$C T^p$$, where $$p$$ is the abscissa of convergence of a Dirichlet series. The proof is based first on the study of the linear action of the matrices on the projective space, secondly on thermodynamic formalism applied to a Poincaré series.
The authors also describe the distribution of the orbits $$\{A^l v$$, $$A \in \Gamma\}$$ on the projective space. Their result could be viewed as an analogue of the Patterson-Sullivan measure for a hyperbolic manifold.

MSC:

 37C35 Orbit growth in dynamical systems 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
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References:

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