Radial Limit Sets of Graph Directed Markov Systems associated to Free Groups

Details Radial Limit Sets of Graph Directed Markov Systems associated to Free Groups Radial Limit Sets of Graph Directed Markov Systems associated to Free Groups

2012-03-20

arxiv.org/abs/1203.4301v1

Mathematics

Dynamical Systems

Scientific paper

For an arbitrary conformal graph directed Markov system $\Phi$ associated to the free group $\F_{d}$ with $d\ge2$, we investigate the radial limit set $L_{r}(N,\Phi)$ of a normal subgroup $N$ of $\F_{d}$ with respect to $\Phi$. We show that if $\Phi$ is weakly symmetric, then the Hausdorff dimensions of $L_{r}(N,\Phi)$ and $L_{r}(\F_{d},\Phi)$ coincide if and only if $\F_{d}/N$ is amenable. This extends results by Brooks and by Bishop and Jones for Kleinian groups to the framework of conformal graph directed Markov systems. Moreover, we prove that $\dim_{H}(L_{r}(N,\Phi))\ge\dim_{H}(L_{r}(\F_{d},\Phi))/2$ provided that $\Phi$ is weakly symmetric.

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