Orbit-counting for nilpotent group shifts

Abstract:

We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens’ theorem for the full $G$-shift for a finitely-generated torsion-free nilpotent group $G$. Using bounds for the Möbius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape \[ \sum _{\vert \tau \vert \le N}\frac {1}{e^{h\vert \tau \vert }}\sim CN^{\alpha }(\log N)^{\beta } \] where $\vert \tau \vert$ is the cardinality of the finite orbit $\tau$ and $h$ denotes the topological entropy. For the usual orbit-counting function we find upper and lower bounds, together with numerical evidence to suggest that for actions of non-cyclic groups there is no single asymptotic in terms of elementary functions.