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 G -shift for a finitely-generated torsion-free nilpotent group G 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 \[ ∑ | τ | ≤ N 1 e h | τ | ∼ C N α ( log ⁡ N ) β \sum _{\vert \tau \vert \le N}\frac {1}{e^{h\vert \tau \vert }}\sim CN^{\alpha }(\log N)^{\beta } \] where | τ | \vert \tau \vert is the cardinality of the finite orbit τ \tau and h 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.