A Dichotomy in Orbit-growth for Commuting Automorphisms

We consider asymptotic orbit-counting problems for certain expansive actions by commuting automorphisms of compact groups. A dichotomy is found between systems with asymptotically more periodic orbits than the topological entropy predicts, and those for which there is no excess of periodic orbits. Let G be a countable group acting on some set X, with the action written x 7→ g.x. Let L = L(G) denote the poset of finite index subgroups of G, and write an(G) = |{L ∈ L | [G : L] = n}|. We assume that L is locally finite (a finiteness assumption on G, guaranteed if G is finitely generated). For L ∈ L, the set of L-periodic points in X under the action is F(L) = {x ∈ X | g.x = x for all g ∈ L}. An L-periodic orbit τ is the orbit of a point with stabilizer L, and the length of the orbit is denoted [L] = [G : L], the index of L in G. We always assume that there are only finitely many orbits of length n for each n > 1 (a finiteness assumption on the action, guaranteed if the action is expansive). The number of L-periodic orbits is O(L) = 1 [L] |{x ∈ X | g.x = x ⇐⇒ g ∈ L}| Orbit growth may be studied via the asymptotic behaviour of the orbitcounting function π(N) = ∑