Orbit Growth for Algebraic Flip Systems

An algebraic flip system is an action of the infinite dihedral group by automorphisms of a compact abelian group X. In this paper, a fundamental structure theorem is established for irreducible algebraic flip sytems, that is, systems for which the only closed invariant subgroups of X are finite. Using irreducible systems as a foundation, for expansive algebraic flip systems, periodic point counting estimates are obtained that lead to the orbit growth estimate Ae 6 π(N) 6 Be where π(N) denotes the number of orbits of length at most N , A and B are positive constants and h is the topological entropy. 1. Background and main results Stemming from the seminal works of Ja. Sinăı [25] and G. Margulis [16], periodic orbits in dynamical systems have been investigated using orbit growth functions, with entropy featuring as a constant controlling the exponential growth. An extensive body of work now exists for both flows and discrete time dynamical systems (see for example, [21], [20], [27], [8], [9]). The study of orbit growth functions for dynamical group actions in general was pursued in [18] and [19]. This context encompasses the case of a single invertible transformation which corresponds to an action of the group G = Z. Let G be a finitely generated group acting on some set X, with the action written as x 7→ g · x. The set L = L(G) of finite index subgroups of G becomes a locally finite poset with the order arising from inclusion. For L ∈ L, the number of L-periodic points in X is F(L) = |{x ∈ X : g · x = x for all g ∈ L}|. (1) An L-periodic orbit 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. Assuming that there are only finitely many orbits of length n for each n > 1, the number of L-periodic orbits is O(L) = 1 [L] |{x ∈ X : g · x = x⇐⇒ g ∈ L}| , (2) Date: March 23, 2014. 2010 Mathematics Subject Classification. 37A45,37B05,37C25,37C35,37C85,22D40. 1