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 systems, 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 AehN 6 π(N) 6 BehN, 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