m at h . D S ] 1 J ul 1 99 9 ARITHMETIC AND GROWTH OF ORBITS : WHAT IS POSSIBLE ?

Given a sequence φ = (φn) of non-negative integers, we examine two related questions about dynamical realizations of φ. The sequence is exactly realizable if there is a dynamical system (X, T) (a homeomorphism of a compact metric space) for which fn(T) = #{x ∈ X | T n x = x} = φn for all n ≥ 1, and is realizable in rate if there is a dynamical system (X, T) for which fn(T)/φn −→ 1 as n → ∞. We give necessary and sufficient conditions for a sequence to be exactly realizable and discuss sufficient conditions for a sequence to be realizable in rate. Using these conditions, we show that (for example) no non-constant polynomial is exactly realizable, and show that a binary recurrence sequence with non-square discriminant can only be exactly realized with a unique ratio of initial terms. Realization in rate is always possible for sufficiently rapidly-growing sequences , and is never possible for sufficiently slowly-growing sequences. For powers of n, we show that (⌊n α ⌋) is realizable in rate if α is large, and is not if α is small. Finally, we discuss the relationship between the growth of fn(T) and the growth in the number of orbits of length n under T .