Arithmetic and growth of periodic orbits

We give necessary and sufficient conditions for a sequence to be exactly realizable as the sequence of numbers of periodic points in a dynamical system. Using these conditions, we show that no non-constant polynomial is realizable, and give some conditions on realizable binary recurrence sequences. Realization in rate is always possible for sufficiently rapidly-growing sequences, and is never possible for slowly-growing sequences. Finally, we discuss the relationship between the growth rate of periodic points and the growth rate of points with specified least period.