Arithmetic and growth of periodic orbits

Abstract

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.

Revised version with new content; to appear in J.Integer Sequences

Cited by in corpus (17)

• Differential equations, mirror maps and zeta values
• Dynamical invariants for group automorphisms
• Towards a Polya-Carlson dichotomy for algebraic dynamics
• The formal series Witt transform
• Dold sequences, periodic points, and dynamics
• A note on the dynamical zeta function of general toral endomorphisms
• Functorial orbit counting
• Witt vectors. Part 1
• Orbit counting with an isometric direction
• Realizability of integer sequences as differences of fixed point count sequences
• A note on the relation between fixed point and orbit count sequences
• Stirling number and periodic points
• Closed orbits in quotient systems
• Number of cycles in the graph of 312-avoiding permutations
• Boolean automata isolated cycles and tangential double-cycles dynamics
• Group automorphisms with few and with many periodic points
• Integer sequences counting periodic points