Stochastic-like characteristics of arithmetic dynamical systems: the Collatz hailstone sequences

Abstract

The numerical hailstone sequences, or orbits, generated by the Collatz map have been disclosed to present relevant features commonly associated with complex systems. It is so despite the extreme simplicity of the arithmetic dynamical system iteration rule. Indeed, for a positive integer n , the Collatz map f reads $f(n) = n/2$ ( $f(n) = 3 n + 1$ ) for n even (odd). Seeking to elucidate this surprising fact, here we unveil distinct characteristics of stochastic-like behavior for collections of Collatz orbits by considering methods commonly employed to temporal series, as cryptography tests, power-spectrum, detrended fluctuation, auto-correlation and entropy measure. Besides confirming previous predictions that the Collatz orbits display some global properties of geometric Brownian motion, our results are likewise able to explain, at least heuristically, the reasons for so. In special, we show by means of comprehensive analysis that our findings cannot be ascribed to standard chaotic evolution. Moreover, we identify novel short- and mid-range correlations in the Collatz orbits. The Collatz map is hence a paradigmatic example of an arithmetic dynamical system which could also be regarded as displaying key characteristics of an arithmetic statistical physics system, explaining its dynamical richness.

Keywords

• arithmetic dynamics
• Collatz sequences
• stochastic behavior
• time series analysis
• complexity