Modelling evolution of the 3x+l dynamical system via Marl partitions

The representation of nonlinear systems via Markov partitions is a well defined method of reducing their study as linear dynamical systems. We apply this method for the 3x+1 dynamical system an important discrete nonlinear dynamical system related to the Collatz problem. This system although simple in its description possesses many interesting properties and may be used as benchmark for testing the various methodologies of examination of nonlinear systems. For every n ∈ N we consider the Markov partition of N = A0 ∪ A2 ∪ … ∪ An − 1 where Ak = {pn + k : p ∈ N}. This partition gives rise to a linear dynamical system with system matrix M (n) ∈ R n×n representing the 3x+1 dynamical system. As n increases, the resolution of the Markov partition becomes finer and the representation of the 3x+1 system via M(n) richer. Here we explore the evolution of the properties of M(n) as n varies. The ultimate purpose is to examine which properties are inherited, stay invariant or created as n increases. This may produce a new point of view and new tools to facilitating in finding new properties or solving the Collatz problem where the 3x+1 system originates (Lagarias 1985). This extended abstract outlines the problem and the main idea behind its solution which will be developed properly subsequently.