On generalized convergence in collatz systems and δ k + β dynamical systems

Abstract

The Collatz Conjecture belongs to the generalized class of Dynamical system δk + β, where δ = 3, β = 1.This paper establishes that such a system converges to β as long as the conditions are suitable. A full de- scription of the Collatz system is given and formula derived for the reverse number, synchronized elements(those with similar Collatz Sequence) and closed forms that can help ease with the computation. One of the most important discoveries of this paper is that the Collatz Space is bounded by 1 at the end and by an odd multiple of 3 on the other end. From this a proof of the elusive conjecture can then be derived easily