A dynamical approach towards Collatz conjecture

Abstract

The present work focuses on the study of the renowned Collatz conjecture, also known as the $3x+1$ problem. The analysis rests on the dynamics of an iterative map in binary form, defined on the dyadic rationals of $[1/2,1)$. We estimate how the length of the binary expansion of the iterated numbers changes, and we regard the iterative map as a perturbation of a circle map without periodic points, with an explicit bound for the error between both maps. A condition on the periods of periodic orbits is given; it is the classical argument of Crandall, and it recovers the lower bound of Eliahou for the length of nontrivial cycles. This version corrects several errors of the first one; they are listed in the Addendum.

arXiv comment10 pages, 3 figures, 2 tables