The Dynamical System Generated by the 3n+1 Function

Günther J. Wirsching (auth.)3540639705, 9783540639701

The 3n+1 function T is defined by T(n)=n/2 for n even, and T(n)=(3n+1)/2 for n odd. The famous 3n+1 conjecture, which remains open, states that, for any starting number n>0, iterated application of T to n eventually produces 1. After a survey of theorems concerning the 3n+1 problem, the main focus of the book are 3n+1 predecessor sets. These are analyzed using, e.g., elementary number theory, combinatorics, asymptotic analysis, and abstract measure theory. The book is written for any mathematician interested in the 3n+1 problem, and in the wealth of mathematical ideas employed to attack it.

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Table of contents :
Introduction….Pages 1-9
Some ideas around 3n+1 iterations….Pages 10-30
Analysis of the Collatz graph….Pages 31-75
3-adic averages of counting functions….Pages 76-95
An asymptotically homogeneous Markov chain….Pages 96-122
Mixing and predecessor density….Pages 123-140