The Dynamical System Generated by the 3n+1 Function

1. Front Matter

   Pages i-vii

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2. Introduction

   • G. J. Wirsching

   Pages 1-9

3. Some ideas around 3n+1 iterations

   • Günther J. Wirsching

   Pages 10-30

4. Analysis of the Collatz graph

   • Günther J. Wirsching

   Pages 31-75

5. 3-adic averages of counting functions

   • Günther J. Wirsching

   Pages 76-95

6. An asymptotically homogeneous Markov chain

   • Günther J. Wirsching

   Pages 96-122

7. Mixing and predecessor density

   • Günther J. Wirsching

   Pages 123-140

8. Back Matter

   Pages 141-158

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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.

• 3n+1 problem
• asymptotic analysis
• calculus
• collatz graph
• dynamical systems on N
• number theory
• p-adic distribution

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