| Introduction |
| Some ideas around 3n + 1 iterations / Chapter I： |
| The problem / 1： |
| About the origin of the problem / 2： |
| Empirical investigations and stochastic models / 3： |
| Related functions and generalizations / 4： |
| Some formulae describing the iteration / 5： |
| Numbers with finite stopping time / 6： |
| Asymptotics of predecessor sets / 7： |
| Consecutive numbers with the same height / 8： |
| Cycles / 9： |
| Binary sequences and 2-adic analysis / 10： |
| Reduction to residue classes and other sets / 11： |
| Formal languages / 12： |
| Functional equations / 13： |
| A continuous extension to the real line / 14： |
| Analysis of the Collatz graph / Chapter II： |
| Directed graphs and dynamical systems on N |
| Directed graphs |
| The Collatz graph |
| The size of a subset of N |
| Encoding of predecessors by admissible vectors |
| Encoding a path in the Collatz graph |
| Concatenation of integer vectors |
| Tracing back integer vectors in the rationals |
| Admissible integer vectors |
| Some properties of admissible vectors |
| Recognizing admissible vectors |
| Extending admissible vectors |
| Similar integer vectors |
| Recurrent patterns in the Collatz graph |
| Counting functions and an estimate |
| Counting functions for admissible vectors |
| Counting predecessors of given size |
| The error of the estimate |
| Some restricted predecessor sets |
| The odd predecessors |
| The pruned Collatz graph |
| Pruned counting functions |
| Inductive construction of the pruned counting functions |
| Odd predecessors in the pruned Collatz graph |
| Comparison with other approaches |
| Uniform bounds |
| Crandall's approach |
| Crandall's estimate |
| Sander's estimate |
| Minorant vectors of Applegate and Lagarias |
| 3-adic averages of counting functions / Chapter III： |
| Basics of 3-adic numbers |
| The estimating series |
| Counting functions on 3-adic numbers |
| The sequence of estimating series |
| Ill-behaviour of the estimating series |
| The averaged estimating series |
| A formula for 3-adic averages |
| Maximal terms |
| The candidate for the maximal term |
| An estimate for the remaining terms |
| First order asymptotics of maximal terms |
| Asymptotic behaviour of the averaged sums |
| A naive approach |
| The theorem |
| Why 3n + 1 and not pn + 1? |
| An asymptotically homogeneous Markov chain / Chapter IV： |
| Small vectors and a Cauchy product |
| The structure of similarity classes |
| Partitions |
| Counting functions for small admissible vectors |
| A Cauchy product |
| Renormalization |
| Construction of the second factor of the state space |
| Construction of the first factor |
| The state space and the pull-backs |
| The normalization factor |
| Transition probabilities |
| Basic notions for Markov chains |
| Domains of dependence and domains of transition |
| The integral kernels |
| Vague convergence of the transition measures |
| The limiting transition probability |
| The invariant density |
| A relation to Cantor's set |
| Some further remarks |
| Mixing and predecessor density / Chapter V： |
| Locally covering triples |
| The basic estimate for locally covering triples |
| The normalized remainder map |
| 3-adic balls and spheres |
| Globally covering triple |
| A predecessor density criterion |
| Consequences |
| A sufficient condition for positive density |
| Uniform positive density |
| Non-existence of globally optimal sequences |
| The reduction theorem |
| Bibliography |
| Index of authors |
| List of symbols |
| Index |
| Introduction |
| Some ideas around 3n + 1 iterations / Chapter I： |
| The problem / 1： |
図書
