| Introduction | p. 1 |
| Some ideas around 3n + 1 iterations | p. 10 |
| The problem | p. 10 |
| About the origin of the problem | p. 11 |
| Empirical investigations and stochastic models | p. 12 |
| Related functions and generalizations | p. 13 |
| Some formulae describing the iteration | p. 17 |
| Numbers with finite stopping time | p. 19 |
| Asymptotics of predecessor sets | p. 21 |
| Consecutive numbers with the same height | p. 21 |
| Cycles | p. 22 |
| Binary sequences and 2-adic analysis | p. 24 |
| Reduction to residue classes and other sets | p. 27 |
| Formal languages | p. 27 |
| Functional equations | p. 28 |
| A continuous extension to the real line | p. 29 |
| Analysis of the Collatz graph | p. 31 |
| Directed graphs and dynamical systems on N | p. 33 |
| Directed graphs | p. 33 |
| The Collatz graph | p. 36 |
| The size of a subset of N | p. 37 |
| Encoding of predecessors by admissible vectors | p. 38 |
| Encoding a path in the Collatz graph | p. 38 |
| Concatenation of integer vectors | p. 39 |
| Tracing back integer vectors in the rationals | p. 40 |
| Admissible integer vectors | p. 42 |
| Some properties of admissible vectors | p. 45 |
| Recognizing admissible vectors | p. 45 |
| Extending admissible vectors | p. 45 |
| Similar integer vectors | p. 47 |
| Recurrent patterns in the Collatz graph | p. 48 |
| Counting functions and an estimate | p. 51 |
| Counting functions for admissible vectors | p. 51 |
| Counting predecessors of given size | p. 53 |
| The error of the estimate | p. 55 |
| Some restricted predecessor sets | p. 56 |
| The odd predecessors | p. 56 |
| The pruned Collatz graph | p. 58 |
| Pruned counting functions | p. 59 |
| Inductive construction of the pruned counting functions | p. 60 |
| Odd predecessors in the pruned Collatz graph | p. 61 |
| Comparison with other approaches | p. 63 |
| Uniform bounds | p. 63 |
| Crandall's approach | p. 65 |
| Crandall's estimate | p. 67 |
| Sander's estimate | p. 71 |
| Minorant vectors of Applegate and Lagarias | p. 73 |
| 3-adic averages of counting functions | p. 76 |
| Basics of 3-adic numbers | p. 77 |
| The estimating series | p. 79 |
| Counting functions on 3-adic numbers | p. 79 |
| The sequence of estimating series | p. 80 |
| Ill-behaviour of the estimating series | p. 81 |
| The averaged estimating series | p. 83 |
| A formula for 3-adic averages | p. 83 |
| The averaged estimating series | p. 84 |
| Maximal terms | p. 85 |
| The candidate for the maximal term | p. 86 |
| An estimate for the remaining terms | p. 88 |
| First order asymptotics of maximal terms | p. 90 |
| Asymptotic behaviour of the averaged sums | p. 91 |
| A naive approach | p. 92 |
| The theorem | p. 92 |
| Why 3n + 1 and not pn + 1? | p. 95 |
| An asymptotically homogeneous Markov chain | p. 96 |
| Small vectors and a Cauchy product | p. 97 |
| The structure of similarity classes | p. 98 |
| Partitions | p. 98 |
| Counting functions for small admissible vectors | p. 100 |
| A Cauchy product | p. 102 |
| Renormalization | p. 103 |
| Construction of the second factor of the state space | p. 103 |
| Construction of the first factor | p. 104 |
| The state space and the pull-backs | p. 106 |
| The normalization factor | p. 107 |
| Transition probabilities | p. 107 |
| Basic notions for Markov chains | p. 108 |
| Domains of dependence and domains of transition | p. 109 |
| The integral kernels | p. 111 |
| Vague convergence of the transition measures | p. 112 |
| The limiting transition probability | p. 117 |
| The invariant density | p. 117 |
| A relation to Cantor's set | p. 119 |
| Some further remarks | p. 120 |
| Mixing and predecessor density | p. 123 |
| Locally covering triples | p. 124 |
| The basic estimate for locally covering triples | p. 124 |
| The normalized remainder map | p. 125 |
| 3-adic balls and spheres | p. 127 |
| Globally covering triple | p. 129 |
| A predecessor density criterion | p. 130 |
| Consequences | p. 134 |
| A sufficient condition for positive density | p. 134 |
| Uniform positive density | p. 135 |
| Non-existence of globally optimal sequences | p. 136 |
| The reduction theorem | p. 137 |
| Bibliography | p. 141 |
| Index of authors | p. 146 |
| List of symbols | p. 149 |
| Index | p. 152 |
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