**Mathematical Surveys and Monographs**

Volume: 210;
2016;
280 pp;
Hardcover

MSC: Primary 11; 14; 37;

Print ISBN: 978-1-4704-2408-4

Product Code: SURV/210

List Price: $110.00

AMS Member Price: $88.00

MAA member Price: $99.00

**Electronic ISBN: 978-1-4704-2908-9
Product Code: SURV/210.E**

List Price: $110.00

AMS Member Price: $88.00

MAA member Price: $99.00

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#### Supplemental Materials

# The Dynamical Mordell–Lang Conjecture

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*Jason P. Bell; Dragos Ghioca; Thomas J. Tucker*

The Dynamical Mordell-Lang Conjecture is an analogue of the classical Mordell-Lang conjecture in the context of arithmetic dynamics. It predicts the behavior of the orbit of a point \(x\) under the action of an endomorphism \(f\) of a quasiprojective complex variety \(X\). More precisely, it claims that for any point \(x\) in \(X\) and any subvariety \(V\) of \(X\), the set of indices \(n\) such that the \(n\)-th iterate of \(x\) under \(f\) lies in \(V\) is a finite union of arithmetic progressions. In this book the authors present all known results about the Dynamical Mordell-Lang Conjecture, focusing mainly on a \(p\)-adic approach which provides a parametrization of the orbit of a point under an endomorphism of a variety.

#### Readership

Graduate students and research mathematicians interested in algebraic geometry and its arithmetic applications.

#### Reviews & Endorsements

...[S]uitable for experts working on problems related to the dynamical Mordell-Lang conjecture. It may also be of interest to anyone who is interested in dynamics or number theory.

-- Liang-Chung Hsia, Mathematical Reviews

#### Table of Contents

# Table of Contents

## The Dynamical Mordell-Lang Conjecture

- Cover Cover11
- Preface xi12
- Chapter 1. Introduction 116
- Chapter 2. Background material 1126
- Chapter 3. The Dynamical Mordell-Lang problem 4762
- Chapter 4. A geometric Skolem-Mahler-Lech Theorem 6782
- Chapter 5. Linear relations between points in polynomial orbits 85100
- 5.1. The main results 85100
- 5.2. Intersections of polynomial orbits 89104
- 5.3. A special case 91106
- 5.4. Proof of Theorem 5.3.0.2 93108
- 5.5. The general case of Theorem 5.3.0.1 98113
- 5.6. The method of specialization and the proof of Theorem 5.5.0.2 98113
- 5.7. The case of Theorem 5.2.0.1 when the polynomials have different degrees 102117
- 5.8. An alternative proof for the function field case 107122
- 5.9. Possible extensions 110125
- 5.10. The case of plane curves 110125
- 5.11. A Dynamical Mordell-Lang type question for polarizable endomorphisms 113128

- Chapter 6. Parametrization of orbits 117132
- Chapter 7. The split case in the Dynamical Mordell-Lang Conjecture 127142
- Chapter 8. Heuristics for avoiding ramification 143158
- Chapter 9. Higher dimensional results 153168
- Chapter 10. Additional results towards the Dynamical Mordell-Lang Conjecture 167182
- Chapter 11. Sparse sets in the Dynamical Mordell-Lang Conjecture 179194
- 11.1. Overview of the results presented in this chapter 179194
- 11.2. Sets of positive Banach density 182197
- 11.3. General quantitative results 185200
- 11.4. The Dynamical Mordell-Lang problem for Noetherian spaces 189204
- 11.5. Very sparse sets in the Dynamical Mordell-Lang problem for endomorphisms of (\bP¹)^{𝑁} 193208
- 11.6. Reductions in the proof of Theorem 11.5.0.2 198213
- 11.7. Construction of a suitable 𝑝-adic analytic function 199214
- 11.8. Conclusion of the proof of Theorem 11.5.0.2 202217
- 11.9. Curves 205220
- 11.10. An analytic counterexample to a 𝑝-adic formulation of the Dynamical Mordell-Lang Conjecture 207222
- 11.11. Approximating an orbit by a 𝑝-adic analytic function 209224

- Chapter 12. Denis-Mordell-Lang Conjecture 217232
- Chapter 13. Dynamical Mordell-Lang Conjecture in positive characteristic 231246
- 13.1. The Mordell-Lang Conjecture over fields of positive characteristic 232247
- 13.2. Dynamical Mordell-Lang Conjecture over fields of positive characteristic 233248
- 13.3. Dynamical Mordell-Lang Conjecture for tori in positive characteristic 234249
- 13.4. The Skolem-Mahler-Lech Theorem in positive characteristic 237252

- Chapter 14. Related problems in arithmetic dynamics 249264
- 14.1. Dynamical Manin-Mumford Conjecture 249264
- 14.2. Unlikely intersections in dynamics 252267
- 14.3. Zhang’s conjecture for Zariski dense orbits 254269
- 14.4. Uniform boundedness 257272
- 14.5. Integral points in orbits 258273
- 14.6. Orbits avoiding points modulo primes 259274
- 14.7. A Dynamical Mordell-Lang conjecture for value sets 261276

- Chapter 15. Future directions 263278
- Bibliography 267282
- Index 277292
- Back Cover Back Cover1297