The Dynamical Mordell–Lang Conjecture

Cover
Preface
Notation
Chapter 1. Introduction
1.1. Overview of the problem
1.2. Linear recurrence sequences
1.3. Polynomial-exponential Diophantine equations
1.4. Linear algebra
1.5. Arithmetic geometry
1.6. Plan of the book
Chapter 2. Background material
2.1. Algebraic geometry
2.2. Dynamics of endomorphisms
2.3. Valuations
2.4. Chebotarev Density Theorem
2.5. The Skolem-Mahler-Lech Theorem
2.6. Heights
Chapter 3. The Dynamical Mordell-Lang problem
3.1. The Dynamical Mordell-Lang Conjecture
3.2. The case of rational self-maps
3.3. Known cases of the Dynamical Mordell-Lang Conjecture
3.4. The Mordell-Lang conjecture
3.5. Denis-Mordell-Lang conjecture
3.6. A more general Dynamical Mordell-Lang problem
Chapter 4. A geometric Skolem-Mahler-Lech Theorem
4.1. Geometric reformulation
4.2. Automorphisms of affine varieties
4.3. Étale maps
4.4. Proof of the Dynamical Mordell-Lang Conjecture for étale maps
Chapter 5. Linear relations between points in polynomial orbits
5.1. The main results
5.2. Intersections of polynomial orbits
5.3. A special case
5.4. Proof of Theorem 5.3.0.2
5.5. The general case of Theorem 5.3.0.1
5.6. The method of specialization and the proof of Theorem 5.5.0.2
5.7. The case of Theorem 5.2.0.1 when the polynomials have different degrees
5.8. An alternative proof for the function field case
5.9. Possible extensions
5.10. The case of plane curves
5.11. A Dynamical Mordell-Lang type question for polarizable endomorphisms
Chapter 6. Parametrization of orbits
6.1. Rational maps
6.2. Analytic uniformization
6.3. Higher dimensional parametrizations
Chapter 7. The split case in the Dynamical Mordell-Lang Conjecture
7.1. The case of rational maps without periodic critical points
7.2. Extension to polynomials with complex coefficients
7.3. The case of “almost” post-critically finite rational maps
Chapter 8. Heuristics for avoiding ramification
8.1. A random model heuristic
8.2. Random models and cycle lengths
8.3. Random models and avoiding ramification
8.4. The case of split maps
Chapter 9. Higher dimensional results
9.1. The Herman-Yoccoz method for periodic attracting points
9.2. The Herman-Yoccoz method for periodic indifferent points
9.3. The case of semiabelian varieties
9.4. Preliminaries from linear algebra
9.5. Proofs for Theorems 9.2.0.1 and 9.3.0.1
Chapter 10. Additional results towards the Dynamical Mordell-Lang Conjecture
10.1. A ?-adic analytic instance of the Dynamical Mordell-Lang Conjecture
10.2. A real analytic instance of the Dynamical Mordell-Lang Conjecture
10.3. Birational polynomial self-maps on the affine plane
Chapter 11. Sparse sets in the Dynamical Mordell-Lang Conjecture
11.1. Overview of the results presented in this chapter
11.2. Sets of positive Banach density
11.3. General quantitative results
11.4. The Dynamical Mordell-Lang problem for Noetherian spaces
11.5. Very sparse sets in the Dynamical Mordell-Lang problem for endomorphisms of (\bP¹)^{?}
11.6. Reductions in the proof of Theorem 11.5.0.2
11.7. Construction of a suitable ?-adic analytic function
11.8. Conclusion of the proof of Theorem 11.5.0.2
11.9. Curves
11.10. An analytic counterexample to a ?-adic formulation of the Dynamical Mordell-Lang Conjecture
11.11. Approximating an orbit by a ?-adic analytic function
Chapter 12. Denis-Mordell-Lang Conjecture
12.1. Denis-Mordell-Lang Conjecture
12.2. Preliminaries on function field arithmetic
12.3. Proof of our main result
Chapter 13. Dynamical Mordell-Lang Conjecture in positive characteristic
13.1. The Mordell-Lang Conjecture over fields of positive characteristic
13.2. Dynamical Mordell-Lang Conjecture over fields of positive characteristic
13.3. Dynamical Mordell-Lang Conjecture for tori in positive characteristic
13.4. The Skolem-Mahler-Lech Theorem in positive characteristic
Chapter 14. Related problems in arithmetic dynamics
14.1. Dynamical Manin-Mumford Conjecture
14.2. Unlikely intersections in dynamics
14.3. Zhang’s conjecture for Zariski dense orbits
14.4. Uniform boundedness
14.5. Integral points in orbits
14.6. Orbits avoiding points modulo primes
14.7. A Dynamical Mordell-Lang conjecture for value sets
Chapter 15. Future directions
15.1. What is known?
15.2. What is next?
15.3. Varieties with many rational points
15.4. A higher dimensional Dynamical Mordell-Lang Conjecture
Bibliography
Index
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