The Dynamical Mordell–Lang Conjecture

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. 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 v-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)^{N} 11.6. Reductions in the proof of Theorem 11.5.0.2 11.7. Construction of a suitable p-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 p-adic formulation of the Dynamical Mordell-Lang Conjecture 11.11. Approximating an orbit by a p-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 Back Cover The Dynamical Mordell-Lang Conjecture is an analogue of the classical Mordell-Lang conjecture in the context of arithmetic dynamics. This volume presents all known results of 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.