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The dynamical Mordell-Lang conjecture. (English) Zbl 1362.11001
Mathematical Surveys and Monographs 210. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2408-4/hbk; 978-1-4704-2908-9/ebook). xiii, 280 p. (2016).
The dynamical Mordell-Lang (DML) conjecture lies at the interface of algebraic geometry and dynamics. Given a discrete-time dynamical system over a variety, this conjecture predicts how the points of the orbits should intersect sub-varieties. These intersections are characterised by a sequence of return times to the sub-variety, and the conjecture states that such return times form a finite union of arithmetic progressions. (By allowing progressions with zero common difference, one accounts for one-off visits to the variety.)
The DML conjecture is attracting considerable attention, and this is the first research monograph on this topic. The name derives from an extension of the classical Mordell conjecture (on rational points on curves), which states that if \(X\) is a semiabelian variety defined over \(\mathbb{C}\), and \(V\) is a subvariety, then the intersection of \(V(\mathbb{C})\) and a finitely generated subgroup \(\Gamma\) of \(X(\mathbb{C})\) is a finite union of cosets of subgroups of \(\Gamma\).
The story begins with a tour of seemingly disconnected arithmetical questions (on linear recurrence sequences, Diophantine equations, arithmetic geometry, etc.), re-stated in terms of recurrence times; the latter are proved – or believed – to be the union of finitely many arithmetic progressions. Such a simple unifying theme provides ample motivations. The rigid constraint on recurrence times may puzzle a dynamicist; in a system with positive entropy, recurrence times to closed subsets are fairly unconstrained, and examples may be found where any pre-assigned sequence of return times to certain subsets is matched by an orbit. But then one quickly learns that here ‘topological space’ is intended in the sense of Zariski, which an algebraic geometer hardly needs pointing out.
An early chapter supplies the necessary background in algebraic geometry, valuation theory, and analysis; it also devotes considerable attention to the Skolem-Mahler-Lech theorem, which features finitely many progressions in the return times to 0 for the elements of linear recurrence sequences. The DML conjecture proper is spelled out in a subsequent chapter, with an exhaustive account of its many incarnations and interconnections.
The idea of parametrisation of the points of an orbit pervades the book; it first appears in a reformulation of the Skolem-Mather-Lech theorem in geometric language, which brings it closer to the setting of the DML conjecture. Starting from an explicit formula for the terms of a linear sequence, one derives a \(p\)-adic analytic parametrisation of the orbit, and a representation of the recurrence times as zeros of \(p\)-adic analytic functions. The same idea is then extended and generalised, first to a class of regular maps (étale endomorphisms), then for rational maps on the projective line (building on work of Rivera-Letelier), and ending with parametrisations of some endomorphisms of higher-dimensional varieties, connected to older results of Herman and Yoccoz. Proofs of special cases or variants of the DML conjecture (for coordinate-wise action of \(N\) one variable rational maps, for Drinfeld modules, and in positive characteristic), are sprinkled across several chapters, together with a large number of accessory results, including heuristics on the limits of \(p\)-adic approach, and a survey of related problems in arithmetic dynamics.
Unavoidably, the exposition is somewhat fragmented in places, as it aims to provide a detailed account of many works on many fronts. It suffices to say that over a third of the 230 items in the bibliography date 2010 or later. These include recent results by Xie on the DML conjecture for endomorphisms of \(\mathbb{A}^2\), which use a real analytic parametrisation. Even though Xie’s works appeared when this monograph was about to go to press, the authors managed to squeeze in an account of it. This episode gives an idea of the fast pace of development in this area of research, to which this volume provides a timely, welcome, and comprehensive contribution.
I have one minor complaint: to number definitions, theorems, equations, and examples, the authors adopt a unified four-part integer code, which calls to mind IP addresses. Thus, at p. 214, I read: “But then ( and ( contradict condition (4) in theorem”. This numbering system is frighteningly logical, but it made me long for “Theorem A”, or “the Key Lemma”.

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11G25 Varieties over finite and local fields
11G35 Varieties over global fields
14G20 Local ground fields in algebraic geometry
14G25 Global ground fields in algebraic geometry
37P55 Arithmetic dynamics on general algebraic varieties
37P20 Dynamical systems over non-Archimedean local ground fields
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