Powers in orbits of rational functions: cases of an arithmetic dynamical Mordell-Lang conjecture

Cahn, Jordan; Jones, Rafe; Spear, Jacob

doi:10.48550/arxiv.1512.03085

Cited by 2 publications

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“…There is an extensive literature proving various special cases of Conjecture 7.1; see for example [18,36,45,84,85,86,87,91,93,144,181,217,233,234,235,237]. Here are two notable examples.…”

Section: Intersections Of Orbits and Subvarietiesmentioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

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Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from p-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics. 1 2 BENEDETTO, ET AL. 22. Local-Global Questions in Dynamics 61 References 63 TRENDS AND PROBLEMS IN ARITHMETIC DYNAMICS 3background on dynamical systems, number theory, and algebraic geometry that we will need. As noted at the end of Section 3, we have attempted to make this article accessible by generally restricting attention to maps of projective space, thereby obviating the need for more advanced material from algebraic geometry. Our survey of arithmetic dynamics then commences in Section 4 and concludes in Section 22. These nineteen sections are mostly independent of one another, albeit liberally sprinkled with instructive cross-references.A Brief Timeline of the Early Days. The study of Galois groups of iterated polynomials was initiated by Odoni in a series of papers [177,178,179] during the mid-1980s. Beyond this, it appears that the first paper to describe dynamical analogues for a wide range of classical problems in arithmetic geometry did so not for polynomial maps or projective space, but rather for K3 surfaces admitting an automorphism of infinite order. The 1991 paper of Silverman [204] discusses dynamical analogues of various problems, including: (1) Uniform boundedness of periodic points;(2) Periodic points on subvarieties;(3) Lower bounds for canonical heights; (4) Open image of Galois groups; (5) Integral points in orbits. The rest of the 1990s saw an explosion of papers in which numerous researchers began to study a wide variety of problems in arithmetic dynamics. The serious study of p-adic dynamics starts with a 1983 paper of Herman and Yoccoz [107], two of the world's leading complex dynamicists, in which they investigate a p-adic analogue of a classical problem in complex dynamics. Later in the 1980s and 90s, the physics literature includes several papers [6,7,21,230] that discuss p-adic dynamics for polynomial maps, but the deeper study of p-adic dynamics really took off with the Ph.D. theses of Benedetto [24, 25] in 1998 and Rivera-Letelier [190, 191] in 2000. The dynamics of iterated p-adic power series was investigated by Lubin [156] in a 1994 paper. We also mention that there is an extensive literature on ergodic theory over p-adic fields which, although not within the purview of this survey, dates back to at least 1975 [180]. Abstract Dynamical SystemsA discrete dynamical system consists of a set S and a self-map

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“…There is an extensive literature proving various special cases of Conjecture 7.1; see for example [18,36,45,84,85,86,87,91,93,144,181,217,233,234,235,237]. Here are two notable examples.…”

Section: Intersections Of Orbits and Subvarietiesmentioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

Self Cite

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“…More generally, if A, B and X satisfy (3), then curves (1) have a factor of genus zero for any rational function U , which is a "compositional left factor" of the function A •l •X for some l ≥ 0, where by a compositional left factor of a holomorphic map f : R 1 → R 2 between Riemann surfaces, we mean any holomorphic map g : R ′ → R 2 such that f = g • h for some holomorphic map h : R 1 → R ′ . Indeed, it follows from (3) and…”

Section: Introductionmentioning

confidence: 97%

“…Our approach to Problem 1.1 is based on the ideas and methods described in these papers. Roughly speaking, our main result states that, unless A belongs to a special family of functions, all corresponding solutions U of Problem 1.1 can be obtained as described above from some fixed semiconjugacy (3), where X is a Galois covering that depends only on A. Moreover, for "most" rational functions A, this Galois covering X A is equal simply to the identity map.…”

Section: Introductionmentioning

confidence: 99%

“…Describe the pairs of complex rational functions A and U of degree at least two such that for every d ≥ 1, the algebraic curve The motivation for this problem comes from the arithmetic dynamics. Specifically, in [3], the following problem was investigated: which rational functions A defined over a number field K have a K-orbit containing infinitely many distinct mth powers of elements from K ? If such an orbit exists, then for every d ≥ 1, the algebraic curve (2) A •d (x) − y m = 0 has infinitely many K-points, implying by the Faltings theorem that it has a factor of genus zero or one.…”

Section: Introductionmentioning

confidence: 99%

“…The paper [3], based on painstaking calculations of the possible ramifications of rational functions A such that every curve (2) has a factor of genus zero or one, provides a very explicit description of such functions. In contrast, our approach is more geometric and provides an answer in the general case in terms of semiconjugacies and Galois coverings.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic Curves A ∘ l ( x ) − U ( y ) = 0 and Arithmetic of Orbits of Rational Functions

Pakovich¹

2020

MMJ

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We give a description of pairs of complex rational functions A and U of degree at least two such that for every d ≥ 1 the algebraic curve A •d (x) − U (y) = 0 has a factor of genus zero or one. In particular, we show that if A is not a "generalized Lattès map", then this condition is satisfied if and only if there exists a rational function V such that U • V = A •l for some l ≥ 1. We also prove a version of the dynamical Mordell-Lang conjecture, concerning intersections of orbits of points from P 1 (K) under iterates of A with the value set U (P 1 (K)), where A and U are rational functions defined over a number field K.

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Powers in orbits of rational functions: cases of an arithmetic dynamical Mordell-Lang conjecture

Cited by 2 publications

References 0 publications

Current trends and open problems in arithmetic dynamics

Current trends and open problems in arithmetic dynamics

Algebraic Curves A ∘ l ( x ) − U ( y ) = 0 and Arithmetic of Orbits of Rational Functions

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