Small Dynamical Heights for Quadratic Polynomials and Rational Functions

Benedetto, Robert L.; Chen, Ruqian; Hyde, Trevor; Kovacheva, Yordanka; White, Colin

doi:10.1080/10586458.2014.938203

Cited by 6 publications

(6 citation statements)

References 26 publications

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“…We note that there are examples in [34,39] showing that the values 7, 9, and 14 in Conjecture 4.7 are optimal.…”

Section: Uniform Boundedness Of (Pre)periodic Pointsmentioning

confidence: 81%

“…(D, N, d) and c 2 (D, N, d) such that for all number fields K/Q with [K : Q] ≤ D, all degree d morphisms f : P N → P N defined over K, and all points P ∈ P N (K) whose orbit O f (P ) is Zariski dense in P N , we haveĥ N, d). See [35,34] for numerical evidence supporting Conjecture 16.3 in the case N = D = 1 and d ∈ {2, 3}, including conjectural values for the constants. We also note that the Zariski density assumption in Conjecture 16.3 is necessary.…”

Section: Conjecture 162 (Dynamical Lehmer Conjecturementioning

confidence: 99%

“…It is a generalization of a conjecture that Serge Lang formulated for elliptic curves [140] and provides a different quantitative converse to (16.4). See [35,34] for numerical evidence supporting Conjecture 16.3 in the case N = D = 1 and d ∈ {2, 3}, including conjectural values for the constants. We also note that the Zariski density assumption in Conjecture 16.3 is necessary.…”

Section: Canonical Heightsmentioning

confidence: 99%

“…Conjecture 4.7 (Benedetto, Chen, Hyde, Kovacheva, White, 2014, [34]; Blanc-Canci-Elkies, 2015, [39]). Let f ∈ End 1 2 (Q) be a rational map of degree 2 defined over Q.…”

Section: Uniform Boundedness Of (Pre)periodic Pointsmentioning

confidence: 99%

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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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“…We note that there are examples in [34,39] showing that the values 7, 9, and 14 in Conjecture 4.7 are optimal.…”

Section: Uniform Boundedness Of (Pre)periodic Pointsmentioning

confidence: 81%

Section: Conjecture 162 (Dynamical Lehmer Conjecturementioning

confidence: 99%

Section: Canonical Heightsmentioning

confidence: 99%

“…Conjecture 4.7 (Benedetto, Chen, Hyde, Kovacheva, White, 2014, [34]; Blanc-Canci-Elkies, 2015, [39]). Let f ∈ End 1 2 (Q) be a rational map of degree 2 defined over Q.…”

Section: Uniform Boundedness Of (Pre)periodic Pointsmentioning

confidence: 99%

See 2 more Smart Citations

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…In 2007, Benedetto [2] proved for the case of polynomial maps of degree d ≥ 2 that | PrePer(φ, K)| is bounded by O(|S| log |S|) , where S is the set of places of K at which φ has bad reduction, including all archimedean places of K. The big-O is essentially d 2 −2d+2 log d for large |S|. Many other results have been proven in recent years [3], [6] , [11], [15].…”

Section: Conjecture 11 (Uniform Boundedness Conjecture)mentioning

confidence: 98%