Attracting cycles in p-adic dynamics and height bounds for postcritically finite maps

Benedetto, Robert L.; Ingram, Patrick; Jones, Rafe; Levy, Alon Y.

doi:10.1215/00127094-2804674

Cited by 33 publications

(59 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also note that an important corollary of Theorem 16.6 is that outside of the Lattès locus, the set of PCF maps in M 1 d is a set of bounded height. This weaker result had been proven earlier by Benedetto, Ingram, Jones, and Levy [22]. See also [83] for a recent quantitative version of Theorem 16.6.…”

Section: Conjecture 162 (Dynamical Lehmer Conjecturesupporting

confidence: 57%

“…In words, it says that if f has good reduction, then reduction commutes with both iteration and evaluation. 22 We discuss later why this definition is "naive." Proposition 12.5.…”

Section: Good Reduction Of Maps and Orbitsmentioning

confidence: 99%

“…For example, the Gauss point is fixed and wild for the map f (z) = z p ∈ C p (z), and every periodic residue class D(a, 1) is an attracting component. By contrast, if there are no wild points in P 1 Berk , for example if d < p or p = 0, then there are at most 2d − 2 cycles of attracting components; see [22,Corollary 1.6].…”

Section: P-adic and Nonarchimedean Dynamicsmentioning

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.

Self Cite

View full text Add to dashboard Cite

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

show abstract

Section: Conjecture 162 (Dynamical Lehmer Conjecturesupporting

confidence: 57%

“…In words, it says that if f has good reduction, then reduction commutes with both iteration and evaluation. 22 We discuss later why this definition is "naive." Proposition 12.5.…”

Section: Good Reduction Of Maps and Orbitsmentioning

confidence: 99%

Section: P-adic and Nonarchimedean Dynamicsmentioning

confidence: 99%

See 1 more Smart Citation

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The action of GL 2 (K) also preserves the logarithmic path distance: ρ(γ(x), γ(y)) = ρ(x, y) for all x, y ∈ H Berk and all γ ∈ GL 2 (K). For these and other facts, see ( [2]) and ( [3], [4], [11], [12], [14], [23]). …”

mentioning

confidence: 94%

“…This means there is a well-defined function ordRes ϕ (·) on the type II points in P Berk , given by (3) ordRes ϕ (γ(ζ G )) := ordRes(ϕ γ ) .…”

mentioning

confidence: 99%

The minimal resultant locus

Rumely

2015

Acta Arith.

View full text Add to dashboard Cite

Abstract. Let K be a complete, algebraically closed, nonarchimedean valued field, and let ϕ(z) ∈ K(z) have degree d ≥ 2. We give an algorithm to determine whether ϕ has potential good reduction over K, based on a geometric reformulation of the problem using the Berkovich Projective Line. We show the minimal resultant is is either achieved at a single point in P 1 Berk , or on a segment, and that minimal resultant locus is contained in the tree in P 1 Berk spanned by the fixed points and poles of ϕ. When ϕ is defined over Q the algorithm runs in probabilistic polynomial time. If ϕ has potential good reduction, and is defined over a subfield2 such that ϕ has good reduction over L.

show abstract