A classification of degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math> semi-stable rational maps <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>ℙ</mml:mi> <mml:mn>2</mml:mn> </mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>ℙ</mml:mi> <mml:mn>2</mml:mn> </mml:msup></mml:mrow></mml:math> with large finite dynamical automorphism group

Silverman, Joseph H.

doi:10.5802/afst.1614

Cited by 4 publications

(5 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…d corresponding to maps f whose automorphism group Aut(f ) is non-trivial. See [168] for a detailed analysis for N = 1 and [159] for some partial results for N = d = 2. For example,…”

Section: Dynamical Moduli Spacesmentioning

confidence: 99%

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

“…d corresponding to maps f whose automorphism group Aut(f ) is non-trivial. See [168] for a detailed analysis for N = 1 and [159] for some partial results for N = d = 2. For example,…”

Section: Dynamical Moduli Spacesmentioning

confidence: 99%

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

“…since the only change between the definition (8) of End N d [P] and the definition (43) of W is that we have dropped the requirement that the points P v be distinct. The use of the inverse image Ψ −1 v,Φ(v) (∆) in (43) still forces the geometric points of W to satisfy (45).…”

Section: Formal Models and Stability Of End N D [P]mentioning

confidence: 99%

“…Example 9.5. Let W be the scheme defined by (43). We proved that End N d [P] is contained in W , but we want to give an example showing that they need not be equal.…”

Section: Formal Models and Stability Of End N D [P]mentioning

confidence: 99%

See 1 more Smart Citation

Moduli Spaces for Dynamical Systems with Portraits

Doyle,

Silverman

2018

Preprint

Self Cite

View full text Add to dashboard Cite

A portrait P on P N is a pair of finite point sets Y ⊆ X ⊂ P N , a map Y → X, and an assignment of weights to the points in Y . We construct a parameter space End N d [P] whose points correspond to degree d endomorphisms f : P N → P N such that f : Y → X is as specified by a portrait P, and prove the existence of the GIT quotient moduli spacerelative to an appropriately chosen line bundle. We also investigate the geometry of M N d [P] and give two arithmetic applications.

show abstract

“…(c) It is a standard fact that if Γ ⊂ P 2 is a curve with the property that f (Γ) is a point, then necessarily Γ ⊆ Crit(f ); see for example [6,Lemma 23(c)]. We compute…”

Section: A Bounded Height Examplementioning

confidence: 99%

Degeneration of dynamical degrees in families of maps

Silverman

Call

2018

Acta Arith.

Self Cite

View full text Add to dashboard Cite

The dynamical degree of a dominant rational map f : P N P N is the quantity δ(f ) := lim(deg f n ) 1/n . We study the variation of dynamical degrees in 1-parameter families of maps f T . We make a conjecture and ask two questions concerning, respectively, the set of t such that:and δ(g t ) < δ(g T ) for "independent" families of maps. We give a sufficient condition for our conjecture to hold and prove that the condition is true for monomial maps. We describe non-trivial families of maps for which our questions have affirmative and negative answers.

show abstract

A classification of degree 2 semi-stable rational maps ℙ 2 →ℙ 2 with large finite dynamical automorphism group

Cited by 4 publications

References 24 publications

Current trends and open problems in arithmetic dynamics

Current trends and open problems in arithmetic dynamics

Moduli Spaces for Dynamical Systems with Portraits

Degeneration of dynamical degrees in families of maps

Contact Info

Product

Resources

About