Preperiodic points and unlikely intersections

Baker, Matthew; DeMarco, Laura

doi:10.1215/00127094-1384773

Cited by 85 publications

(176 citation statements)

References 28 publications

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“…The following lemma appears as [GHT1,Proposition 7.6], but we include a proof for completeness. The arguments are similar to our proof of [BD,Lemma 3.2].…”

Section: Preliminary Definitionssupporting

confidence: 76%

“…In our case, the first few terms in the expansion of ψ are for some constant depending on s, where g s and R both depend on t. Now, recall that a i (t) lies in the domain of ϕ t for all t sufficiently large, by Lemma 5.2 (and the comment following the proof). From the traditional distortion arguments applied to ϕ t (applying the estimate |b 2 | ≤ 2; see [BD,Lemma 3.2] or [BH1, Section 3]), the region {z: G t (z) > d M(f t )} lies outside the disk of radius R t for all t large. And so we may assume that a i (t) lies outside the disk of radius R t for all t large.…”

Section: Preliminary Definitionsmentioning

confidence: 99%

“…Theorem 1.3 is an extension of the results [BD,Theorem 1.1] and [GHT1,Theorem 2.3], where stronger hypotheses guaranteed that the symmetries {h t } must be trivial. The article [GHT2] is closely related, showing that (1) ⇐⇒ (2) for certain families of rational functions.…”

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confidence: 94%

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Special Curves and Postcritically Finite Polynomials

Baker¹,

Marco²

2013

Forum math. Pi

Self Cite

106

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We study the postcritically finite maps within the moduli space of complex polynomial dynamical systems. We characterize rational curves in the moduli space containing an infinite number of postcritically finite maps, in terms of critical orbit relations, in two settings: (1) rational curves that are polynomially parameterized; and (2) cubic polynomials defined by a given fixed point multiplier. We offer a conjecture on the general form of algebraic subvarieties in the moduli space of rational maps on P 1 containing a Zariski-dense subset of postcritically finite maps.2010 Mathematics Subject Classification: 37F45 (primary); 11G50, 30C10 (secondary) OverviewIn this paper we address the question 'Which algebraic subvarieties of the moduli space M d of degree-d rational maps contain a Zariski-dense set of special points?' Here, a special point is the conjugacy class of a postcritically finite map f : P 1 → P 1 ; that is, every critical point of f has finite forward orbit under iteration. Postcritically finite (PCF) maps play an important role in complex dynamics, and in recent years there has been an explosion of work around them. It has been known since the foundational work of Thurston based on Teichmüller theory that PCF maps come in two flavors, the flexible Lattès maps (associated with elliptic curves and arising in one-dimensional families) and the rest (which are rigid). The rigid PCF maps form a countable Zariski-dense subset of the moduli space M d . From an arithmetic point of view, PCF maps are in several ways similar to elliptic curves with complex multiplication, and one can therefore view the above question as a dynamical analog of the André-Oort conjecture in arithmetic geometry. We formulate a conjectural answer to the above question: the special subvarieties V having a dense set of special points should M. Baker and L. DeMarco 2 be those for which the number of dynamically independent critical points does not exceed the dimension of V.As evidence for our general conjecture, we study rational curves within the parameter space of critically marked, monic, centered, degree-d polynomials. Our main result provides an explicit description of the polynomially parameterized rational curves that are special: they are those for which there is exactly one active critical orbit, up to polynomial symmetries. (We in fact prove a more general result about marked but not necessarily critical points that are simultaneously preperiodic.) We provide examples showing that the 'up to symmetries' condition is necessary, and we illustrate how one can check that a given curve is special. We also study the family of curves Per 1 (λ) inside the space of cubic polynomials. First introduced by Milnor, Per 1 (λ) is defined as the set of maps with a fixed point of multiplier λ. The curve Per 1 (0), defined by the condition that one critical point is fixed, is special, and we prove that Per 1 (λ) is not special for all λ = 0.The proofs of these results rely on several ingredients, including: (1) an arithmetic equidistr...

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“…The following lemma appears as [GHT1,Proposition 7.6], but we include a proof for completeness. The arguments are similar to our proof of [BD,Lemma 3.2].…”

Section: Preliminary Definitionssupporting

confidence: 76%

Section: Preliminary Definitionsmentioning

confidence: 99%

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Special Curves and Postcritically Finite Polynomials

Baker¹,

Marco²

2013

Forum math. Pi

Self Cite

106

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“…Theorem 11.1 (Baker-DeMarco, 2011, [9]). For t ∈ C, let f t (x) = x 2 + t be the usual quadratic polynomial.…”

Section: Unlikely Intersections In Dynamicsmentioning

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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“…Rather than listing these analogues case by case, we instead describe one recent beautiful result due to Baker and DeMarco [3] that is a dynamical analogue of Masser and Zannier's result on unlikely intersections in varying families. For c ∈ C * , let f c (z) = z 2 + c, and write f…”

Section: Book Reviewsmentioning

confidence: 99%

Book Review: Some problems of unlikely intersections in arithmetic and geometry

Silverman¹

2012

Bull. Amer. Math. Soc.

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What makes an intersection likely or unlikely? A simple dimension count indicates that two curves in the plane are likely to intersect, while two curves in three-space are unlikely to intersect. Indeed, in the complex projective plane, Bezout's theorem says that two algebraic curves always intersect. Moving down a dimension, it is unlikely that a curve C in the plane will intersect a given point P . More generally-and this is where the real interest begins-the intersection of C with a countable collection of "special" points {P i } i≥1 is unlikely to be large unless C itself is also special in some way.A prototype for the subject of unlikely intersections is the following theorem in which the special points are those whose coordinates are roots of unity and the special curves are translates of subgroups of the torus C * 2 . This result was conjectured by Serge Lang and proved by Ihara, Serre, and Tate in the 1960s.

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Preperiodic points and unlikely intersections

Cited by 85 publications

References 28 publications

Special Curves and Postcritically Finite Polynomials

Special Curves and Postcritically Finite Polynomials

Current trends and open problems in arithmetic dynamics

Book Review: Some problems of unlikely intersections in arithmetic and geometry

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