Wreath Products and Proportions of Periodic Points

Juul, Jamie; Kurlberg, Pär; Madhu, Kalyani; Tucker, Thomas J.

doi:10.1093/imrn/rnv273

Cited by 38 publications

(46 citation statements)

References 19 publications

(19 reference statements)

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“…As explained in [105], it seems unlikely that such a result holds in general, so we pose the problem of characterizing those maps for which there is such a result. There are a handful of published results giving upper bounds for the size of the orbit Of (P mod p), including [105,129,201]. To illustrate, we quote an easily stated corollary from one of these papers.…”

Section: Question 182 (Vague Motivating Question) To What Extent Domentioning

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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Section: Question 182 (Vague Motivating Question) To What Extent Domentioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

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“…For Corollary 2, we observe as above that the condition of the theorem holds under the assumption (3). The choice r = [(log log p)/(log 4)] − 1 will satisfy (3) when p A,C 1 and the theorem then yields # f r (F p ) p/r .…”

mentioning

confidence: 79%

“…His result does not include an explicit dependence on r . Juul et al [3] handled general rational functions rather than restricting to quadratic polynomials. Their emphasis is on the reductions Before discussing the implications of the theorem, let us examine the condition that f i (0) = f j (0) for 0 i < j r .…”

Section: Iteration Of Quadratic Polynomials Over Finite Fields 1043mentioning

confidence: 99%

Iteration of Quadratic Polynomials Over Finite Fields

Heath‐Brown¹

2017

Mathematika

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Abstract. For a finite field of odd cardinality q, we show that the sequence of iterates of a X 2 +c, starting at 0, always recurs after O(q/log log q) steps. For X 2 +1, the same is true for any starting value. We suggest that the traditional "birthday paradox" model is inappropriate for iterates of X 3 + c, when q is 2 mod 3. §1. Introduction. Let f (X ) ∈ F q [X ] and define the iterates f j (X ) by setting f 0 (X ) = X and f j+1 (X ) = f ( f j (X )). Let m ∈ F q and consider the sequence of values f 0 (m), f 1 (m), f 2 (m), . . . . Since the field F q is finite, the sequence eventually recurs and one enters a closed cycle. We are interested in the questions: how long is it before one enters the cycle? How long is the cycle? In general we can construct a directed graph f = f (F q ), whose vertices are the elements m of F q and with edges (m, f (m)). The trajectory f 0 (m), f 1 (m), f 2 (m), . . . then consists of a pre-cyclic "tail", followed by a cycle.Linear polynomials are easily handled. When f (X ) = X + b, one has f j (X ) = X + bj, so that if b = 0 the cycles are singleton sets and, if b = 0, then f is a union of cycles of length p, the characteristic of the field. For linear polynomials f (X ) = a X + b with a = 0, 1, one finds that

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“…[50,38]. Let us mention one further result [29,Thm. 1.5 & Example 7.2]: for the graph of a quadratic polynomial with integer coefficients, the value of lim inf p→+∞ #{x ∈ F p belongs to a cycle of D f mod p }/p is 0 for x 2 + 1 but 1/4 for x 2 − 2.…”

Section: Introductionmentioning

confidence: 97%

Dynamically affine maps in positive characteristic

Byszewski¹,

Cornelissen²,

Houben³

2020

Dynamics: Topology and Numbers

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We study fixed points of iterates of dynamically affine maps (a generalisation of Lattès maps) over algebraically closed fields of positive characteristic p. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function of the dynamical system: it is either rational or non-holonomic, depending on specific characteristics of the map. We also study the algebraicity of the so-called tame zeta function, the generating function for periodic points of order coprime to p. We then verify these hypotheses for dynamically affine maps on the projective line, generalising previous work of Bridy, and, in arbitrary dimension, for maps on Kummer varieties arising from multiplication by integers on abelian varieties.

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Wreath Products and Proportions of Periodic Points

Cited by 38 publications

References 19 publications

Current trends and open problems in arithmetic dynamics

Current trends and open problems in arithmetic dynamics

Iteration of Quadratic Polynomials Over Finite Fields

Dynamically affine maps in positive characteristic

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