On Functional Graphs of Quadratic Polynomials

Mans, Bernard; Sha, Min; Shparlinski, Igor E.; Sutantyo, Daniel

doi:10.1080/10586458.2017.1391725

Cited by 8 publications

(4 citation statements)

References 30 publications

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“…See also [19] for an affirmative answer to Question 18.7 for N = 1 assuming an independence hypothesis. For material on the connectivity of the graphs Γ x 2 +c as c ranges over F q , see [136]; and for results on the number of non-isomorphic graphs Γ f as f ranges over polynomials in F q [x] of fixed degree, see [160].…”

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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“…. .. Mans et al [2017] find the average number of cyclic points for quadratic polynomials to be about the average size of a longest cycle, namely, close to 2n/π. For n = 500 000, this evaluates to about 564.…”

Section: Introductionmentioning

confidence: 85%

Iteration entropy

Gathen¹

2018

Math. Comp.

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“…Later, Mans et al [5] provided algorithms on quadratic polynomials over finite fields to approximate the number of connected functional graphs, the number of graphs having a maximal cycle, the number of cycles of fixed size and the number of components of fixed size.…”

mentioning

confidence: 99%

“…Mans et al [5] suggested that almost all of the functional graphs generated by the polynomial f (t) = t 2 + a ∈ F q [t] are weakly connected.…”

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confidence: 99%

Hamiltonian cycles of equational graphs over finite fields

Chaifongsri

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Let F q be the finite field of q elements. For any equation E(X, Y ) = 0 over F q , the equational graph of this equation is a digraph whose vertex set is F q and for x, y ∈ F q there is the edge from x to y if E(x, y) = 0. In this work, we assume that q − 1 ≥ k and work on the equational graph G (k) (λ, f ) associated with the equationwith variables X and Y , where f (t) ∈ F q [t] and λ is an element in F × q of order at least k. We study strongly connectivity and the existence of Hamiltonian cycle of the graph G (k) (λ, f ). Moreover, we classify the equational graph G (3) (λ, f ) up to isomorphism where f (t) is a permutation polynomial in F q [t] and find some conditions for the existence of components with small vertices.

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On Functional Graphs of Quadratic Polynomials

Cited by 8 publications

References 30 publications

Current trends and open problems in arithmetic dynamics

Current trends and open problems in arithmetic dynamics

Iteration entropy

Hamiltonian cycles of equational graphs over finite fields

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