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Small Dynamical Heights for Quadratic Polynomials and Rational Functions.1.1 Experimental Mathematics, 23:433–447, 2014 Copyright C Taylor & Francis Group, LLC ISSN: 1058-6458 print / 1944-950X online DOI: 10.1080/10586458.2014.938203 Small Dynamical Heights for Quadratic Polynomials and Rational Functions Robert L. Benedetto1, Ruqian Chen1, Trevor Hyde2, Yordanka Kovacheva3, and Colin White1 1Department of Mathematics and Statistics, Amherst College, Amherst, Massachusetts, USA 2Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA 3Department of Mathematics, University of Chicago, Chicago, Illinois, USA CONTENTS Let φ ∈ Q(z) be a polynomial or rational function of degree 2. A spe- 1. Introduction cial case of Morton and Silverman’sdynamical uniform boundedness 2. Background conjecture states that the number of rational preperiodic points of φ 3. Quadratic Polynomials is bounded above by an absolute constant. A related conjecture of 4. Quadratic Rational Functions Silverman states that the canonical height hˆφ (x) of a nonpreperiodic Acknowledgments rational point x is bounded below by a uniform multiple of the height Funding of φ itself. We provide support for these conjectures by computing References the set of preperiodic and small-height rational points for a set of degree-2 maps far beyond the range of previous searches. 1. INTRODUCTION In this paper, we consider the dynamics of a rational func- tion φ(z) ∈ Q(z) acting on P1(Q). The degree of φ = f/g is deg φ := max{deg f, deg g}, where f, g ∈ Q[z]haveno common factors. Define φ0(z) = z, and for every n ≥ 1, let Downloaded by [] at 10:38 03 August 2015 φn(z) = φ ◦ φn−1(z); that is, φn is the nth iterate of φ un- der composition. In this context, the automorphism group PGL (2, Q)ofP1(Q) acts on Q(z) by conjugation. The forward orbit of a point x ∈ P1(Q) is the set of iterates n O(x) = Oφ(x):={φ (x):n ≥ 0}. The point x is said to be periodic under φ if there is an inte- ger n ≥ 1 such that φn(x) = x. In that case, we say that x is n-periodic, we call the orbit O(x)ann-cycle, and we call n the period of x, or of the cycle. The smallest period n ≥ 1 of a pe- riodic point x is called the minimal period of x, or of the cycle. More generally, x is preperiodic under φ if there are integers n > m ≥ 0 such that φn(x) = φm(x). Equivalently, φm(x)is periodic for some m ≥ 0; also equivalently, the forward orbit 2000 AMS Subject Classification: Primary 37P35; Secondary 37P30, 11G50 O(x) is finite. We denote the set of preperiodic points of φ in Keywords: canonical height, arithmetic dynamics, preperiodic points P1(Q) by Preper(φ,Q). Address correspondence to Robert L. Benedetto, Department of Mathematics and Statistics, Amherst College, Amherst, MA 01002, USA. Using the theory of arithmetic heights, it was proved in Email: [email protected] [Northcott 50] that if deg φ ≥ 2, then φ has only finitely many 433 434 Experimental Mathematics, Vol. 23 (2014) preperiodic points in P1(Q). (In fact, Northcott proved a far infinitely many with a 2-periodic cycle having a tail of length more general finiteness result, for morphisms of PN over an 6; for the moment, however, these finiteness questions remain arbitrary number field.) Then a dynamical uniform bounded- open. ness conjecture was proposed in [Morton and Silverman 94, Meanwhile, in their study of the space of degree-two ra- Morton and Silverman 95]; for φ ∈ Q(z) acting on P1(Q), it tional functions with a rational point of period 6, Blanc et al. says the following. announced an infinite family of quadratic maps with 14 Q- rational preperiodic points; see [Blanc et al. 13, Lemma 4.7]. Conjecture 1.1. [Morton and Silverman 94] For every d ≥ 2, This family uses two separate orbits: a 6-cycle and a fixed there is a constant M = M(d) such that for every φ ∈ Q(z) of point, together with the preimages of all seven periodic points. degree d, Besides its preperiodic orbits, every rational function φ ∈ Q ≥ φ,Q ≤ . (z)ofdegreed 2 has an associated canonical height.The # Preper( ) M 1 canonical height is a function hˆφ : P (Q) → [0, ∞) satisfying the functional equation hˆφ(φ(z)) = d · hˆφ(z), and it has the Only partial results toward Conjecture 1.1 have been property that hˆφ(x) = 0 if and only if x is a preperiodic point proven, including nonuniform bounds of various strengths, of φ; see Section 2. as well as conditions under which certain preperiodic For a nonpreperiodic point y, on the other hand, hˆφ(y) mea- orbit structures are possible or impossible. See, for exam- sures how fast the standard Weil height h(φn(y)) of the iterates ple, [Benedetto 07, Call and Goldstine 97, Flynn et al. 97, of y increases with n. By analogy with Lang’s height lower Manes 07, Morton 92, Morton 98, Morton and Silverman 94, bound conjecture for elliptic curves, Silverman has asked how Morton and Silverman 95, Narkiewicz 89, Pezda 94, small hˆφ(y) can be for nonpreperiodic points y. More precisely, Poonen 98, Zieve 96], as well as [Silverman 07, Section 4.2]. considering φ as a point in the appropriate moduli space, and A sharper version of the conjecture for the special case of defining h(φ) to be the Weil height of that point, he stated the quadratic polynomials over Q was later stated in [Poonen 98]. following conjecture; see [Silverman 07, Conjecture 4.98] for a more general version. Conjecture 1.2. [Poonen 98] Let φ ∈ Q[z] be a polynomial of degree 2. Then # Preper(φ,Q) ≤ 9. Conjecture 1.3. [Silverman 07] Let d ≥ 2. Then there is a pos- itive constant M = M (d) > 0 such that for every φ ∈ Q(z) If true, Conjecture 1.2 is sharp; for example, z2 − 29/16 of degree d and every point x ∈ P1(Q) that is not preperiodic and z2 − 21/16 each have exactly nine rational preperiodic for φ, we have hˆφ(x) ≥ M h(φ). points, including the point at ∞. However, even though it is the simplest case of Conjecture 1.1, a proof of Conjecture 1.2 Conjecture 1.3 essentially says that the height of a non- Downloaded by [] at 10:38 03 August 2015 seems to be very far off at this time. preperiodic rational point must start to grow rapidly within a As little as we know about the uniform boundedness con- bounded number of iterations. Some theoretical evidence for jecture for quadratic polynomials, we know even less about the Conjecture 1.3 appears in [Baker 06, Ingram 09], and compu- conjecture for rational functions. The first systematic attack on tational evidence for polynomials of degree d = 2, 3 appears in preperiodic points of quadratic rational functions was made in [Benedetto et al. 09, Doyle et al. 13, Gillette 04]. The smallest [Manes 07], including a conjecture that # Preper(φ,Q) ≤ 12 known value of hˆφ(x)/h(φ) when φ is a polynomial of degree when φ(z) ∈ Q(z) has deg φ = 2. In this paper, we give exam- 2 occurs for x = 7/12 and φ(z) = z2 − 181/144. The first few ples with 14 rational preperiodic points, showing that Manes’s iterates of this pair (x,φ), first discovered in [Gillette 04], are conjecture is false. 7 →−11 → 5 →−13 →−1 →−5 → 11 On the one hand, we found a single map with a rational 7- 12 12 12 12 12 4 36 cycle, along with the immediate preimages of all seven points; →−377 → 2445 →··· . see equation (1–1). On the other hand, we found many maps 324 26244 φ6 with a rational point x whose sixth iterate (x) is 2-periodic; The small canonical height ratio the immediate preimages of all those preperiodic points again ˆφ ( / ) give a total of 14 points. We also found a single map with a h 7 12 ≈ . φ 0 0066 rational point x for which φ5(x) is 3-periodic, again giving a h( ) total of 14 points. See Table 1 for examples. It would appear makes precise the observation that although the numerators that there are only finitely many maps with a 7-cycle or with a and denominators of the iterates eventually explode in size, it 3-periodic cycle with a tail of length 5, while there seem to be takes several iterations for the explosion to get underway. Benedetto et al.: Small Dynamical Heights for Quadratic Polynomials and Rational Functions 435 φ Orbit Tail Period Total Length 330z2 − 187z − 143 1 11 3 55 13 3 ∞, 1, 0, − , − , − , − , − , − 53 8 330z2 + 1217z + 429 3 15 5 114 44 5 21z2 − 84z + 63 7 1 3 ∞, 1, 0, −3, , − , −7, − , −7628 21z2 − 16z − 21 3 3 2 52z2 − 30z − 22 1 3 4 9 4 ∞, 1, 0, − , − , −1, − , − , − 62 8 52z2 + 245z + 88 4 8 7 26 7 120z2 − 98z − 22 1 2 1 12 1 12 ∞, 1, 0, − , − , − , − , − , − 62 8 120z2 + 749z + 132 6 9 5 65 12 65 30z2 − 10z − 20 2 10 2 6 10 6 ∞, 1, 0, , , , , , 62 8 30z2 + 7z − 30 3 9 5 7 3 7 33z2 − 429z + 396 11 3 ∞, 1, 0, 3, , 5, 33, , 33 6 2 8 33z2 − 197z + 132 3 4 176z2 + 1397z − 1573 11 11 11 55 55 ∞, 1, 0, , − , − , , 2, 62 8 176z2 + 500z − 1144 8 2 4 16 16 1350z2 − 837z − 513 3 9 3 72 1 72 ∞, 1, 0, − , − , − , − , − , − 62 8 1350z2 + 5585z + 1710 10 10 5 175 6 175 700z2 − 95z − 605 11 5 7 5 11 5 ∞, 1, 0, − , − , − , − , − , − 62 8 700z2 + 1336z + 880 16 7 11 6 70 6 784z2 − 416z − 368 4 2 8 20 1 20 ∞, 1, 0, − , − , − , − , , − 62 8 784z2 + 3885z + 644 7 21 7 49 12 49 1428z2 − 1668z + 240 4 4 10 4 12 4 ∞, 1, 0, , − , , − , , − 62 8 1428z2 − 1723z + 900 15 21 21 7 17 7 308z2 + 19292z − 19600 28 28 308 40 308 ∞, 1, 0, − , −14, − , − , − , − 62 8 308z2 + 1937z + 7700 11 5 17 11 17 9009z2 − 17094z + 8085 7 77 35 63 77 63 ∞, 1, 0, − , , , , − , 62 8 9009z2 − 18454z − 10395 9 27 11 31 78 31 5712z2 − 5937z + 225 1 1 3 117 2 117 ∞, 1, 0, , , , , , 62 8 5712z2 − 137612z + 5400 24 26 68 2992 35 2992 Downloaded by [] at 10:38 03 August 2015 51480z2 + 910z − 52390 13 26 91 780 13 780 ∞, 1, 0, , − , − , , , 62 8 51480z2 + 275477z − 120900 30 5 11 253 36 253 24255z2 − 277830z + 253575 35 5 49 105 15 105 ∞, 1, 0, , − , − , , − , 62 8 24255z2 + 314788z + 65205 9 18 3 13 154 13 ∞,φ ∈ PM Q TABLE 1.
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