Primitive Divisors in Generalized Iterations of Chebyshev Polynomials

Transcription

1 University of Colorado, Boulder CU Scholar Mathematics Graduate Theses & Dissertations Mathematics Spring Primitive Divisors in Generalized Iterations of Chebyshev Polynomials Nathan Paul Wakefield University of Colorado at Boulder, Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Wakefield, Nathan Paul, "Primitive Divisors in Generalized Iterations of Chebyshev Polynomials" (2013). Mathematics Graduate Theses & Dissertations This Dissertation is brought to you for free and open access by Mathematics at CU Scholar. It has been accepted for inclusion in Mathematics Graduate Theses & Dissertations by an authorized administrator of CU Scholar. For more information, please contact

2 Primitive Divisors in Generalized Iterations of Chebyshev Polynomials by Nathan Paul Wakefield B.S., Metropolitan State College of Denver, 2006 M.A., University of Northern Colorado, 2008 M.S., University of Colorado, 2012 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics 2013

3 This thesis entitled: Primitive Divisors in Generalized Iterations of Chebyshev Polynomials written by Nathan Paul Wakefield has been approved for the Department of Mathematics Su-Ion Ih Katherine Stange Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.

4 iii Nathan Paul Wakefield, (Ph.D., Mathematics) Primitive Divisors in Generalized Iterations of Chebyshev Polynomials Thesis directed by Prof. Su-Ion Ih Let (g i ) i 1 be a sequence of Chebyshev polynomials, each with degree at least two, and define (f i ) i 1 by the following recursion: f 1 = g 1, f n = g n f n 1, for n 2. Choose α Q such that {g n 1 (α) : n 1} is an infinite set. The main result is as follows: Let γ {0, ±1}, if f n(α) = An B n written in lowest terms, then for all but finitely many n > 0, the numerator, A n, has a primitive divisor; that is, there is a prime p which divides A n but does not divide A i for any i < n. In addition to the main result, several of the tools developed to prove the main result may be of interest. A key component of the main result was the development of a generalization of canonical height. Namely: If [f] is a set of rational maps, all commuting with a common function f, and f = (f i ) i=1 is a generalized iteration of rational maps formed by f n(x) = g n (f n 1 (x)) with g i coming from [f], then there is a unique canonical height function ĥf : K R which is identical to the canonical height function associated to f. Another key component of the main result was proving that under certain circumstances, being acted upon by a Chebyshev polynomial does not lead to significant differences between the size of the numerator and denominator of the result. Specifically, let γ {0, ±1, ±2} be fixed, and g i be a sequence of Chebyshev polynomials. Let f given by the following recurrence f 1 (z) = g 1 (z), and f i = g i (f i 1 (z)) for i 2. Pick any α Q with α + γ < 2, such that α + γ is not pre-periodic for one hence any Chebyshev polynomial. Write f n (α + γ) γ = An B n log A n lim n log B n = 1. Finally, some areas of future research are discussed. in lowest terms. Then is

5 iv Acknowledgements While mere words cannot describe the gratitude I have for all those who have helped me, I would like to take this opportunity to especially thank a few people. I would like to start by thanking my wife for her support, advice, and encouragement. I owe a lot of debt to my family for spending time working with the child who could not read. Thank you for your patience and taking the time to work one-on-one with me. I would also like to thank my advisor, Su-Ion Ih, for his patience, insight, and help. I am especially thankful for Katherine Stange who has served as my second reader and provided many useful comments and suggestions. I would also like to thank Robert Tubbs, Eric Stade, and Juan Restrepo for serving on my committee. Finally I would like to thank the Author and Finisher of my faith.

6 Contents Chapter 1 Introduction Dynamics Primitive Divisors Summary of Results Generalized Iterations and Canonical Heights Heights Canonical Heights in Generalized Iterations Chebyshev Polynomials Chebyshev Polynomials Heights, Canonical Heights, and Generalized Iteration of Chebyshev Polynomials The Chordal Metric Approximating Algebraic Numbers Wandering Points Rate of Growth of Height Under Generalized Iteration of Chebyshev Polynomials The Growth of Numerator and Denominator Some Bounds

7 vi 5 Primitive Divisors in Random Arithmetic Dynamics of Chebyshev Polynomials Prime Divisors and Chebyshev Polynomials Some Bounds When Subtracting a Constant Some Manipulations The Zsigmondy-type Theorem A Slight Extension Closing Remarks, Further Research, and Possible Extensions Lucas Sequences and Chebyshev polynomials The Case of Chebyshev Polynomials Shifted Chebyshev Polynomials With Non-Zero Shift Further Research Commuting Maps and Reduction Total Ramification and Commuting Maps A Direction For Further Generalizations of the Tools Closing Bibliography 87 Appendix A Sample Mathematica Code 89 A.0.1 Basic Functions A.0.2 Generating Examples

8 vii Tables Table 3.1 The first 12 Chebyshev Polynomials Upper bounds on ord p ( ˆT d (x))

9 viii Figures Figure 1.1 Diagram of Dependencies Plot of successive values of h(f n (α)) given α = 1/2 for 1 n Plot of successive values of h(f n (1/2))/ deg(f n ) for 1 n Orbit of a generalized iteration given α = 1/ Plot of 6.1 Graph of y = 6.2 Graph of y = 6.3 Graph of y = log An log B n for α = 1/ log An log B n for α = 1/2, and T n (α) = An B n log An log B n for α = 7/3 and T n (α) = An B n log An log B n for α = 11/5, and T n (α) = An B n

10 Chapter 1 Introduction This thesis explains some arithmetic properties of Chebyshev polynomials, specifically, arithmetic properties of compositions of Chebyshev polynomials. The Chebyshev polynomials are a collection of orthogonal polynomials which can be defined recursively. A common use of Chebyshev polynomials is in approximation theory. However, Chebyshev polynomials also come from the multiplicative group of a field. The Chebyshev polynomials have a lot of underlying structure and provide a convenient domain in which examples, and tests of conjectures can be made. This thesis is the direct result of testing one such conjecture in the field of arithmetic dynamics. 1.1 Dynamics Let X be a set and φ a map from X to X. A dynamical system is the set X together with the function φ. In the study of Arithmetic Dynamics one studies arithmetic properties of a dynamical system. Let n be a positive integer, denote by φ n the n-fold composition of φ with itself. That is φ n := φ φ... φ, }{{} n and φ 0 is simply the identity map. We define the forward orbit of a point α under φ to be the set {α, φ(α), φ 2 (α),...}. In dynamics it is common to classify points into three groups based on their orbits. A point α is called a fixed point for φ (denoted by Fix(φ)), if α = φ(α); a point is called a periodic point of φ (denoted by Per(φ)), if there is some n 1 such that φ n (α) = α. One may note

11 2 that a fixed point is, in fact, a periodic point with n = 1. A point is called pre-periodic for φ (denoted by PrePer(φ)), if there is some i j both positive integers such that φ i (α) = φ j (α). Finally, a point α is called a wandering point for φ if it is not pre-periodic. One should note that Fix(φ) Per(φ) PrePer(φ), thus a wandering point is a point which is not fixed, periodic or pre-periodic. Wandering points will be the primary points of interest for this thesis. 1.2 Primitive Divisors The questions we will be addressing have to do with primitive divisors. If A = (A n ) n 1 is a sequence of integers, then a prime divisor p is said to be primitive for A m if p A m but p A i for all 1 i < m. Zsigmondy sets give a means of describing where primitive divisors do not occur. The Zsigmondy set of a sequence A is the set of positive integers n for which A n does not have a primitive divisor, Z(A) := {n 0 : A n does not have a primitive divisor }. Toward the end of the 19th century Bang [2] and Zsigmondy [27] showed that for integers a, b with a > b > 0, the Zsigmondy set attached to the sequence (a n b n ) n 1 is a finite set. Carmichael [4] considered primitive divisors of Lucas sequences, and later Schinzel [21] further addressed primitive divisors of Lucas sequences over algebraic number fields. Primitive divisors have been an important area of study in the field of arithmetic dynamics ([6][7][8][11][16][19][24]). In general, given a sequence x i = φ i (x i ), and x 1 given, primitive divisors of the numerator of x i are studied. However, there has been some research on primitive divisors of the denominator of a sequence [7]. In the case of Chebyshev polynomials, the denominator is not of interest because the denominator will have the form of a power map. In [7] the authors examine primitive divisors in the sequence x n+1 x n following the proof used in [11], yet avoiding the use of Roth s Theorem. As an interesting application Faber and Granville use their result to show that given q an odd prime, there are infinitely many primes of the form q n k + 1.

12 3 Doerksen and Haensch [6] looked at the appearance of primitive divisors in the orbit of zero under iteration of φ(x) = x d + c for c, d Z. The authors look at questions that were addressed by Rice [19] in the case of polynomials. In [19], Rice indicated an effective bound on the terms of the sequence without a primitive divisor could be computed when considering quadratic polynomials. However, Rice does not compute these bounds. Doerksen and Haensch find bounds in the special case of φ(x) = x d +c for c, d Z. In particular, they were able to show that if zero was a wandering point, then after only two iterations primitive divisors always appeared. A tool in the work of Doerksen and Haensch is rigid divisibility. For a more complete treatment on rigid divisibility we direct the reader to [20]. Krieger [16] further extends primitive divisors of φ(x) = x d + c by allowing c Q. Krieger shows that the number of terms without a primitive divisor in the zero orbit of φ is bounded above by 23. In [11] it was proved that under certain conditions the Zsigmondy set attached to iteration was finite. Specifically, Ingram and Silverman proved the following theorem: Theorem (Ingram, Silverman). Let φ(z) Q(z) be a rational function of degree d 2 such that φ(0) = 0, but φ does not vanish to order d at z = 0. Let α Q be a point with infinite orbit under iteration of φ. For each n 1 and write φ n (α) = A n B n Q as a fraction in lowest terms. Then the dynamical Zsigmondy set Z((A n ) n 0 ) is finite. Motivated by the results of Ingram and Silverman one may want to ask if the results can be generalized any further. One generalization by Gratton, Nguyen, and Tucker has been approached using the abc conjecture [8]. The authors address primitive divisors in fields in which the abc conjecture holds. Specifically, the authors show that in an abc field, a primitive divisor theorem holds without the restriction that φ(0) = 0. In [24], Silverman further generalizes the ideas in [8]. In remarks following one conjecture, Silverman introduces the idea of primitive divisors of the

13 4 sequence of numerators of φ n ψ n, where φ and ψ are two maps. The introduction of multiple maps into arithmetic dynamics allows for many more generalizations. One such generalization is what if the map φ is changed at each iteration? i.e. let g i be a sequence of maps, pick an initial value x 1, and define x n = g n (x n 1 ); what conditions on g i and x 1 ensure that a primitive divisor exists? Are there interesting examples where such a primitive divisor theorem exists? The following examples help to illustrate that there are examples where a primitive divisor theorem exists. Example: Let g i be defined as follows: g 1 = x 3 3x, g 2 = x 4 4x 2 + 2, g 3 = x 2 2, g 4 = x 2 2, g 5 = x 5 5x 3 + 5x, g 6 = x 4 4x 2 + 2, g 7 = x 6 6x 4 + 9x 2 2, g 8 = x 4 4x 2 + 2, g 9 = x 2 2, g 10 = x 6 6x 4 + 9x 2 2, g 11 = x 6 6x 4 + 9x 2 2 Furthermore, let x 1 = 1/2, and A n = Numerator(x n ), then A 1 = 1 11 A 2 = A 3 = A 4 = A 5 = In particular, each number in red represents a primitive divisor. Similarly, if g 1 = x 5 5x 3 + 5x, g 2 = x 3 3x, g 3 = x 4 4x 2 + 2, g 4 = x 2 2, g 5 = x 4 4x 2 + 2, g 6 = x 2 2, g 7 = x 4 4x 2 + 2, g 8 = x 2 2, g 9 = x 6 6x 4 + 9x 2 2, g 10 = x 5 5x 3 + 5x, g 11 = x 5 5x 3 + 5x,

14 5 and x 1 = 1/2, then A 1 =61 A 2 = A 3 = A 4 = A 5 = The numerical evidence seems to suggest a primitive divisor theorem may exist in these specific cases, and there are some theoretical reasons that one might expect a primitive divisor theorem to exist. To understand the theoretical reasons we look at the proof of Theorem 1.2.1, and ask what were the key steps involved? The proof of Theorem has two main components. The first component is to prove that the sequence grows rapidly. The second component is to prove that once a prime divides a specific element of the sequence, a very large power of that prime cannot divide a further element of the sequence. In the case of Theorem the original sequences are sequences of rational numbers, the proof requires analyzing the growth, not of the rational number, but of the numerator. To this end Ingram and Silverman use height and canonical height functions. However, if φ and ψ are commuting maps then they, in fact, have identical canonical height. A natural extension of the above theorem is to form an analogue for a set of commuting maps. Conjecture (Ih-Wakefield). Let K be a number field and φ(x) K(x) be a nonzero rational function of degree greater than 1. Define [φ] K to be the set of nonzero rational functions of degree greater than 1 commuting with φ under composition, and defined over K. Suppose that α K is a wandering point for φ, and γ K is a pre-periodic point of φ. Further suppose that no element

15 6 of [φ] K is totally ramified at γ. For any ϕ [φ] K, let the ideal (ϕ(α) γ) = A ϕ B 1 ϕ be written as a quotient of relatively prime integral ideals of O K (if ϕ(α) = then let A ϕ = (1) and B ϕ = (0)). Then the dynamical Zsigmondy set Z((A ϕ ) ϕ [φ]k ) is finite. More precisely, for any ϕ [φ] K with deg ϕ 1, A ϕ has a primitive prime divisor, i.e., a prime divisor not dividing A ϕ for any ϕ [φ] K with deg ϕ < deg ϕ. In addition to the conjecture there are several additional questions which may be of interest: It is also interesting to ask for a slightly different requirement, i.e., the existence of a prime divisor of A ϕ not dividing A ϕ for any ϕ [φ] K {ϕ} with deg ϕ deg ϕ. We can also appropriately change the definition of [φ] K to enlarge this set to an interesting set, e.g., the set of all rational maps of degree at least 2 defined over K whose pre-periodic points coincide with those of φ. Then we predict a similar statement of the primitive divisor property for future study. In the case of replacing the pre-periodic point γ with a wandering point like α itself, many things remain open. Although, this does seem to be of the flavor of [24] and [8] in the context of generalized iteration. In any case neither the conjecture nor any of the questions following the conjecture have been completely analyzed. However, Ritt and Eremenko ([23]) have previously shown that two rational commuting maps of degree at least two are either Common iterates of each other, Both power maps, Both Lattes maps, or Both Chebyshev Polynomials.

16 The main theorem of this thesis will be to prove a primitive divisor theorem in the case of Chebyshev Polynomials. Specifically, we set out to prove the following. 7 Theorem (Main Theorem). Let f = (f i ) i=1 be a generalized iteration of Chebyshev polynomials, and α Q a wandering point. Zsigmondy set Z((A n ) n 1 ) is finite. If f n (α) = An B n is written in lowest terms, then the dynamical Summary of Results To prove the main theorem several key steps from the results of [11] will need to be generalized. In Chapter 2 we develop the tools needed to generalize the notion of canonical heights. For a beginning treatment of heights and canonical heights the reader may want to reference [23]. The key result of Chapter 2 is given in the following theorem: Theorem. If [f] is a set of rational maps all commuting with a given map f, and f = (f i ) i=1 is a generalized sequence of rational maps formed by f n (x) = g n (f n 1 (x)) with g i [f], then there is a unique canonical height function ĥf : K R. Furthermore, this canonical height function is equal to the canonical height ĥf associated with f. The proof of this result requires defining the notion of a bounded sequence of maps. With a notion of a bounded sequence of maps in hand, generalized iterations of commuting maps are shown to be bounded. This allows the use of [15] to find a canonical height. Chapter 3 is concerned with properties of Chebyshev polynomials, and how Chebyshev polynomials behave under generalized iteration. An early but important result of chapter 3 is the proof that the canonical height developed in chapter 2 is given by h(f n (β)) ĥ(β) := lim n deg(f n ) where f n is a generalized iteration of Chebyshev polynomials. Chapter 3 concludes with a series of results that together show the following: Theorem. Let f = (f i ) i=1 be a generalized iteration of Chebyshev polynomials, then not only is there a rational number α such that f i (α) f j (α) for i j, but {α : f i (α) = f j (α)} = {0, ±1, ±2}.

17 8 In chapter 4 it is shown that when a generalized iteration of Chebyshev polynomials acts on a rational number with absolute value less than two, then the numerator and denominator grow at approximately the same rate. Specifically: Theorem. Let γ {0, ±1, ±2} be fixed, and g i be a sequence of Chebyshev polynomials. Let f be given by the following recurrence f 1 (z) = g 1 (z), and f i = g i (f i 1 (z)) for i 2. Pick any α Q with α < 2, and no i j exist where f i (α) = f j (α). Write f n (α + γ) γ = An B n in lowest terms. Then the limit log A n lim n log B n = 1. This result is analogous to Theorem C of [22]. The proof relies on Roth s Theorem, and as such a rate of convergence is unknown. The method of proof is similar to that of Theorem 3.48 of [23]. Chapter 5 contains the proof of the main result. The proof of the main result can be essentially reduced too two main statements. (1) The size of the numerator of the sequence grows quickly. (2) Once a prime number divides an element of the sequence, that prime raised to a very large power cannot divide a later element of the sequence. The first statement comes essentially in chapter 4 and the second statement is proved within chapter 5. In chapter 6 further results and future research is discussed. In particular, questions of analogous results in the case of other families of commuting maps are discussed. Theorem. Figure 1.1 below gives a flow chart showing dependencies of the various elements of the Main

18 9 Lem Cor Lem Lem Prop Thm Prop Lem Cor Lem Lem Thm Thm Lem Lem Lem Lem Cor Cor Thm Lem Lem Lem Lem Lem Lem Thm Table 5.1 Eqn Lem Lem Thm Lem Lem Lem Eqn Lem Eqn Eqn Main Theorem Figure 1.1: Diagram of Dependencies

19 Chapter 2 Generalized Iterations and Canonical Heights 2.1 Heights The ultimate goal of this thesis will be to prove a Bang-Zsigmondy result. However, to accomplish this goal a method of measuring the size of a point is required. The usual absolute value on C,, is in fact a measure of size. However, does not necessarily provide an accurate measure of the arithmetic complexity of a number. Given α Q and writing α = a/b with a, b relatively prime integers, we can consider max{ a, b } to be a measure of the arithmetic complexity of α. Over Q, this gives rise to the following definition. Definition Given α = a/b Q with a, b relatively-prime integers we define H(α) = max{ a, b }. Generalizing this definition to an arbitrary number field K creates a complication. The ring of integers is not always a principle ideal domain in an arbitrary number field K. The absence of a principle ideal domain creates problems when trying to generalize α = a/b with (a, b) = 1. A solution to the principle ideal domain problem comes from the theory of absolute values. A full discussion of absolute values is outside the scope of this material; however, for completeness a short discussion is included. Following [23], we begin with the absolute values on Q and then proceed to general number fields. The standard absolute values on Q will be denoted by M Q. Once an embedding of Q into R has been chosen, we associate one Archimedean absolute value

20 to Q denoted by x := max{x, x}, that is the usual absolute value on R. The other absolute values in M Q are each associated to a prime p. For a Z let 11 ord p (a) := exponent of the highest power of p dividing a. The non-archimedean absolute values in M Q are defined, for each prime p, as where a, b Z. a := b p 1 p ordp(a) ordp(b), For a K/Q a number field, the set of standard absolute values on K can be found by looking at all the absolute values on K which when restricted to Q are contained in M Q. These absolute values are simply denoted as M K. This gives rise to the notion of local degree as in [23]. For any absolute value ν M K on K, let K ν denote the completion of K at ν. The local degree of ν, denoted by n ν, is the quantity n ν = [K ν : Q ν ]. Using local degree we are in a position to define the height of a point P K. Let K be a number field, P K, and write the ideal (P ) as (P ) = UB 1 for relatively prime integral ideals U, B. Then the height of P relative to K is the quantity H K (P ) := max{1, P ν } nν. ν M K This now allows us to define the absolute height of a point P Q to be the quantity H(P ) := H K (P ) 1/[K:Q] where K is any number field such that P K. In [23], this quantity is shown to be well-defined independent of the choice of K. Using the absolute height, a relationship between the height of a point P before and after being acted upon by a rational function φ can be found. That is the following theorem holds.

21 Theorem (See, e.g. [23]). Let K be a number field and φ K(z) be a rational function of degree d. Then there are constants C 1, C 2 > 0, depending on φ, such that 12 C 1 H(P ) d H(φ(P )) C 2 H(P ) d for all P K. This theorem shows that the absolute height is a sort of multiplicative function. To work with an additive function, we introduce the following: Definition The absolute logarithmic height is the function h : Q R given by h(p ) = log H(P ) for P Q. Using the absolute logarithmic height we can begin to analyze the arithmetic dynamics of certain iterations. The words absolute logarithmic in absolute logarithmic height will be dropped when no confusion should arise. That is, we may simply refer to h(p ) as the height of P. Arithmetic dynamics is concerned with number theoretic questions with respect to the sequence f f... f where f is some function. In general the notation f n (x) represents the n-fold composition of the function f. An important tool in arithmetic dynamics is the canonical height ĥ f given by h(f n (P )) ĥ f (P ) := lim n deg(f) n where P is a point in the domain of f. In [23] the canonical height is shown to exist and satisfy ĥ f (P ) = h(p ) + O(1) (2.1.1) and ĥ f (f(p )) = dĥ(p ) (2.1.2)

22 where d is the degree of φ. Furthermore, it is shown that f i (P ) = f j (P ) for some i j integers 13 if and only if ĥf (P ) = 0. That is the canonical height gives an arithmetic characterization of pre-periodic points. 2.2 Canonical Heights in Generalized Iterations Using heights and canonical heights we can begin to analyze a special sequence coming from a family of rational maps. Let g i be an indexed family of rational maps; we do not disallow g i = g j for some i j. As in [3], a generalized iteration is an iteration of the form g k (g k 1 (...(g 1 (x)))). From this point forward let f = (f i ) i=1 denote such a sequence of rational maps formed by f n (x) = g n (f n 1 (x)) for n 2 and f 1 = g 1. Definition A sequence f = (f i ) i=1 is said to be bounded if is finite. c(f) := sup sup i 1 x K 1 h(f i (x)) h(x) deg f i Arithmetic dynamics has been traditionally concerned with iteration of a single function. The canonical height associated to a single function can be used to analyze the pre-periodicity of a point. We will show later that if f and g are two rational functions satisfying f g = g f, then ĥ f = ĥg, that is commuting maps have similar arithmetic dynamical properties. In [13], Kawaguchi developed the following theorem which will give a notion of canonical height on such a generalized iteration. Theorem (Kawaguchi). There is a unique way to attach to each bounded sequence f = (f i ) i=1 a height function ĥf : K R such that and sup ĥf (x) h(x) 2c(f) x K ĥ f (f 1 (x)) = deg(f 1 )ĥf (x)

23 14 The proof of this result does not require commutativity, and does depend on f = (f i ) i=1 being a bounded sequence of maps. We will focus on showing that commutativity gives this bound. From this point on ĥf (x) will be called the canonical height function for f = (f i ) i=1. The goal is now to show that when f = (f i ) i=1 comes from a set of commuting functions the canonical height function does in fact exist. To this end we wish to develop a bound for a generalized sequence of commuting functions, this will in turn show that canonical heights exist for a generalized iteration of commuting functions. Definition Imitating [15], the arithmetic distance between two rational functions φ, ψ will be defined to be the quantity ˆδ(φ, ψ) = sup ĥφ(p ) ĥψ(p ). P Q In [15] it is shown that ˆδ(φ, ψ) obeys the triangle inequality. That is for φ, ψ, ν rational maps, ˆδ(φ, ψ) ˆδ(φ, ν) + ˆδ(ν, ψ). We first remark that this quantity is in fact finite. If φ and ψ are to maps then by equation (2.1.1) ĥ φ (P ) = h(p ) + O φ (1), and ĥψ(p ) = h(p ) + O ψ (1). Therefore, ˆδ(φ, ψ) = sup ĥφ(p ) ĥψ(p ) P Q = sup h(p ) + O φ (1) h(p ) O ψ (1) P Q O φ (1) O ψ (1). We will further show that for commuting functions f and g, ˆδ(f, g) = 0. Equivalently, we prove that when f and g commute ĥf = ĥg. Lemma Suppose that f and g are two rational functions of degrees d f, d g 2 and f g = g f, then for x Q ĥ f (x) = ĥg(x).

24 Proof. Suppose that f and g are two rational functions with deg(f), deg(g) 2 and f g = g f, let x PrePer(f), then f l (x) = f m (x) for some l m. Thus 15 f l (g(x)) = g(f l (x)) = g(f m (x)) = f m (g(x)) Thus g(x) is also a pre-periodic point for f. Similarly for every k, g k (x) is pre-periodic for f, and so L = {x, g(x), g 2 (x)...} must have the property that L PrePer(f). However, by Northcott s theorem ([23] Theorem 3.12) PrePer(f) is a finite set. Thus L must be finite and so x PrePer(g). An identical argument gives PrePer(g) PrePer(f). So it must be that PrePer(f) = PrePer(g). Now [26] Theorem 3.1.2, states that if f, and g have identical pre-periodic points then ĥf (x) = ĥg(x). Therefore, ĥ f (x) = ĥg(x). Corollary Suppose that f and g are two rational functions of degree greater than or equal to two and f g = g f, then ˆδ(f, g) = 0. The arithmetic distance will play an important role in finding the bound in Theorem However, the arithmetic distance function requires two functions; we wish to find a distance function which is dependent only on a single function. To find such a function we turn to power functions, x x n. If φ is a power function then from the definition of height we have h(φ(α)) = deg(φ)h(α), and so in particular ĥφ(α) = h(α). Therefore, for any rational function f and point P. ˆδ(f, φ) = sup P = sup P ĥf (P ) ĥφ(p ) ĥf (P ) h(p ) That is the arithmetic distance between f and φ is independent of the degree of φ. This motivates the following definition.

25 Definition The arithmetic complexity of f, denoted ˆδ(f), is defined to be the quantity 16 ˆδ(f) := ˆδ(f, ψ) where ψ is any power map. A constant multiple of the arithmetic complexity of a map f will give the upper bound needed in Theorem However, the upper bound in Theorem was a function of f = (f i ) i=1 not any one particular map. To this end we show that for a commuting set {g i : i 1} ˆδ(g i ) = ˆδ(g j ). Lemma Suppose that f and g are two rational functions of degree greater than or equal to two, and f g = g f, then ˆδ(f) = ˆδ(g). Proof. Suppose that f and g are two rational functions and f g = g f, then ˆδ(f, g) = 0. Let ψ be a power map. Then by the triangle inequality ˆδ(f) = ˆδ(f, φ) ˆδ(f, g) + ˆδ(g, φ) = ˆδ(f, g) + ˆδ(g) = ˆδ(g) However, by an identical argument, ˆδ(g) ˆδ(f), thus ˆδ(f) = ˆδ(g). We now establish some notation regarding commuting functions. In particular, we define a set of functions commuting with a common function. The set of rational maps of degree at least two defined over some field K commuting with a function f will be denoted by [f] K := {g K(x) : g f = f g, deg(g) 2}. If the choice of K is clearly understood from context, we will simply write [f] for [f] K. Theorem Let K be a number field. If [f] is a set of rational maps and f = (f i ) i=1 is a generalized sequence of rational maps formed by f n (x) = g n (f n 1 (x)) with g i coming from [f], then there is a unique canonical height function ĥf : K R.

26 Proof. Let f = (f i ) i=1 be a generalized sequence of rational maps from [f]. Then for each f i f we have ˆδ(f) = ˆδ(f i ), thus sup h(f i (P )) P deg(f i ) h(p ) = sup h(p ) + ĥf i (P ) ĥf i (P ) h(f i(p )) P deg(f i ) = sup P h(p ) + ĥf i (f i (P )) deg(f i ) ĥf i (P ) h(f i(p )) deg(f i ) ĥf i (f i (P )) h(f i (P )) sup h(p ) ĥf i (P ) + sup P P deg(f i ) ĥf i (P ) h(p ) sup ĥf i (P ) h(p ) + sup P deg(f i ) ˆδ(f i ) + ˆδ(f i ) deg(f i ) ˆδ(f i ) + ˆδ(f i ) = 2ˆδ(f i ) = 2ˆδ(f) P 17 Thus c(f) is finite i.e. f is bounded, and so by Theorem there is a unique canonical height function ĥf : K Q. At this point we have established the existence of a canonical height function for a generalized sequence of rational maps commuting with a common function. Recall that for commuting functions f and g it holds that ĥf = ĥg, intuitively one might suspect that for f = (f i ) i=1, our generalized sequence, some similar relationship exists. Proposition Let f = (f i ) i=1 be a generalized sequence of rational maps from [f] whose degrees go to as the index goes to infinity. Then ĥf = ĥf. Proof. Let f = (f i ) i=1 be a generalized sequence of rational maps from [f] and c(f) the bound from Theorem Let α be given. Since f 1 f = f f 1 we have that ĥf 1 (α) = ĥf (α). Set d = deg(f 1 ).

27 18 Now let ɛ > 0 and M = max{ 2c(f) ɛ, 1}. Then for any i > M we have ĥf (α) h(f 1 i(α)) = 1 d i d i ĥ f (α) h(f1(α)) i d i = 1 d i ĥ f (f1(α)) i h(f1(α)) i by Theorem d i 2c(f) M d i ɛ < ɛ Thus lim i h(f i 1 (α)) d i = ĥf (α) and so ĥf (α) = ĥf 1 (α) = ĥf (α). For α K the quantity ĥf (α) is independent of f = (f i ) i=1, where f = (f i) i=1 is a generalized sequence of rational maps from [f] whose degrees go to infinity as the index goes to infinity. This motivates the following definition. Definition For α K we define ĥ[f] := ĥf (α) where f = (f i ) i=1 is a generalized sequence of rational maps from [f] whose degrees go to infinity as the index goes to infinity. In general one might simply work with ĥf (α), however, we use ĥ[f] to remind the reader that the canonical height function associated with general iteration is independent of the specific sequence. The canonical height only depends on the set of commuting maps from which the generalized iteration is formed. As in [23] the canonical height plays an important role in the analysis of the arithmetic of a generalized iteration. What follows will be an analysis of the arithmetic in the special case where each g i in the generalized iteration is a Chebyshev polynomial.

28 Chapter 3 Chebyshev Polynomials 3.1 Chebyshev Polynomials We start by defining T d (x) Z[x] to be the monic polynomial of degree d satisfying T d (x + x 1 ) = x d + x d. In [23] Proposition 6.6 these polynomials are shown to exist, be monic, commute under composition, and have the property that T d (T e (x)) = T de (x). The classical normalization T d ( x+x 1 2 ) = xd +x d 2 of the Chebyshev polynomials in [23] is related to these polynomials by the relationship T d (x) = 1 2 T d(2x). The classically normalized Chebyshev polynomials fail to be monic and in particular drop in degree modulo 2, thus it is desirable to work with the Chebyshev polynomials satisfying T d (x + x 1 ) = x d + x d. From this point on these polynomials will simply be called Chebyshev polynomials. That is, we have the following definition. Definition a) A Chebyshev Polynomial of degree d is the unique, monic polynomial of degree d satisfying T d (x + x 1 ) = x d + x d. b) For ease of notation, define C to be the set of Chebyshev polynomials. The first few Chebyshev polynomials are listed in Table 3.1, When studying iteration it is sometimes useful to be able to explicitly state how a map acts on a specific domain. In fact, this is possible for the Chebyshev polynomials. One important property of Chebyshev polynomials is that the set of points with absolute value less than two is mapped into

29 Table 3.1: The first 12 Chebyshev Polynomials 20 T 2 (x) = x 2 2 T 3 (x) = x 3 3x T 4 (x) = x 4 4x T 5 (x) = x 5 5x 3 + 5x T 6 (x) = x 6 6x 4 + 9x 2 2 T 7 (x) = x 7 7x x 3 7x T 8 (x) = x 8 8x x 4 16x T 9 (x) = x 9 9x x 5 30x 3 + 9x T 10 (x) = x 10 10x x 6 50x x 2 2 T 11 (x) = x 11 11x x 7 77x x 3 11x T 12 (x) = x 12 12x x 8 112x x 4 36x itself. This results follows from the characterization T d (α) = cos(d arccos(x)) for x [ 1, 1] found in [23]. We have the following lemma. Lemma If T d (x) is a Chebyshev polynomial of degree d, is the usual absolute value over C and α 2, then T d (α) 2. Proof. Let α 2, then T d (α) = T d (2α/2) = 2 T d (α/2) = 2 cos(d arccos(α/2)) 2 We now recall the idea of a generalized iteration, and apply the idea to C, the set of Chebyshev polynomials. Definition Fix an arbitrary sequence of Chebyshev polynomials, say (g i ) i 1. From this

30 21 sequence recursively define the following sequence of composite functions. f 1 = g 1 f i = g i f i 1, for i 2 Such a sequence is called a generalized iteration of Chebyshev polynomials. For notation we will write f := (f i ) i 1 when no confusion will arise. Given a generalized iteration of Chebyshev polynomials f, let α Q be a rational number with the property that f i (α) f j (α) for any i j. We now set out to prove the following result. Theorem (Main Theorem). Let f = (f i ) i=1 be a generalized iteration of Chebyshev polynomials. Let α Q be a rational number with the property that f i (α) f j (α) for any i j. If f n (α) = An B n is written in lowest terms then the dynamical Zsigmondy set Z((A n ) n 1 ) is finite. As described in [11], Zsigmondy-type theorems when reduced to essentials have two main components. The first component is to prove that the sequence grows rapidly. The second component is to prove that once a prime divides a specific element of the sequence a very large power of that prime cannot divide a further element of the sequence. Lemma will give a lower bound on elements of our sequence and thus prove that the sequence grows rapidly. The proof of lemma is itself relatively simple. However, the proof will rely heavily on Theorem and Theorem 4.1.2, these two theorems are analogues of results found in [22] for the case of generalized iterations of Chebyshev Polynomials. Lemma will give a method for controlling the power of the prime dividing elements of the sequence. The result in Lemma is very specific to Chebyshev Polynomials. We will now prove a series of lemmas and theorems toward the goal of proving the Main Theorem. The Main Theorem will be restated and proved at the conclusion of this paper. Before proceeding we need to establish some results involving heights and chordal metrics.

31 3.2 Heights, Canonical Heights, and Generalized Iteration of Chebyshev Polynomials 22 The final result will make significant use of height functions. For β Q we can write β = A B for relatively prime integers A and B. Using β = A B it is easy to see h(β) = log B + log max{1, β }. (3.2.1) Where h is the logarithmic height on Q (see Section 2.1). For the purposes of our results we will further use A = p p ordpa, when A is an integer, and is the usual absolute value on C. The canonical height function of Chapter 2 will eventually allow us to relate the height of a point to the dynamics of a generalized iteration. Before proceeding, consider the following example. Example: Recall the previous example where g i was defined as g 1 = x 3 3x, g 2 = x 4 4x 2 + 2, g 3 = x 2 2, g 4 = x 2 2, g 5 = x 5 5x 3 + 5x, g 6 = x 4 4x 2 + 2, g 7 = x 6 6x 4 + 9x 2 2, g 8 = x 4 4x 2 + 2, g 9 = x 2 2, g 10 = x 6 6x 4 + 9x 2 2, g 11 = x 6 6x 4 + 9x 2 2, g 12 = x 6 6x 4 + 9x 2 2 g 13 = x 3 3. Then for α = 1/2, and f n (α) a generalized iteration, Figure 3.1 shows how the height grows Figure 3.1: Plot of successive values of h(f n (α)) given α = 1/2 for 1 n 10.

32 23 One may notice the the height of f n (1/2) is growing rapidly. However, the degree of f 10 (α) is 276, 480. In other words while the height is growing rapidly the degree is also growing rapidly. In fact, this agrees with Theorem 3.11 from [11], specifically, if φ : P N (K) P M (K) is a morphism of degree d then there are constants C 1, C 2 > 0 depending only on φ, such that C 1 + dh(p ) h(φ(p )) C 2 + dh(p ) for all P P N (K). We recall that by Theorem there should be a canonical height function for this sequence. The standard canonical height for a single map φ can be given by h(φ n (α)) lim n deg(φ) n. Therefore, it would be convenient if the canonical height function associated to generalized iteration comes from something similar to h(f n (β)) lim n deg(f n ). The proof of this result will come shortly. However, a graphical representation is illustrative of the result. By dividing by the degree of the function at each step we can plot the limiting behavior of the canonical height, (Figure 3.2) Figure 3.2: Plot of successive values of h(f n (1/2))/ deg(f n ) for 1 n 12. With the example above in mind, we next establish the existence of a canonical height function for a generalized iteration of Chebyshev Polynomials. More generally, we establish canonical height

33 for any generalized sequence of commuting polynomials, and prove that the limit definition of this canonical height is appropriate. 24 Lemma Let f = (f i ) i=1 be a generalized iteration of polynomials from some set of commuting polynomials G. Then there is a canonical height function associated to f, ĥ G : Q R such that for any β Q, ĥg is given by The canonical height satisfies h(f n (β)) ĥ G (β) = lim n deg(f n ). (3.2.2) ĥ G = h + O(1) and (3.2.3) ĥ G (f(β)) = deg(f)ĥg(β), (3.2.4) where h is the logarithmic height, and f is any function in f. Proof. Consider the canonical height given by Definition and Theorem Let f be an element of the sequence f, d = deg(f), and β Q. Then ĥ G = ĥf, (3.2.5) where h(f n (α)) ĥ f (α) = lim n deg(f) n. Thus ĥ G (f(β)) = ĥf (f(β)) by (3.2.5), = dĥf (β) by Theorem 2.2.1, = dĥg(β) by (3.2.5).

34 Thus equation (3.2.4) is valid. Furthermore, Theorem ensures that equation (3.2.3) is 25 valid. What remains to be shown is that h(f n (β)) ĥ G (β) = lim n deg(f n ). From equation (3.2.3) we know that h(β) = ĥg(β) + O(1), so in particular for any f i we have h(f n (β)) deg(f n ) = ĥg(f n (β)) + O(1) deg(f n ) deg(f n ) = deg(f n)ĥg(β) deg(f n ) + O(1) deg(f n ) by (3.2.4). Thus, h(f n (β)) lim n deg(f n ) = lim deg(f n )ĥc(β) + 0 n deg(f n ) = ĥg(β). One might also note that we could also have called this ĥf. Since the specific sequence used is not important (see Definition 2.2.4) we adopt the notation ĥg to emphasize that the canonical height is independent of the specific sequence, but instead is dependent on the set G. 3.3 The Chordal Metric We recall that for an arbitrary field K, the projective line P 1 (K) can be constructed via the relationship P 1 (K) := K2 \{(0,0)} where (X, Y ) (X, Y ) if there is a u K such that X = ux and Y = uy. Definition We define the Chordal metric for both Archimedean and non-archimedean absolute values. a) The chordal metric associated to an Archimedean absolute value is given by ρ ([X 1, Y 1 ], [X 2, Y 2 ]) := X 1 Y 2 X 2 Y 1 X1 2 + Y 1 2 X2 2 + Y 2 2

35 26 b) The chordal metric associated to a non-archimedean absolute value ν is given by ρ ν ([X 1, Y 1 ], [X 2, Y 2 ]) := X 1 Y 2 X 2 Y 1 ν max{ X 1 2 ν, Y 1 2 ν} max{ X 2 2 ν, Y 2 2 ν} In [23], the Archimedean chordal metric is shown to satisfy the triangle inequality, and the non-archimedean chordal metric is shown to be an ultrametric. Let ρ be the chordal metric P 1 (C) associated to an Archimedean absolute, x, y C and suppose that ρ(x, y) 1 2ρ(y, 0). We set out to show ρ(x, y) 1 2 ρ(y, 0) = ρ(x, y) x y x y 1 2 ρ(y, 0)2. If x = 0 and y 0 then the result is vacuously true. If x = 0 and y = 0 then the result is trivially true. So assume x, y 0. By hypothesis ρ(x, y) 1 2ρ(y, 0), so, in particular, ρ(y, 0) ρ(x, y) 1 ρ(y, 0). (3.3.1) 2 Note that ρ(x, 0) = 1 ρ(x, y) = ( ) and ρ(x,0) 1 x = x x y x y x Thus = x y ρ(x, 0)ρ(y, 0) x y x y x y (ρ(y, 0) ρ(x, y)) ρ(y, 0) from the triangle inequality, x y 1 x y 2 ρ(y, 0)2 from equation Finally, multiplying x y on both sides gives ρ(x, y) x y x y 1 2 ρ(y, 0)2 Thus for all x, y C, ρ(x, y) 1 2 ρ(y, 0) = ρ(x, y) x y x y 1 2 ρ(y, 0)2. (3.3.2) A similar argument does in fact give the same result for the chordal metric associated to a non-archimedean absolute value. For the sake of completeness the argument is included here.

36 27 Let x, y C and suppose that ρ(x, y) 1 2ρ(y, 0). If x = 0 and y 0 then the result is vacuously true. If x = 0 and y = 0 then the result is trivially true. So assume x, y 0. Write x = x 1 /x 2, y = y 1 /y 2 with x 1, x 2, y 1, y 2 non-zero integers Note that, ρ(x, 0) = x 1 max{ x 1, x 2 } thus ρ(x, y) = x 1 y 2 y 1 x 2 max{ x 1, x 2 } max{ y 1, y 2 } = x 1y 2 y 1 x 2 ρ(x, 0)ρ(y, 0) x 1 y 1 = x y ρ(x, 0)ρ(y, 0). x y Now we can use the ultra-metric triangle inequality to get ρ(x, y) x y x y max{ρ(y, 0), ρ(x, y)}ρ(y, 0) and since ρ(x, y) 1 2ρ(y, 0) we have ρ(x, y) x y 1 x y 2 ρ(y, 0)2. Finally, since x y 0 we have ρ(x, y) x y x y 1 2 ρ(y, 0)2. Thus it is also the case that if ρ is the chordal metric associated to a non-archimedian absolute value on P 1 (C). Then for all x, y C ρ(x, y) 1 2 ρ(y, 0) = ρ(x, y) x y x y 1 2 ρ(y, 0)2. Using the chordal metric we can now talk about how the distance between points is affected by a map. In particular, given P, Q and a rational map φ we wish to better understand how the distance between P and Q φ 1 (Q) is related to the distance between φ(p ) and Q. The following lemma says that if φ(p ) and Q are close, then there is a point Q φ 1 (Q) that is close to P. How close depends on the ramification.

37 28 The ramification index (e α (φ)) of a map φ at a point α is given by e α (φ) := ord α (φ(z) φ(α)), assuming neither α = nor φ(α) =. Lemma (Lemma 3.51 from [23] ). Let φ : P 1 (C) P 1 (C) be a rational map of degree d 2, let ρ be the chordal metric, and for Q P 1 (C) define e Q (φ) := max e Q φ 1 Q (φ) (Q) to be the maximum of the ramification indices of the points in the inverse image of Q. Then there is a constant C = C(φ, Q), depending on φ and Q, such that min ρ(p, Q φ 1 (Q) Q ) eq(φ) Cρ(φ(P ), Q) for all P P 1 (C). To make use of Lemma we will need to understand the ramification of Chebyshev polynomials. In particular, we show that the Chebyshev polynomials are un-ramified at zero, replacing Lemma with a stronger statement for Chebyshev polynomials when applied to Q = 0. Lemma Let T d be a Chebyshev polynomial of degree at least two. Then e 0 (T d ) = 1. Proof. Let deg(t d ) = d > 1, and Q T 1 (0). Then d ( e Q (T d ) = ord Q Td (x) T d (Q ) ) = ord Q (T d (x) 0) = ord Q (T d (x)). To complete the proof it is enough to show that T d has d distinct roots. Let T d (x) = (x a) l h(x), where l 1, h(x) is a polynomial, and (x a) h(x). Recall that the classical normalization of the

38 29 Chebyshev polynomials is related to our Chebyshev polynomials by T d (x) = 1 2 T d(2x). However, it is well known that the classical Chebyshev Polynomials (normalized so they have bad reduction at 2, [23] page 322) are orthogonal with respect to the inner product (f, g) = x 2 f(x)g(x)dx. Furthermore, orthogonal polynomials have distinct roots. Thus T d (x) has d distinct real roots, see for example pages of [1]. Therefore, l = 1 and so e 0 (T d ) = 1. In fact, there are other related polynomials for which we can find a bound on e 0. Let γ {±1, ±2}, then the polynomials given by T d (x + γ) γ also satisfy a condition which is nearly as strong. Lemma Let γ {0, ±1, ±2}. If ˆTd (x) := T d (x + γ) γ, and α is a zero of ˆTd (x) with α + γ < 2, then e α ( ˆT d ) 2. Proof. Let α + γ < 2. For contradiction suppose that e α ( ˆT d ) > 2. As a transformation of a polynomial ˆTd (x) is itself a polynomial, therefore, ˆTd (α) = ˆT d (α) = ˆT d (α) = 0. However, ˆT d (x) = T d (x + γ) γ, and so in particular, ˆT d (x) = T d (x + γ) ˆT d (x) = T d (x + γ). Now recall the classical normalization of the Chebyshev Polynomials T d (x) on 1 x 1, is given by T d (cos(θ)) = cos(dθ). Furthermore, on all of R, T d (x) = 2T d ( 1 2 x).

39 30 Thus, ˆT d (x) = T d( 1 (x + γ)) 2 ˆT d T (x) = d( 1 2 (x + γ)). 2 Therefore, if ˆT d (α) = ˆT d (α) = 0, then it must be the case that T d( 1 (α + γ)) = 0 2 T d( 1 (α + γ)) = 0. 2 Now implicitly differentiating T d (cos(θ)) = cos(dθ) we have T d(cos(θ)) sin(θ) = d sin(dθ). Thus on 0 < θ < π, or 1 < cos(θ) < 1, T d(cos(θ)) = d sin(dθ) sin(θ). However, on 0 < θ < π, sin(dθ) has d 1 distinct zeros while sin(θ) has no zeros. Thus α+γ 2 can only have multiplicity 1 for T d(x) and so multiplicity 2 for T d (x). This contradicts T d( 1 2 (α+γ)) = 0. This, in fact, ensures that e 0 ( ˆT d ) Approximating Algebraic Numbers A primary tool in our proof of the main theorem is going to be to show that when a generalized iteration of Chebyshev maps acts on certain rational numbers α Z, then the numerator and denominator of the associated sequence grow at approximately the same rate. More specifically, we will eventually need to prove that when f n (α) = An B n log A n lim n log B n = 1. written in lowest terms the limit The proof of the above statement will follow by assuming the statement to be untrue and proceeding to show that this, in fact, contradicts Roth s Theorem. We first prove a result that is similar to Roth s theorem for the purely rational case and then proceed to state Roth s Theorem.

40 Lemma Fix c d with c Z and d 0 Z, then for all rational numbers a b where a 0 Z 31 with a b c d, we have a b c C d b 3 for some C only depending on d. Proof. Let c, d be fixed integers. Suppose that a b c d, then in particular ad bc 1 and so a b c = ad bc d bd 1 bd since ad bc 1 = 1 1 b d 1 1 b 3 d since b 1, C where C = 1 d is fixed. b 3 Theorem (Roth s Theorem [9]). Let K be a number field, let S M K be a finite set of absolute values on K, and assume that each absolute value in S has been extended in some way to K. Let α K and ɛ > 0 be given. Then there are only finitely many β K satisfying the inequality min{ β α ν, 1} ν S 1 H K (β) 2+ɛ. Corollary Let K be a number field and ν be an archimedian absolute value on K. Fix α K. Then there is some C > 0 depending only on α such that for any β K with β α we have β α ν C H K (β) 3. The above Corollary will ensure that two distinct points cannot be too close. We will use log A this result to reach a contradiction if lim n n log B n 1. There are, however, some obvious cases where this will not be true. For example if α Z, then T d (α) Z for any d. Further, if f = (f i ) i=1