The Arithmetic of Dynamical Systems Joseph H. Silverman Brown University Kuwait Lecture | Cambridge University Tuesday, February 20, 2007 0 What Is Dynamics? What Is Dynamics? Abstractly, a (Discrete) Dynamical System is a map Á : S ¡! S from a set to itself. Dynamics is the study of the be- havior of the points in S under iteration of the map Á. We write n Á = |Á ± Á ±{zÁ ¢ ¢ ¢ Á} n iterations for the nth iterate of Á and © 2 3 ª OÁ(®) = ®; Á(®);Á (®);Á (®);::: for the (forward) orbit of ® 2 S. A primary goal in the study of dynamics is to classify the points of S according to the behavior of their orbits. The Arithmetic of Dynamical Systems { 1{ What Is Dynamics? Rational Maps In this talk I will concentrate on rational functions d d¡1 F (z) adz + ad¡1z + ¢ ¢ ¢ + a1z + a0 Á(z) = = e e¡1 G(z) bez + be¡1z + ¢ ¢ ¢ + b1z + b0 that de¯ne maps Á : P1 ¡! P1 on the projective line, although I will also mention maps Á : PN ¡! PN on higher dimensional projective spaces, and possibly maps Á : V ! V on other algebraic varieties. The degree of the rational map Á is the larger of d and e, where ad 6= 0 and be 6= 0. The rational maps in this talk will be assumed to have degree at least 2. The Arithmetic of Dynamical Systems { 2{ A Soup»conof Classical Dynamics A Soup»conof Classical Dynamics Periodic and Preperiodic Points A Periodic Point is a point that returns to where it started, Á Á Á Á ® ¡! Á(®) ¡! Á2(®) ¡!¢ ¢ ¢ ¡! Án(®) = ®: The smallest such n is the Period of ® Another interesting set of points consists of those with a ¯nite orbit, #OÁ(®) < 1: These are called Preperiodic Points. We write Per(Á) and PrePer(Á) for the sets of periodic and preperiodic points of Á. Periodic and preperiodic points are de¯ned by purely algebraic properties. Things get more interesting when we add in a topology or a metric. The Arithmetic of Dynamical Systems { 3{ A Soup»conof Classical Dynamics The Chordal Metric on P1(C) A natural metric on the complex projective plane is the chordal metric, which is de¯ned by identifying P1(C) with the Riemann sphere. (0,0,1) z* z Each point z 2 P1(C) = C [ 1 is identi¯ed with a point z¤ on the sphere. The chordal metric is 1 jz ¡ wj ½(z; w) = jz¤ ¡ w¤j = p p : 2 jzj2 + 1 jwj2 + 1 The Arithmetic of Dynamical Systems { 4{ A Soup»conof Classical Dynamics The Fatou and Julia Sets Some points ® 2 P1(C) have the property that if ¯ starts close to ®, then all of the iterates Án(¯) stay close to the corresponding Án(®). Informally, the Fatou Set of Á is the set of all ® 2 P1(C) satisfying ¯ close to ® =) Án(¯) close to Án(®) for all n ¸ 0: The Julia Set of Á is the complement of the Fatou set. Points in the Julia set tend to move away from one an- other under iteration of Á. They behave chaotically. More formally, the Fatou set Fatou(Á) is the largest 1 n open subset of P (C) on which the set of iterates fÁ gn¸0 is an equicontinuous family of maps. A major goal of complex dynamics is to describe the Fatou and Julia sets for various types of maps. The Arithmetic of Dynamical Systems { 5{ What Is Arithmetic Dynamics? What Is Arithmetic Dynamics? Arithmetic Dynamics and Arithmetic Geometry Arithmetic Dynamics is the study of number the- oretic properties of dynamical systems. The transposi- tion of classical results from arithmetic geometry and the theory of Diophantine equations to the setting of discrete dynamical systems leads to a host of new and interesting questions. Although there is no precise dic- tionary connecting the two areas, the following associa- tions give a flavor of the correspondence: Diophantine Dynamical Equations Systems rational and integral rational and integral Ã! points on varieties points in orbits torsion points on periodic and preperiodic Ã! abelian varieties points of rational maps The Arithmetic of Dynamical Systems { 6{ What Is Arithmetic Dynamics? Two Motivating Problems in Arithmetic Dynamics Let Á 2 Q(z). Rationality of Periodic Points To what extent can the periodic or preperiodic points of Á be in Q? The Diophantine analogue is Q-ration- ality of torsion points on elliptic curves (Mazur, Merel) and abelian varieties. Integral Points in Orbits To what extent can orbit OÁ(®) contain in¯nitely many integer points? The Diophantine analogue is integral points on curves (Siegel) and on higher dimensional va- rieties (Faltings, Vojta). More generally, one can ask the same questions for mor- phisms Á : PN ¡! PN de¯ned over a number ¯eld K. The Arithmetic of Dynamical Systems { 7{ Rationality of Periodic Points Rationality of Periodic Points Northcott's Theorem Theorem. (Northcott 1949) Let Á : PN ! PN be a morphism de¯ned over a number ¯eld K. Then PrePer(Á) \ PN(K) is ¯nite. In particular, a rational function Á(z) 2 Q(z) of degree d ¸ 2 has only ¯nitely many rational peri- odic points. The proof is an easy application of the theory of heights. Northcott's theorem is the dynamical analog of the fact that if A=K is an abelian variety, then the torsion sub- group A(K)tors is ¯nite. The Arithmetic of Dynamical Systems { 8{ Rationality of Periodic Points The Uniform Boundedness Conjecture It is natural to ask: How large can we make the set PrePer(Á) \ PN(K)? (¤) It is easy to make (¤) large if we allow deg(Á) or N or K to grow. Otherwise we have the. Uniform Boundedness Conjecture. (Morton{ Silverman) There is a constant C = C(D; d; N) so that for all ¯elds K=Q of degree D and all mor- phisms Á : PN ¡! PN of degree d de¯ned over K we have # PrePer(Á) \ PN(K) · C: The Arithmetic of Dynamical Systems { 9{ Rationality of Periodic Points Consequences of the Uniform Boundedness Conjecture The Uniform Boundedness Conjecture for (D; d; N) = (1; 4; 1) implies Mazur's Theorem: There is an absolute constant C so that for all elliptic curves E=Q, #E(Q)tors · C: The implication is trivial, using¡ the¢ rational¡ map¢ Á de- termined by the equation¡ Á¢x(P ) = x [2]P and the fact that Etors = PrePer [2] . Fakhruddin has shown the Uniform Boundedness Con- jecture implies uniform boundedness for abelian varieties: Conjecture. For all ¯elds K=Q of degree D and all abelian varieties A=K of dimension N, #A(K)tors · C(D; N): The Arithmetic of Dynamical Systems { 10{ Rationality of Periodic Points 2 Uniform Boundedness for Ác(z) = z + c Quadratic polynomials 2 Ác(z) = z + c provide the ¯rst nontrivial cases of dynamical systems and have been used as a testing ground since the genesis of the subject. Even for this family, little is known about rationality of periodic points. Here is our meagre state of knowledge: n Ác may have Q-rational point of period n? 1 YES | 1 parameter family 2 YES | 1 parameter family 3 YES | 1 parameter family 4 NO (Morton) 5 NO (Flynn{Poonen{Schaefer) ¸ 6 ??? The Arithmetic of Dynamical Systems { 11{ Rationality of Periodic Points Dynamical Modular Curves Consider the moduli problem of classifying pairs © 2 ª (c; ®): ® has exact period n for Ác(z) = z + c : This is analogous to the classical moduli problem of classifying pairs (E; P ), where E is an elliptic curve and P is a point of exact period n on E. dyn As in the classical case, there is an a±ne curve Y1 (n) that (almost) solves the dynamical moduli problem. We can summarize the table on the previous slide: dyn » dyn » dyn » 1 Y1 (1) = Y1 (2) = Y1 (3) = P ; dyn dyn Y1 (4)(Q) = Y1 (5)(Q) = ;: Unfortunately, there does not seem to be a dynamical analog for the ring of Hecke operators, which is so im- portant in studying elliptic modular curves. The Arithmetic of Dynamical Systems { 12{ Integral Points in Orbits Integral Points in Orbits Integral Points in Orbits A famous theorem of Siegel says that a polynomial equa- tion f(x; y) = 0 has only ¯nitely many solutions (x; y) 2 Z if the asso- ciated Riemann surface has genus at least 1. A natural dynamical analog: Given Á(z) 2 Q(z) and ® 2 Q, can ¡ ¢ # OÁ(®) \ Z = 1? The answer is clearly \Yes". For example, if Á(z) 2 Z[z] and ® 2 Z, then OÁ(®) ½ Z. Similarly, if 2 Á(z) = 1=zd, then Á2(z) = zd , so OÁ(®) may contain in¯nitely many integer points. The Arithmetic of Dynamical Systems { 13{ Integral Points in Orbits Integral Points in Orbits More generally, there may be in¯nitely many integer n points in OÁ(®) if any iterate Á (z) is a polynomial. However, it turns out. Proposition. Let Á(z) 2 C(z). If Án(z) 2 C[z] for some n ¸ 1, then already Á2(z) 2 C[z]. The proof is a nice exercise using the Riemann-Hurwitz genus formula. Ruling out polynomial iterates leads to the correct state- ment: Theorem. Let Á(z) 2 Q(z) with Á2(z) 2= Q[z] and let ® 2 Q. Then ¡ ¢ # OÁ(®) \ Z is ¯nite. The Arithmetic of Dynamical Systems { 14{ Integral Points in Orbits A Re¯ned Integrality Result Let Á(z) 2 Q(z) and ® 2 Q, and write A Án(®) = n 2 Q Bn as a fraction in lowest terms. Theorem. Assume that Á2(z) 2= Q[z] and that 1=Á2(1=z) 2= Q[z]: Then log jA j lim n = 1: n!1 log jBnj The proof is an adaptation of Siegel's argument, with extra complications due to the fact that the map Á is rami¯ed.