A Number Theorist's Perspective on Dynamical Systems

A Number Theorist's Perspective on Dynamical Systems

A Number Theorist's Perspective on Dynamical Systems Joseph H. Silverman Brown University <[email protected]> Frontier Lectures Texas A&M, Monday, April 4{7, 2005 Rational Functions and Dynamical Systems 1. Rational Functions and Dynamical Systems A rational function is a ratio of polynomials d d¡1 F (z) adz + ad¡1z + ¢ ¢ ¢ + a1z + a0 Á(z) = = e e¡1 : G(z) bez + be¡1z + ¢ ¢ ¢ + b1z + b0 ² The degree of Á is the larger of d and e, where ad 6= 0 and be 6= 0. ² The subject of Dynamical Systems is the study of iteration of func- tions Án(z) = Á ± Á ± Á ¢ ¢ ¢ Á(z): | {z } n iterations ² More precisely, start with a number ® and look at its orbit © 2 3 ª OÁ(®) = ®; Á(®);Á (®);Á (®);::: : ² One studies the iterates of Á by classifying and describing the dif- ferent sorts of orbits ² We generally assume that deg(Á) ¸ 2 . Frontier Lectures|April 4{7, 2005|Page 1 Rational Functions and Dynamical Systems A Simple Example Consider the function Á(z) = z2: ² Some points have orbits that head out to in¯nity, OÁ(2) = f2; 4; 16; 256;:::g; ² while others head in towards zero, ¡ ¢ © ª 1 1 1 1 OÁ 2 = 2 ; 4 ; 16 ;::: : ² Some points are ¯xed, OÁ(0) = f0; 0; 0;:::g and OÁ(1) = f1; 1; 1;:::g; ² while other points are eventually ¯xed OÁ(¡1) = f¡1; 1; 1; 1; 1;:::g: ² And if we use complex numbers, there are points that cycle, ³ p ´ n p p p p o ¡1+ ¡3 ¡1+ ¡3 ¡1¡ ¡3 ¡1+ ¡3 ¡1¡ ¡3 OÁ 2 = 2 ; 2 ; 2 ; 2 ;::: : Frontier Lectures|April 4{7, 2005|Page 2 Rational Functions and Dynamical Systems Fixed, Periodic, and Preperiodic Points ² A point ® is called periodic if Án(®) = ® for some n ¸ 1. The smallest such n is called the period of ®. ² If Á(®) = ®, then ® is a ¯xed point. ² A point ® is preperiodic if some iterate Ái(®) is perioidic. Equiva- lently, ® is preperiodic if its orbit OÁ(®) is ¯nite. ² A point ® that has in¯nite orbit is called a wandering point. The Example Á(z) = z2 1 ² 2 and 2 are wandering points. ² 0 and 1 are ¯xed points. ² ¡1 is a preperiodic point that is not periodic. p ¡1+ ¡3 ² 2 is a periodic point of period 2. Frontier Lectures|April 4{7, 2005|Page 3 Rational Functions and Dynamical Systems That Pesky Point \At In¯nity" ² For rational functions, the orbit of a point may include \in¯nity." For example, ² z2 + 1 Á(z) = has Á(1) = 1: (¤) z2 ¡ 1 ² But then what is Á2(1) = Á(1)? It is natural to set Á(1) = lim Á(z): z!1 ² So for the example (¤), we have Á(1) = 1 and OÁ(1) = f1; 1g: Thus 1 and 1 are periodic points of period 2. Similarly OÁ(¡1) = f¡1; 1; 1; 1; 1;:::g; so ¡1 is preperiodic. ² We want to treat this extra point \at in¯nity" exactly the same as every other point. In particular, points that are \close to in¯nity" should be close to one another. Frontier Lectures|April 4{7, 2005|Page 4 Rational Functions and Dynamical Systems One-Point Compacti¯cation of R and C ² There are many ways to describe the (one point) compacti¯cation of the real line (or of the complex plane). ² A nice pictorial method is to identify R with the points of the unit circle excluding the point (0; 1). The point (0; 1) then plays the role of the point at in¯nity. H (0;1) ³ 2 ´ '$Hr ¤ 2z z ¡1 H ¤ z = ; Hr z z2+1 z2+1 HHrz - HH &% ² It is not important to worry about the precise transformation for- mula. Just remember that a rational map Á : R ! R gives a map Á : R [ f1g ¡! R [ f1g; and that a small interval around a point in R is no di®erent from a small interval around the point 1. Frontier Lectures|April 4{7, 2005|Page 5 Dynamics and Chaos 2. Dynamics and Chaos ² Consider again the function Á(z) = z2: ² If we start with a point 0 < ® < 1, then the orbit 2 4 8 OÁ(®) = f®; ® ; ® ; ® ;:::g has the property that lim Án(®) = 0: n!1 ² Further, if we start with a point ¯ that is close to ®, then Án(¯) remains close to Án(®) as n ! 1. ² Similarly, if we start with a point ® > 1, then lim Án(®) = 1; n!1 and if we take a point ¯ that is close to ®, then Án(¯) remains close to Án(®) as n ! 1. [N.B. In our one-point compacti¯cation, points that are close to 1 are also close to each other.] Frontier Lectures|April 4{7, 2005|Page 6 Dynamics and Chaos Chaos and the Julia and Fatou Sets ² But look what happens if we take ® = 1 for Á(z) = z2. ² The point ® = 1 has a very simple orbit, since it is a ¯xed point. ² However, no matter how close we choose ¯ to ®, eventually Án(¯) moves far away from Án(®). ² This is an example of chaotic behavior. ² Informal De¯nition: A point ® is a chaotic point for Á if points that are close to ® do not remain close to one another when we apply the iterates of Á. ² The Julia set of Á is the set of chaotic points. Its complement is the Fatou set of Á. They are denoted J (Á) = f® where Á is chaoticg; F(Á) = f® where Á is not chaoticg: ² Formal De¯nition: The Fatou set is the largest open set on which the set of iterates fÁ; Á2;Á3;:::g is equicontinuous. Frontier Lectures|April 4{7, 2005|Page 7 Dynamics and Chaos The Julia Set and the Mandelbrot Set Theorem. (a) The Julia set is a closed set (b) The Julia set J (Á) is never empty (if we use complex numbers). In other words, every rational map has chaotic points. (c) All but ¯nitely many of the periodic points of Á are in the Julia set. Example. Even very simple functions such 2 Ác(z) = z + c have very complicated Julia sets. For example, there are some c values for which J (Ác) is connected (but usually fractal looking), while for other c values it is totally disconnected. The famous Mandelbrot set is the set M = fc 2 C : J (Ác) is connectedg: Another way to describe the Mandelbrot set is as the set of c such that 2 the orbit OÁc (0) = f0;Ác(0);Ác (0);:::g is bounded. Frontier Lectures|April 4{7, 2005|Page 8 A Number Theorist's View of Periodic Points 3. A Number Theorist's View of Periodic Points For a dynamicist, the periodic points of Á are the (complex) numbers satisfying an equation Án(z) = z for some n = 1; 2; 3;:::. A number theorist asks: What sorts of numbers may appear as periodic points? For example: Question: Can periodic points be rational numbers? The answer is obviously Yes. We've seen several examples. Question: How many periodic points can be rational numbers? That's a more interesting question. There are always in¯nitely many complex periodic points, and in many cases there are in¯nitely many real periodic points. But among the in¯nitely many periodic points, how many of them can be rational numbers? Frontier Lectures|April 4{7, 2005|Page 9 A Number Theorist's View of Periodic Points Northcott's Theorem Theorem. (Northcott 1949) A rational function Á(z) 2 Q(z) has only ¯nitely many periodic points that are rational numbers. Proof. Since every math talk should have one proof, I'll sketch for you the (relatively elementary) proof of Northcott's result. An important tool is the height of a rational number p=q written in lowest terms: ³p´ © ª H = max jpj; jqj : q Notice that for any constant B, there are only ¯nitely many rational numbers ® 2 Q with height H(®) · B. This makes the height a useful tool for proving ¯niteness results. Lemma. If Á(z) has degree d, then there is a constant C = CÁ > 0 so that ¡ ¢ H Á(¯) ¸ C ¢ H(¯)d for all rational numbers ¯ 2 Q. This is intuitively reasonable if you write out Á(z) as a ratio of polyno- mials. The tricky part is making sure there's not too much cancellation. Frontier Lectures|April 4{7, 2005|Page 10 A Number Theorist's View of Periodic Points Proof of Northcott's Theorem We apply the lemma repeatedly: ¡ ¢ H Á(®) ¸ C ¢ H(®)d ¡ ¢ ¡ ¢d 2 H Á2(®) ¸ C ¢ H Á(®) ¸ C1+d ¢ H(®)d ¡ ¢ ¡ ¢d 2 3 H Á3(®) ¸ C ¢ H Á2(®) ¸ C1+d+d ¢ H(®)d . ¡ ¢ ¡ ¢d 2 n¡1 n H Án(®) ¸ C ¢ H Án¡1(®) ¸ C1+d+d +¢¢¢+d ¢ H(®)d Now suppose that ® is periodic with period n, so Án(®) = ®. Then we get ¡ ¢ n n H(®) = H Án(®) ¸ C(d ¡1)=(d¡1)H(®)d : A little bit of algebra yields H(®) · C¡1=(d¡1): This proves that the rational periodic points have bounded height, hence there are only ¯nitely many of them. QED Frontier Lectures|April 4{7, 2005|Page 11 A Number Theorist's View of Periodic Points Rational Periodic Points All right, we now know that Á(z) has only ¯nitely many rational periodic points. This raises the question: Question: How many rational periodic points can Á(z) have? If we don't restrict the degree of Á, then we can get as many as we want. Simply take Á(z) to have large degree, set Á(0) = 1;Á(1) = 2;Á(2) = 3;:::;Á(n ¡ 1) = 0; and treat these as n linear equations for the coe±cients of Á.

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    41 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us