1. Preperiodic Points and Unlikely Intersections joint work with Laura DeMarco Matthew Baker Georgia Institute of Technology AMS Southeastern Section Meeting November 6-7, 2010

2. Preperiodic points Let f(z) be a polynomial with complex coefficients. A complex number z 0 is called preperiodic for f if the orbit {z 0, f(z 0 ), f (2) (z 0 ),…} is finite.

3. Iteration of z 2 + c Given a complex number a, let S a be the set of complex parameters c such that a is preperiodic for z 2 + c. When a=0, the points of S 0 belong to the Mandelbrot set M and the closure of S 0 contains the entire boundary of M.

4. The Mandelbrot Set

5. Unlikely intersections Lemma: S a is always a (countably) infinite set. Now fix two complex numbers a and b. What can we say about the set S a  S b ? In other words, for how many parameters c can a and b be simultaneously preperiodic? Note that if a 2 = b 2 then S a = S b so S a  S b is infinite.

6. Are a and b preperiodic?

7. How about now?

8. Main theorem Theorem (B., DeMarco): If a 2 ≠ b 2 then S a  S b is finite. Although the statement is purely about complex dynamics, the proof uses ideas from number theory and involves (generalizations of) p-adic numbers.

9. History of the problem At an AIM workshop in January 2008, Umberto Zannier (motivated by questions of David Masser) asked if S 0  S 1 is finite. Our theorem shows that the answer is yes… though we don’t know what S 0  S 1 is!

10. Some experimental data z 2 : 0 --> 0, 1 --> 1 z 2 - 1: 0 --> -1 --> 0, 1 --> 0 --> -1 --> 0 z 2 - 2: 0 --> -2 --> 2 --> 2, 1 --> -1 --> -1 Conjecture (B., Connelly): S 0  S 1 = {0,-1,-2}.

11. The theorem of Masser & Zannier Theorem (Masser-Zannier): The set of complex numbers such that both A( ) and B( ) are torsion in the Legendre curve is finite. Here we take

12. An illustration of the Masser- Zannier theorem

13. The “local-global principle” Lemma: If f is a polynomial with rational coefficients, then a rational number z 0 is preperiodic for f if and only if the orbit {z 0, f(z 0 ), f (2) (z 0 ),…} is bounded in the real numbers and also in the p-adic numbers for all prime numbers p. Example: For f(z) = z 2 - 1/2, the orbit of 0 is {0, -1/2, -1/4, -7/16,…} which is bounded in the usual topology on the rationals but not in the 2-adic topology.

14. Boundedness loci Given a complex number a, let M a be the set of complex parameters c such that a stays bounded under iteration of z 2 + c. We call M a the generalized Mandelbrot set associated to a. a=0 a=1

15. M 0 and M 1 superimposed

16. Complex potential theory Given a compact subset E of the complex plane and a probability measure  supported on E, define the energy of  to be If there is a probability measure of finite energy supported on E, then there is a unique such measure  E having minimal energy, called the equilibrium measure for E.

17. Capacity The capacity of E is defined to be (or zero if there is no probability measure of finite energy supported on E.) Lemma:

18. Proof of the main theorem: algebraic case Sketch of the proof: Assume a,b are algebraic and that S a  S b = {c 1,c 2,c 3,…} is infinite. Easy lemma: All c n ’s must be algebraic. Let  n be the discrete probability measure on the complex plane supported equally on all Galois conjugates of c n.

19. Proof of the main theorem: algebraic case (cont’d) Using the fact that the c n ’s belong to S a, a suitable arithmetic equidistribution theorem shows that the sequence {  n } converges weakly to the equilibrium measure  a for M a. By symmetry, {  n } also converges to  b. Hence  a =  b. This implies (taking supports and ‘filling in’) that M a = M b. An argument from univalent function theory now shows that a 2 = b 2.

20. What if a and b are transcendental? If a is transcendental, then so is b and a,b, and all c n ’s belong to the algebraic closure of We think of k as the function field of an algebraic curve. Can we mimic the above proof with algebraic numbers replaced by algebraic functions?

21. The product formula The proof of the ‘suitable equidistribution theorem’ uses the product formula for algebraic numbers, which for a nonzero rational number  just says that There is also a product formula for algebraic functions, which says that a rational function on the projective line has the same number of zeros as poles, counting multiplicities.

22. The product formula for algebraic functions Let k be the field of rational functions over some algebraically closed constant field. The “places” (equivalence classes of absolute values) of k are in 1-1 correspondence with points of the projective line over k. If a place v corresponds to a point p of P1(k), we define |f|_v = exp(-ord_p(f)) for any non- zero rational function f in k. Then

23. Absolute values on fields In the algebraic function case, all absolute values appearing in the product formula are non-Archimedean. So in order to mimic our proof from the algebraic case, we need to use some sort of non-Archimedean potential theory. Luckily, such a theory exists! But one needs to use Berkovich’s theory of non- Archimedean analytic spaces.

24. The Berkovich affine line Let K be a complete and algebraically closed non-Archimedean field. The Berkovich affine line over K is a locally compact and path-connected space containing K (which is not locally compact and is totally disconnected) as a dense subspace.

25. The Berkovich affine line The Berkovich affine line over K has the structure of an infinitely branched real tree:

26. Definition of As a set, the Berkovich affine line consists of all real-valued multiplicative semi-norms on K[T] which extend the usual absolute value on K. It is endowed with the weakest topology making the absolute value of a polynomial f in K[T] continuous. Each element of K gives a semi-norm (by evaluation), but there are lots of other semi- norms, e.g. the supremum of |f(x)| as x ranges over some closed disk D in K. (This is multiplicative by `Gauss’ Lemma’.)

27. Potential theory on the Berkovich line The extra points allow one to `connect up’ the totally disconnected space K in a natural way. The tree structure on the Berkovich line allows one to do potential theory. As in the complex case, for compact subsets of the Berkovich line one can define capacities, equilibrium measures, Green’s functions, etc. (B.-Rumely “Potential Theory and Dynamics on the Berkovich Projective Line”, see also Thuillier, Favre-Jonsson)

28. Proof of the main theorem (transcendental case) Sketch of the proof: Assume a,b are transcendental and that S a  S b = {c 1,c 2,c 3,…} is infinite. Fix a place v of Let  n be the discrete probability measure on the v-adic Berkovich affine line which is supported equally on all conjugates of c n.

29. Conclusion of the proof By equidistribution, the v-adic generalized Mandelbrot sets M a,v and M b,v coincide for all places v of k. By a local-global principle for function fields due to Benedetto (2005), it follows that a is preperiodic for z 2 +c iff b is, i.e. S a =S b. By Montel’s theorem, S a (resp. S b ) determines M a (resp. M b ) inside Hence M a = M b and we conclude as in the algebraic case!

30. Variant of the main theorem Similar methods allow us to prove the following result: Theorem (B.,DeMarco): If f,g are rational functions with complex coefficients then Preper(f)  Preper(g) is infinite iff Preper(f) = Preper(g). Remark: This was recently proved independently (and generalized to higher dimensions) by X. Yuan and S. Zhang. When f and g have algebraic coefficients, our result follows from a theorem of A. Mimar.