Moduli Spaces for Dynamical Systems Joseph H. Silverman

Moduli Spaces for Dynamical Systems Joseph H. Silverman Brown University CNTA Calgary Tuesday, June 21, 2016 0

Notation: We fix The Space of Rational Self-Maps of P n 1 Rational Maps on Projective Space n 1 and d 2. Primary Object of Study: Dynamics of } Rat n d {Rational := maps f : P n P n of degree d. Dynamics is the study of Iteration: f k := f } f f {{ f }. k copies The Orbit of a point α P n is O f (α) := { f k (α) : k = 0, 1, 2,... }. The point α is Preperiodic if it has finite orbit.

The Space of Rational Self-Maps of P n 2 A Soupçon of Motivation Arithmetic Dynamics is the study of arithmetic properties of iteration of maps and their orbits. Many problems are inspired by analogy from arithmetic geometry. Some examples: Let f(z) Q(z). Only finitely many α Q are preperiodic (Northcott, 1950). Is there a bound for the number of such points that depends only on deg(f)? (Uniform Boundedness Conjecture) Let f(z) Q(z) and α Q. When can O f (α) contain infinitely many integers? (Dynamical analogue of Siegel s theorem) Let f : P n P n and V P n and α P n (C). When is #O f (α) V =? When is # Per(f) V =? (Dynamical analogues of Bombieri-Lang and Manin- Mumford conjectures)

The Space of Rational Self-Maps of P n 3 A Soupçon of Motivation In arithmetic geometry, rather than studying one variety, it is fruitful to look at the space of all varieties. More precisely, one studies the space of isomorphism classes of varieties having a specified structure, i.e., moduli spaces. The aim of this talk is to describe some of the moduli spaces that come up in dynamics, and to see how they are analogous, in some ways, to moduli spaces such as X 1 (N), M g, and A g that are so important in arithmetic geometry.

The Space of Rational Self-Maps of P n 4 Rational Maps on P 1 We start with n = 1. Rational maps P 1 P 1 look like f(z) = a 0z d + a 1 z d 1 + + a d b 0 z d + b 1 z d 1 + + b d, but we get the same map if multiply the numerator and denominator by any non-zero c. Hence Rat 1 d = { [a 0,..., a d, b 0,..., b d ] } = P 2d+1. Convention: For now we allow degenerate maps where top and bottom have a common factor. To get a map of exact degree d, we need a 0, b 0 not both zero and no common factor. Example: { } Rat 1 2 = a 0 z 2 + a 1 z + a 2 b 0 z 2 : [a 0,..., b 2 ] P 5. + b 1 z + b 2

The Space of Rational Self-Maps of P n 5 Identifying Rat n d with PN In general we write f(x 0,..., X n ) = [ f 1 (X 0,..., X n ),..., f n (X 0,..., X n ) ] with f i K[X 0,..., X n ] homogeneous of degree d. Then we identity f with the list of its coefficients: ( ) f [coeffs. of f] P N n + d with N = (n + 1) 1. d The exact value of N = Nd n is not important, except to note that it gets quite large as d. For example, Nd 1 = 2d + 1, N d 2 = 3d2 + 9d + 4. 2

The Space of Rational Self-Maps of P n 6 Equivalence of Dynamical Systems Question: When are two maps f, g : P n P n Dynamically Equivalent? Answer: When they differ by a change of variables of P n, i.e., when there is an automorphism L : P n P n so that g = f L := L 1 f L, P n L P n g P n L f P n N.B. Conjugation by L commutes with composition: (f L ) k = f L f L = (f f) L = (f k ) L. That s why conjugation is the appropriate equivalence relation for dynamics.

The Space of Rational Self-Maps of P n 7 Equivalence of Dynamical Systems For example, if n = 1, then L = αz + β γz + δ PGL 2 is a linear fractional transformation, and in general L Aut(P n ) = PGL n+1 is given by n + 1 linear forms in n + 1 variables. The conjugation action of Aut(P n ) = PGL n+1 on Rat n d = P N gives a very complicated homomorphism PGL n+1 PGL N+1. Primary Goal: Understand the quotient space Rat n d / PGL n+1-conjugation Really Rat n d / SL n+1 for technical reasons. Will ignore!

We illustrate with the map The Space of Rational Self-Maps of P n 8 An Example Aut(P 1 ) = PGL 2 PGL 6 = Aut(Rat 1 2 ) = P 5. The conjugation action of the linear fractional transformation L = αz+β γz+δ on the vector of coefficients of the quadratic map f = a 0z 2 +a 1 z+a 2 b 0 z 2 +b 1 z+b 2 is given by the matrix α 2 δ αγδ γ 2 δ α 2 β αβγ βγ 2 2αβδ αδ 2 + βγδ 2γδ 2 2αβ 2 αβδ β 2 γ 2βγδ β 2 δ βδ 2 δ 3 β 3 β 2 δ βδ 2 α 2 γ αγ 2 γ 3 α 3 α 2 γ αγ 2 2αβγ αγδ βγ 2 2γ 2 δ 2α 2 β α 2 δ + αβγ 2αγδ γβ 2 βγδ γδ 2 αβ 2 αβδ αδ 2

The Space of Rational Self-Maps of P n 9 Geometric Invariant Theory to the Rescue The quotient space Rat n d /(PGL n+1 -conjugation). makes sense as a set, but algebraically and topologically, it s a mess! Geometric Invariant Theory (GIT, Mumford et al.) explains what to do. GIT says that if we restrict to a good subset of Rat n d, then we get a good quotient.

The Space of Rational Self-Maps of P n 10 Moduli Spaces of (Semi)Stable Rational Maps Theorem. There exist subsets of semi-stable and stable points so that the quotients M n d := (Rat n d )ss / PGL n+1, M n d := (Ratn d )stab / PGL n+1, are nice algebraic varieties. (Rat n d )stab (Rat n d )ss are non-empty Zariski open subsets of Rat n d, i.e., they re pretty big. M n d and Mn d are varieties, with M n d projective. Two maps f, g (Rat n d )stab have the same image in M n d if and only if g = f L for some L PGL n+1. Theorem. (Levy, Petsche Szpiro Tepper) f Rat n d a morphism = f Mn d.

The Space of Rational Self-Maps of P n 11 Dynamical Moduli Space of Maps on P 1 For n = 1, we know a fair amount. Theorem. (Milnor over C, JS as Z-schemes) M 1 2 = A 2 and M 1 2 = P 2. Milnor describes the isomorphism M 1 2 = A 2 very explicitly. I ll discuss this on the next slide. Theorem. (Levy) M 1 d (It is clear that M n d is a rational variety. is unirational.)

The Space of Rational Self-Maps of P n 12 An Explicit Description of M 1 2 Degree 2 rational maps f have three fixed points: f(α 1 ) = α 1, f(α 2 ) = α 2, f(α 3 ) = α 3. Compute the multipliers λ 1 = f (α 1 ), λ 2 = f (α 2 ), λ 3 = f (α 3 ), and take the symmetric functions σ 1 (f) := λ 1 + λ 2 + λ 3, σ 2 (f) := λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3. Fact: σ 1 (f L ) = σ 1 (f) and σ 2 (f L ) = σ 2 (f). Theorem: The map (σ 1, σ 2 ) : M 1 2 A 2 is an isomorphism. Theorem. (McMullen) If one uses enough symmetric functions of multipliers of periodic points, then one usually obtains a finite-to-one map M 1 d P.

The Space of Rational Self-Maps of P n 13 Special Points in M 1 d Let f M 1 d, and write f(z) K(z). A critical point of f is a point α where f (α) = 0. A rational map of degree d has 2d 2 critical points (counted with multiplicities). The map is said to be Post-Critically Finite (PCF) if every critical point is preperiodic. Coincidentally we also have dim M 1 d = 2d 2. Theorem. (Thurston) There are only countably many PCF maps in M 1 d. Properly formulated, they each appear with multiplicity one. Analogy : PCF maps }{{} dynamics CM abelian varieties }{{} arithmetic geometry There s been much work recently (Baker, DeMarco,... ) on dynamical André Oort type results, with PCF in place of CM.

The Space of Rational Self-Maps of P n 14 Adding Level Structure Analogy : (Pre)periodic points }{{} dynamics We can add level structure to M n d point: Torsion points }{{} arithmetic geometry by marking a periodic M n d (N) := { (f, α) : f M n d and α Per N(f) }. Conjecture. M n d (N) is of general type for all sufficiently large N. Theorem. (Blanc, Canci, Elkies) M 1 2 (N) is rational for N 5. M 1 2 (N) is of general type for N = 6.

The Space of Rational Self-Maps of P n 15 The Automorphism Group of a Rational Map General Principle: Given a category C and an object X, the group of automorphisms Aut C (X) = {isomorphisms φ : X X} is an interesting group. Definition: The automorphism group of f Rat n d is the group Aut(f) = { L PGL n+1 : f L = f }. In other words, Aut(f) is the set of fractional linear transformations that commute with f. Observation: Aut(f L ) = L 1 Aut(f)L, so Aut(f) is well-defined (as an abstract group) for f M n d.

The Space of Rational Self-Maps of P n 16 The Automorphism Group of a Rational Map Theorem. If f is a morphism, then Aut(f) is finite. But note that Aut(f) may be infinite for dominant rational maps. The theorem is not hard to prove, for example by using the fact that φ Aut(f) must permute the points of period N. More precise results include: f M n d f M 1 d f M 2 d f M 2 2 = # Aut(f) C(n, d) (Levy), = # Aut(f) max{60, 2d + 2}, = # Aut(f) 6d3 (de = # Aut(f) 21 (Manes JS), Faria Hutz),

The Space of Rational Self-Maps of P n 17 The Automorphism Group of Maps of Degree 2 on P 1 Proposition. Recall that there is a natural isomorphism M 1 2 = A 2. With this identification, { f M 1 2 : Aut(f) is non-trivial } is a cuspidal cubic curve Γ A 2, and we have 1 if f / Γ, Aut(f) = C 2 if f = Γ cusp, S 3 if f = cusp Γ. The non-cusp points of Γ are maps of the form f(z) = bz + 1 z with b 0. The cusp is the map f(z) = z 2.

The Space of Rational Self-Maps of P n 18 Automorphisms and Twists Just as with elliptic curves, the existence of non-trivial automorphisms leads to twists, that is, maps f and f that are equivalent over the algebraic closure of K, but not over K itself. The dynamnics of distinct twists may exhibit different arithmetic properties. The twists of f are classfied, up to K-equivalence, by ( Ker H 1( G K/K, Aut(f) ) H 1( G K/K, PGL n+1 ( K) )). Example: The map f(z) = bz + z 1 on the last slide has Aut(f) = C 2. Its quadratic twists are f c = bz + c z with c K. Notice that L(z) = c z sends f L c (z) = f 1 (z).

The Space of Rational Self-Maps of P n 19 Maps in M 2 2 with Large Automorphism Group Joint work (in progress) with Michelle Manes We classify semi-stable dominant rational maps f : P 2 P 2 of degree 2 with 3 # Aut(f) <. At this point we have a geometric classification. Some interesting factoids gleaned from the classification: Let f M 2 2 with Aut(f) finite. Then Aut(f) is isomorphic to one of the following groups: C 1, C 2, C 3, C 4, C 5, C 2 2, C 3 C 2, C 3 C 4, C 2 2 S 3, C 4 C 2, C 7 C 3. Further, all of these groups occur. The map f = [Y 2, X 2, Z 2 ] has Aut(f) = C 7 C 3. If Aut(f) contains a copy of either C 5 or C 2 C 2, then f M 2 2 M 2 2, so f is not a morphism.

The Space of Rational Self-Maps of P n 20 Proof Idea For a dominant rational map f : P 2 P 2, let: I(f) = Indeterminacy locus of f = {P P 2 : f is not defined at P }, Crit(f) = Critical locus of f = Ramification locus of f. Observation: Every φ Aut(f) leaves I(f) and Crit(f) invariant, i.e., φ ( I(f) ) = I(f) and φ ( Crit(f) ) = Crit(f). Here I(f) is either empty or a finite set of points, and Crit(f) is a cubic curve in P 2. Typical Examples If I(f) = {P 1, P 2 }, then φ fixes or swaps P 1 or P 2. If I(f) = L C, then φ(l) = L and φ(c) = C. If I(f) = L 1 L 2 L 3, then φ permutes the lines.

The Space of Rational Self-Maps of P n 21 Proof Idea Combining information from φ ( I(f) ) = I(f), φ ( Crit(f) ) = Crit(f), φ f = f φ usually puts sufficient restrictions on φ to allow a caseby-case analysis. The cases with I(f) = and Crit(f) = smooth cubic curve are interesting. For these we exploit the fact that φ ( Crit(f) ) = Crit(f) to note that φ preserves (1) The 9 flex points of Crit(f), and (2) Lines. This leads to 432 possible ways that φ can permute the 9 flex points, and each permutation π yields a 6-by-12 matrix M π,φ such that f satisfies the constraint rank M π,φ 5. A short computer program then checked all cases.

The Space of Rational Self-Maps of P n 22 In Conclusion I want to thank the organizers for inviting me to give this talk. But most of all, I want to say... H A P P Y B I R T H D A Y R I C H A R D R D H A P P Y B I R T H D A Y R I C H A H A R D H A P P Y B I R T H D A Y R I C I C H A R D H A P P Y B I R T H D A Y R R I C H A R D H A P P Y B I R T H D A Y A Y R I C H A R D H A P P Y B I R T H D H D A Y R I C H A R D H A P P Y B I R T R T H D A Y R I C H A R D H A P P Y B I B I R T H D A Y R I C H A R D H A P P Y P Y B I R T H D A Y R I C H A R D H A P A P P Y B I R T H D A Y R I C H A R D H H A P P Y B I R T H D A Y R I C H A R D

Moduli Spaces for Dynamical Systems Joseph H. Silverman Brown University CNTA Calgary Tuesday, June 21, 2016