The Geometry of Cubic Maps

Transcription

1 The Geometry of Cubic Maps John Milnor Stony Brook University ( work with Araceli Bonifant and Jan Kiwi Conformal Dynamics and Hyperbolic Geometry CUNY Graduate Center, October 23, 2010 CELEBRATING LINDA KEEN

2 Parameter Space 1. PROBLEM: To study cubic polynomial maps F with a marked critical point which is periodic under F. Normal form: If F is monic and centered, then it is uniquely determined by the marked critical point a and the associated critical value v = F(a): F(z) = (z a) 2 (z + 2a) + v. Thus the parameter space for the family of all such maps F = F a, v is the set of pairs (a, v) C 2, where F (a) = 0 and F (a) = v.

3 The Curve S p The period p curve S p consists of those (a, v) C 2 such that the marked critical point a has period exactly p. A caricature of S p = C(S p ) E 1 E Np The connectedness locus C(S p ) is compact. Each escape region E h is open, with E h = C D. More precisely, E h is naturally a µ h -fold covering of C D, where µ h 1 is called the multiplicity of E h.

4 Computation of the degree The simplest invariant of S p is the degree deg(s p ). Note that the disjoint union n p S n polynomial function (a, v) F p a, v(a) a is the zero set of a of degree 3 p 1. Hence 3 p 1 = n p deg(s n). It follows that deg(s p ) grows exponentially: deg(s p ) 3 p 1 as p. The degree can also be described as the number of escape regions counted with multiplicity: deg(s p ) = N p h=1 µ h.

5 The smooth compactification S p THEOREM: S p is a smooth affine curve (conjecturally always connected). Adding an ideal point at infinity h to each E h, we obtain a smooth compact complex 1-manifold S p = S p 1. N p We would like to compute the genus of this curve. But will settle for computing the Euler characteristic, since we don t know that it is connected.

6 Computation of the Euler characteristic of S p The computation will proceed in six steps: Step 1. S p is a translation surface. In other words, we can construct a nowhere vanishing holomorphic 1-form dt on S p. Step 2. Some classical geometry. Considered as a 1-form on S p, dt is meromorphic, with zeros or poles only at the ideal points. The Euler characteristic can then be expressed as a sum, with one integer contribution from each ideal point. Step 3. The Branner-Hubbard Puzzle for maps in E h. In order to understand asymptotic behavior near h, we must first study the dynamics of maps F E h. Step 4. The Kiwi Puzzle: Non-Archimedian Dynamics. This will convert the Branner-Hubbard dynamic information into asymptotic information about the differences a F j (a). Step 5. Local Computation: The contribution of each h to χ(s p ). Step 6. A Global Identity. This will help piece the complicated local information together into a relatively simple formula.

7 Step 1. S p is a translation surface Define the Hamiltonian function H p : C 2 C by H p (a, v) = F p (a) a, where F = F a,v. This vanishes everywhere on S p, with dh p 0 on S p. Let t ( a, v ) be any solution to the Hamiltonian differential equation da = H p dt v, dv = H p dt a. The local solutions t (a, v) = ( a(t), v(t) ) are holomorphic, with dhp dt = Hp a + Hp v Hence they lie in curves H p = constant. da dt dv dt 0. Those solutions which lie in S p provide a local holomorphic parametrization, unique up to a translation, t t + constant. Equivalently, the holomorphic 1-form dt on S p well defined and non-zero everywhere. is

8 A typical parameter space picture This is a region in the t-plane for the period 4 curve S 4.

9 Step 2. Some Classical Geometry Consider dt as a meromorphic 1-form on the closed manifold S p. It has zeros or poles only at the ideal points h. We can compute χ(s p ) = #(poles) #(zeros). More precisely, if ord(dt, h ) = { the multiplicity of the pole at h, or minus the multiplicity of the zero, then χ(s p ) = N p ord(dt, h ). j=1

10 The winding number LEMMA. There exists a local parameter η for S p around h, and a local integral t of dt, so that t = η w h. (In particular dt = 0.) h Here w h is a non-zero integer to be called the winding number of E h. Then dt = w h η wh 1 dη. Therefore ord(dt, h ) = 1 w h. Hence χ(s p ) = N p ( 1 wh ) = N p w h. j=1 THEOREM: h w h = (p 2) deg(s p ).

11 Euler characteristic of the period p curve S p Period 1: χ = +2 Period 2: χ = +2 Period 3: χ = 0 Period 4: χ = -28 Period 5: χ = -184 (by Laura De Marco) Period 6: χ = -784 Period 7: χ = Period 8: χ = Period 9: χ = Period 10: χ = Period 11: χ = Period 12: χ = Period 13: χ = Period 14: χ = Period 15: χ = Period 16: χ = Period 17: χ = Period 18: χ =

12 Step 3. The Branner-Hubbard Puzzle. Let a j = F j (a)

13 Step 4. The Kiwi Puzzle: Non-Archimedian Dynamics Instead of working over the field of complex numbers, work with a different algebraically closed field which is complete with a well behaved norm: Let S be the field of all formal infinite series z = c 0 ξ r 0 + c 1 ξ r 1 +. Here each c j is to be an algebraic number, c j Q, while r 0 < r 1 < are rational numbers tending to +, and ξ is a formal indeterminate. If c 0 0, then the norm is defined by z = exp( r 0 ) > 0 (or log z = r 0 ); while 0 = 0. This norm is multiplicative z 1 z 2 = z 1 z 2 }, and non-archimedian: z 1 + z 2 max { z 1, z 2.

14 Application to an escape region E h. Replace the marked critical point a C by a constant element a = ξ 1 S, with log a = +1. Define F v : S S by F(z) = (z a) 2 (z + 2a) + v. Here v S must be chosen so that the orbit F : a = a 0 a 1 a 2 has period p. (For each such choice the series are locally convergent, and determine a parametrization of a neighborhood of some j S p.) Define the Green s function G : S [0, ) by G(z) = 1 lim n 3 n log+ F n (z). Then, as usual, G ( F(z) ) = 3 G(z).

15 Construction of the puzzle For the escaping critical point a, it is easy to check that G( a) = +1.) The locus {z S ; G(z) < 1/3 n 1 } is a union of finitely many round balls, which are called puzzle pieces of level n. The region G(z) > 1/3 n in each such puzzle piece is a round annulus A of the form A = { z ; R 1 < z ẑ < R 2 }, with modulus log(r2 /R 1 ). Such an annulus of level n surrounding the point ẑ = a j denoted by A n j. will be There is an associated marked grid, where the grid at level n for a j is marked if and only if A n j = A n 0.

16 Kiwi s Theorem Let F : S S be the formal map corresponding to an escape region E h S p. THEOREM: The Kiwi marked grid for F is identical to the Branner-Hubbard marked grid for any classical map F E h. Furthermore, this marked grid determines and is determined by the sequence of norms a a j. In particular, log a a j = 3 A n j =A n 0 mod(a n 0 ) 1. Here log a a j = r if and only if, for F = F a, v E h, a a j c a r as a, where c 0 is some constant in Q C.

17 Step 5. Local Computation We must compute the winding number w h in terms of the asymptotic behavior of the critical orbit {a j }. The key step in the proof is to estimate the derivative da dt relates the 1-form dt to the critical orbit. Passing to formal power series, a non-trivial computation shows that p 1 da dt = a(a a j ). j=1 which The rest of the argument is just elementary calculus, using the fact that t = local uniformizing parameter w j = a w j /µ j.

18 The local formula: LEMMA. For each escape region E h, with multiplicity µ h, the winding number can be expressed as a sum w h = (p 2)µ h + p 1 µ h log a a j. j=1 Hence N p h=1 w h = (p 2)deg(S p ) + h ( µh log a a j ). j ( LEMMA: h µh log a a j ) = 0 for each j. This will complete the proof that wh = (p 2)deg(S p ).

19 Step 6. The Global Identity For any complex number a 0, consider the intersection of the line {(a, v) ; a = a 0 } with the affine curve S p. Generically, the number of intersection points is equal to deg(s p ). Number them as (a 0, v k ) with 1 k deg(s p ). Each such pair determines a map F a0, v k : C C. Let a 0 a 1 k a 2 k be the period p critical orbit under F a0, v k. LEMMA. For each fixed 0 < j < p, the product (a 0 a j k ) k is a non-zero complex constant, independent of a 0. Proof. This product is holomorphic as a function of a 0, with no zeros or poles. (Perhaps this product always equals +1??)

20 Now assume that a 0 is large. Then the line {(a, v) ; a = a 0 } intersects each escape region E h exactly µ h times. Number the intersection points as (a 0, v h,l ) with 1 l µ h, and number the corresponding orbit points as a j h l. Then N p µ h h=1 l=1 ( ) a aj h l = constant 0. Now pass to formal power series.the norm a a j h l is independent of l, so we can write simply a a j h µ h = 1. Hence h This completes the proof! h µ h log a a j h = 0 for each j.

21 References B. Branner and J.H. Hubbard, The iteration of cubic polynomials II, patterns and parapatterns, Acta Math. 169 (1992) J. Kiwi, Puiseux series polynomial dynamics and iteration of complex cubic polynomials, Ann. Inst. Fourier (Grenoble) 56 (2006) Cubic Polynomial Maps with Periodic Critical Orbit: Part I, in Complex Dynamics Families and Friends, ed. D. Schleicher, A. K. Peters 2009, pp Part II: Escape Regions (with Bonifant and Kiwi), Conformal Geometry and Dynamics 14 (2010) and Part III: External rays (with Bonifant), in preparation.