Puiseux series dynamics and leading monomials of escape regions

Puiseux series dynamics and leading monomials of escape regions Jan Kiwi PUC, Chile Workshop on Moduli Spaces Associated to Dynamical Systems ICERM Providence, April 17, 2012

Parameter space Monic centered cubic polynomials with marked critical points:

Parameter space Monic centered cubic polynomials with marked critical points: where (a, v) C 2. f a,v : z (z a) 2 (z + 2a) + v

Parameter space Monic centered cubic polynomials with marked critical points: f a,v : z (z a) 2 (z + 2a) + v where (a, v) C 2. After identification of (a, v) with ( a, v) one obtains the moduli space of cubic polynomials with marked critical points.

Parameter space Monic centered cubic polynomials with marked critical points: f a,v : z (z a) 2 (z + 2a) + v where (a, v) C 2. After identification of (a, v) with ( a, v) one obtains the moduli space of cubic polynomials with marked critical points. Critical points of f a,v are ±a. Critical value: v = f a,v (+a). Co-critical value 2a, since v = f a,v (+a) = f a,v ( 2a).

Periodic Curves For p 1, the periodic curve S p consists of all (a, v) such that +a has period p under f a,v.

Periodic Curves For p 1, the periodic curve S p consists of all (a, v) such that +a has period p under f a,v. S p C 2 is a smooth affine algebraic curve of degree d p where n p d n = 3 p 1.

Periodic Curves For p 1, the periodic curve S p consists of all (a, v) such that +a has period p under f a,v. S p C 2 is a smooth affine algebraic curve of degree d p where n p d n = 3 p 1. What is its topology?

Periodic Curves For p 1, the periodic curve S p consists of all (a, v) such that +a has period p under f a,v. S p C 2 is a smooth affine algebraic curve of degree d p where n p d n = 3 p 1. What is its topology? Is S p irreducible?

Periodic Curves For p 1, the periodic curve S p consists of all (a, v) such that +a has period p under f a,v. S p C 2 is a smooth affine algebraic curve of degree d p where n p d n = 3 p 1. What is its topology? Is S p irreducible? Bonifant,-,Milnor: the Euler characteristic is S p is (2 p)d p.

Periodic Curves For p 1, the periodic curve S p consists of all (a, v) such that +a has period p under f a,v. S p C 2 is a smooth affine algebraic curve of degree d p where n p d n = 3 p 1. What is its topology? Is S p irreducible? Bonifant,-,Milnor: the Euler characteristic is S p is (2 p)d p. What is the Euler characteristic of the smooth compactification of S p?

Periodic Curves For p 1, the periodic curve S p consists of all (a, v) such that +a has period p under f a,v. S p C 2 is a smooth affine algebraic curve of degree d p where n p d n = 3 p 1. What is its topology? Is S p irreducible? Bonifant,-,Milnor: the Euler characteristic is S p is (2 p)d p. What is the Euler characteristic of the smooth compactification of S p? Requires to compute the number N p of escape regions. (De Marco and Schiff, De Marco-Pilgrim).

Escape Regions The connectedness locus is compact. C(S p ) = { f a,v S p f n a,v( a) }

Escape Regions The connectedness locus C(S p ) = { f a,v S p fa,v( a) n } is compact. The escape locus E(S p ) = { f a,v S p fa,v( a) n } is open and every connected component is unbounded.

Escape Regions The connectedness locus is compact. The escape locus C(S p ) = { f a,v S p f n a,v( a) } E(S p ) = { f a,v S p f n a,v( a) } is open and every connected component is unbounded. A escape region U is a connected component of E(S p ): U is conformally isomorphic to punctured disk. The puncture is at U.

Asymptotics of critical periodic orbit For f a,v U, let a 0 = +a a 1 = v a p 1 a 0.

Asymptotics of critical periodic orbit For f a,v U, let a 0 = +a a 1 = v a p 1 a 0. Dynamical space picture. After conjugacy, f a,v (az)/a, we have:

Asymptotics of critical periodic orbit For f a,v U, let a 0 = +a a 1 = v a p 1 a 0. Dynamical space picture. After conjugacy, f a,v (az)/a, we have: a + o(a) or a j = 2a + o(a).

Asymptotics of critical periodic orbit For f a,v U, let a 0 = +a a 1 = v a p 1 a 0. Dynamical space picture. After conjugacy, f a,v (az)/a, we have: a + o(a) or a j = 2a + o(a). Bonifant and Milnor: Do the leading terms of a j a, for j = 1,..., p 1 determine U uniquely?

Leading monomials There exists µ 1 and a local coordinate ζ for U near such that a = 1 ζ µ.

Leading monomials There exists µ 1 and a local coordinate ζ for U near such that Then, a = 1 ζ µ. a j a a = holomorphic(ζ) = c k ζ k = k k 0 k k 0 c k ( ) k/µ 1. a

Leading monomials There exists µ 1 and a local coordinate ζ for U near such that Then, a = 1 ζ µ. a j a a = holomorphic(ζ) = c k ζ k = k k 0 k k 0 c k ( ) k/µ 1. a Leading monomial m j = c k0 ( 1 a ) k0 /µ.

Theorem. The leading monomial vector m = (m 1,..., m p 1, 0) determines the escape region uniquely.

Theorem. The leading monomial vector m = (m 1,..., m p 1, 0) determines the escape region uniquely. Corollary. (Bonifant,-,Milnor) Assume that the leading monomial vector of U is Then, for all j, (c 1 a k 1/µ,..., c p 1 a k p 1/µ ). a j Q(c 1,..., c p 1 )((a 1/µ )).

Equations For an escape region U, is a solution of v = a 1 = (+a or 2a) + a f p a,v(+a) = +a (*) k k 0 c k ( ) k/µ 1 a in some extension of Q((1/a)).

Field Put t instead of 1/a to get Q((t)) and study solutions of (*) in the algebraic closure of Q((t)): Q a t = Q a ((t 1/m )).

Field Put t instead of 1/a to get Q((t)) and study solutions of (*) in the algebraic closure of Q((t)): Q a t = Q a ((t 1/m )). Q a t is endowed with z = c k t k/m = e ord0(z) = e k0/m. k k 0

Field Put t instead of 1/a to get Q((t)) and study solutions of (*) in the algebraic closure of Q((t)): Q a t = Q a ((t 1/m )). Q a t is endowed with z = c k t k/m = e ord0(z) = e k0/m. k k 0 Complete it to get L.

Non-Archimedean problem For ν L, consider ψ ν (z) = t 2 (z 1)(z + 2) + ν L[z].

Non-Archimedean problem For ν L, consider ψ ν (z) = t 2 (z 1)(z + 2) + ν L[z]. ν is periodic parameter if ω + = +1 is periodic under ψ ν.

Non-Archimedean problem For ν L, consider ψ ν (z) = t 2 (z 1)(z + 2) + ν L[z]. ν is periodic parameter if ω + = +1 is periodic under ψ ν. In this case: ω + = +1 ω + 1 = ν ω+ p 1 ω+.

Non-Archimedean problem For ν L, consider ψ ν (z) = t 2 (z 1)(z + 2) + ν L[z]. ν is periodic parameter if ω + = +1 is periodic under ψ ν. In this case: ω + = +1 ω + 1 = ν ω+ p 1 ω+. It follows, ω + +1 + c t k/m + h.o.t. = j 2 + d t l/n + h.o.t.

Non-Archimedean problem For ν L, consider ψ ν (z) = t 2 (z 1)(z + 2) + ν L[z]. ν is periodic parameter if ω + = +1 is periodic under ψ ν. In this case: ω + = +1 ω + 1 = ν ω+ p 1 ω+. It follows, ω + +1 + c t k/m + h.o.t. = j 2 + d t l/n + h.o.t. The corresponding leading monomial are m(ω + ω + c t k/m ) = j 3 respectively.

Theorem. The leading monomial vector m = (m 1,..., m p 1, 0) determines the periodic parameter ν uniquely.

Dynamical Space The filled Julia set of ψ ν is K(ψ ν ) = {z L ψ n ν(z) }.

Dynamical Space The filled Julia set of ψ ν is K(ψ ν ) = {z L ψ n ν(z) }. ν 1 = ψ n ν(z) when z > 1.

Dynamical Space The filled Julia set of ψ ν is K(ψ ν ) = {z L ψ n ν(z) }. ν 1 = ψ n ν(z) when z > 1. Level 0 dynamical ball D 0 = { z 1}.

Dynamical Space The filled Julia set of ψ ν is K(ψ ν ) = {z L ψ n ν(z) }. ν 1 = ψ n ν(z) when z > 1. Level 0 dynamical ball D 0 = { z 1}. Level n set {ψ n ν(z) D 0 } is a disjoint union of finitely many dynamical balls of level n.

Dynamical Space The filled Julia set of ψ ν is K(ψ ν ) = {z L ψ n ν(z) }. ν 1 = ψ n ν(z) when z > 1. Level 0 dynamical ball D 0 = { z 1}. Level n set {ψ n ν(z) D 0 } is a disjoint union of finitely many dynamical balls of level n. Each level n + 1 ball is contained in, and maps onto, a level n ball.

Branches Maximal open balls in a closed ball D are parametrized by Q a. If ψ ν maps D onto D by degree d, then it induces a polynomial map of degree d in Q a.

Computing ω + j ω +. Fact. If B is a maximal open ball of a dynamical ball D l of level l, then B contains at most one dynamical ball of level l + 1.

Computing ω + j ω +. Fact. If B is a maximal open ball of a dynamical ball D l of level l, then B contains at most one dynamical ball of level l + 1. Consequence. If D l (ω + ) = D l (ω + j ) but D l+1 (ω + ) D l+1 (ω + j ), then ω + j ω + is the diameter of D l (ω + ) = D l (ω + j ).

Parameter space The level 0 parameter ball is D 0 = { ψ ν (ω + ) = ν 1}.

Parameter space The level 0 parameter ball is D 0 = { ψ ν (ω + ) = ν 1}. The parameters of level n { ψ n 1 ν (ω + ) 1} are a disjoint union of closed balls D n called level n parameter balls.

Parameter ball, dynamical balls and branch dynamics Proposition. Assume D l is a level l parameter ball. Then: D l = D l (ν) for all ν D l.

Parameter ball, dynamical balls and branch dynamics Proposition. Assume D l is a level l parameter ball. Then: D l = D l (ν) for all ν D l. The level l + 1 j dynamical ball ψ j ν(d l (ν)) is independent of ν D l.

Parameter ball, dynamical balls and branch dynamics Proposition. Assume D l is a level l parameter ball. Then: D l = D l (ν) for all ν D l. The level l + 1 j dynamical ball ψ j ν(d l (ν)) is independent of ν D l. The ψ ν action on maximal open balls is also independent of ν D l.

Parameter ball, dynamical balls and branch dynamics Proposition. Assume D l is a level l parameter ball. Then: D l = D l (ν) for all ν D l. The level l + 1 j dynamical ball ψ j ν(d l (ν)) is independent of ν D l. The ψ ν action on maximal open balls is also independent of ν D l. If p l is the smallest integer q such that ω + ψν q 1 (D l (ν)), then every periodic parameter in D l has period at least p l.

Centers Proposition. There exists a unique ν in D l which is periodic with period p l.

Centers Proposition. There exists a unique ν in D l which is periodic with period p l. Proof. The map T : D l D l defined by T(ν) = ( ψ p l 1 ν Dl (ν)) 1 (ω + ) is a strict contraction. Such ν is called the center of D l.

Level l + 1 correspondence Proposition. Let B be a maximal open ball of a parameter ball D l and consider any ν D l. B contains a parameter ball of level l + 1 if and only if B contains a dynamical ball of level l + 1. In this case, the level l + 1 balls are unique.

Leading monomials determine parameter balls Assume ν, ν have the same leading monomial vector.

Leading monomials determine parameter balls Assume ν, ν have the same leading monomial vector. Let us prove by induction on l. D l (ν) = D l (ν )

Leading monomials determine parameter balls Assume ν, ν have the same leading monomial vector. Let us prove by induction on l. D l (ν) = D l (ν ) Take ν l the center of level l.

Leading monomials determine parameter balls Assume ν, ν have the same leading monomial vector. Let us prove by induction on l. Take ν l the center of level l. D l (ν) = D l (ν ) There is a unique maximal open ball B in D l (ν l ) whose orbit is compatible with the leading monomial vector.

Leading monomials determine parameter balls Assume ν, ν have the same leading monomial vector. Let us prove by induction on l. Take ν l the center of level l. D l (ν) = D l (ν ) There is a unique maximal open ball B in D l (ν l ) whose orbit is compatible with the leading monomial vector. That is, ν, ν belong to B Thus ν, ν belong to the unique level l + 1 parameter ball contained in B. Hence, D l+1 (ν) = D l+1 (ν ).

Homework Given a sequence D 0 D 1. With centers ν 0, ν 1,. Which lie in L 0 ((t 1/m 0 )) L 1 ((t 1/m 1 )) Branner and Hubbard tell us how to compute m k.

Homework Given a sequence D 0 D 1. With centers ν 0, ν 1,. Which lie in L 0 ((t 1/m 0 )) L 1 ((t 1/m 1 )) Branner and Hubbard tell us how to compute m k. Compute L k.