Admissible Covers and Rescaling Limits Xavier Buff Universit de - - PowerPoint PPT Presentation

Rescaling limits Moduli spaces of rational maps Compactifications

Admissible Covers and Rescaling Limits

Xavier Buff

Université de Toulouse

after Matthieu Arfeux

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Origin of the problem

Rescaling limits appear in the work of Epstein and have been studied by Kiwi, DeMarco and others. The approach we will use is inspired by the work of Epstein

• n the deformation space DefB

A(f) and by work of Selinger

and Koch on Thurston’s theorem. Our contribution is to use admissible covers between marked stable curves.

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

The moduli space ratD

S := P1(C) is the Riemann sphere. RatD is the space of rational maps f : S → S of degree D. The group of Möbius transformations Aut(S) = Rat1 acts on RatD by conjugation. The moduli space ratD is the quotient orbifold RatD/Aut(S). The moduli space ratD is not compact.

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Rescaling limits

Let (τk ∈ ratD) be a divergent sequence. A rescaling for (τk) of period q ≥ 1 is a sequence of representatives (fk ∈ Ratd) such that f ◦q

k

→ g for some rational map g with deg g ≥ 2 (the convergence is locally uniform outside a finite subset of S).

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Example

τε = [fε] ∈ rat3 with fε(z) = z2 + ε/z and ε → 0. (fε) is a rescaling of period 1. Set δ =

3

√ε and gε(w) = 1 δ fε(δ · w) = δ ·

• w2 + 1

w

• .

Then, g◦2

ε (w) = ε ·

• w2 + 1

w 2 + w 1 + w3 . (gε) is a rescaling of period 2. 0 is a multiple fixed point of the limit w → w/(1 + w3).

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Example

3

√ε = .5

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Example

3

√ε = .2

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Example

3

√ε = .1

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Example

Julia sets for z → z2 and w → w/(1 + w3).

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Questions

How many essentially distinct rescaling limits can there be? How can one explain the presence of a multiple fixed point?

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Moduli space of marked Riemann spheres

I is a finite set containing at least three points. Two injective maps x1 : I → S and x2 : I → S are equivalent when there is a Möbius transformation M : S → S such that x2 = M ◦ x1. Mod(I) is the set of equivalence classes of injective maps x : I → S. Mod(I) may be endowed with the structure of a quasiprojective variety of dimension |B| − 3.

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Sarah Koch’s space

f ∈ RatD, V(f) is the set of critical values of f. B, C ⊂ S are finite sets with V(f) ⊆ B, |B| ≥ 3 and C = f −1(B). KB(f) is the set of pairs (y, z) ∈ Mod(B) × Mod(C) for which there are triples (F, y, z) ∈ RatD × SB × SC with [y] = y, [z] = z and: C

z

• f
• S

F

• B

y

S

with degz(c) F = degc f.

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Sarah Koch’s space

Theorem The set KB(f) is a smooth submanifold of Mod(B) × Mod(C). The dimension of KB(f) is |B| − 3. The projection KB(f) → Mod(B) is a finite cover. The projection KB(f) → Mod(C) is a proper embedding.

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Adam Epstein’s space

A, B, C ⊂ S are finite sets with |A| ≥ 3, A ⊆ B ∩ C, B ⊇ V(f) and C = f −1(B). EB

A (f) is the set of pairs (y, z) ∈ KB(f) which admits

representatives y ∈ SB and z ∈ SC such that y|A = z|A: C

z

• f
• S

F

• B

y

S

with degz(c) F = degc f and y|A = z|A.

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Adam Epstein’s space

The conjugacy class of F is determined by the point of EB

A (f).

The space EB

A (f) parameterizes conjugacy classes of

rational maps marked by the dynamics of f on A. Theorem Assume f is not a flexible Lattès example. The set EB

A (f) is a smooth submanifold of KB(f).

The dimension of EB

A (f) is |B| − |A|.

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Compactification of Mod(I)

Mod(I) is the Deligne-Mumford compactification of Mod(I). An I-tree of spheres is a pair (T, Z) where

T is a tree whose leaves are the points of I and whose nodes have valence ≥ 3 and Z is a collection of maps zv : I → S indexed by the nodes of T with zv(i1) = zv(i2) if and only if i1 and i2 are in the same component of T − v.

Mod(I) parameterizes equivalence classes of I-trees spheres.

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Compactification of KB(f)

KB(f) is the closure of KB(f) in Mod(B) × Mod(C). An admissible cover F : (TC, Z) → (TB, Y) is a pair ˆ f, {fv}

• where

ˆ f : TC → TB is a weighted tree map whose restriction to C coincides with f, fv : S − zv(C) → S − yˆ

f(v)(B) is a cover and

fv

• zv(e)
• = fw
• zw(e)
• and degzv(e) fv = degzw(e) fw if v and

w are linked by an edge e.

KB(f) parameterizes equivalence classes of admissible covers.

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Compactification of EB

A (f)

EB

A (f) is the closure of EB A (f) in KB(f).

To each point of EB

A (f), one may associate a dynamical

cover is a triple (F, ιB, ιC) where

F : (TC, Z) → (TB, Y) is an admissible cover, ιB : (TB, Y) → (TA, X) is a contraction, ιC : (TC, Z) → (TA, X) is a contraction.

(TC, Z)

F

• ιC
• (TA, X)

(TB, Y).

ιB

• X. Buff

Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications

Back to rescaling limits

Let (F, ιB, ιC) represent a point in EB

A (f).

Rescaling limits correspond to cycles of spheres for which the first return map has degree ≥ 2. Proposition (Kiwi) There are at most 2D − 2 rescaling limits which are not postcritically finite. Proposition (Arfeux) If e ∈ TA is a periodic edge linking vertices v and w, the product of multipliers at xv(e) and xw(e) is 1.

• X. Buff

Admissible Covers and Rescaling Limits