Finite subdivision rules and rational maps

J. Cannon1 W. Floyd2 W. Parry3

1Department of Mathematics Brigham Young University

2Department of Mathematics Virginia Tech

3Department of Mathematics Eastern Michigan University

Barrett Lectures, May 2010

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 1 / 29 Deﬁnition of a ﬁnite subdivision rule R

A subdivision complex SR, which is the union of its closed 2-cells. Each 2-cell is modeled on an n-gon (called a tile type) with n ≥ 3.

A subdivision R(SR) of SR

A subdivision map σR : R(SR) → SR, which is cellular and takes each open cell homeomorphically onto an open cell.

R-complex: a 2-complex X which is the closure of its 2-cells, together with a structure map h: X → SR which takes each open cell homeomorphically onto an open cell One can use a ﬁnite subdivision rule to recursively subdivide R-complexes. R(X) is the subdivision of X. C-F-P, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153–196 (electronic).

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 2 / 29 Example 1. Pentagonal subdivision rule The subdivision of the tile type and the ﬁrst three subdivisions. (The subdivisions are drawn using Stephenson’s CirclePack).

→

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 3 / 29 Example 2. The barycentric subdivision rule

0 •! 0 •!

e1 e1

3 e e3 #! e e d•! e2 2 e •! e2 3 2 e e2 e a•! •! 1 e1 b 3 e e3 e e3 2 e 2 3 e •! •! •! •! •! 1 e1 c e1 "! 1 e1 "!

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 4 / 29 The ﬁrst two subdivisions of a triangle

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 5 / 29 The next two subdivisions of a triangle

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 6 / 29 Example 3. The dodecahedral subdivision rule The subdivisions of the three tile types. Note that there are two edge types.

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 7 / 29 The second subdivision of the quadrilateral tile type

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 8 / 29 The third subdivision of the quadrilateral tile type

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 9 / 29 This subdivision rule on the sphere at inﬁnity The dodecahedral subdivision rule comes from the recursion at inﬁnity for a Kleinian group. The ﬁgure is from Weeks’s program SnapPea.

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 10 / 29 The shapes of tiles

Given a sequence of subdivisions of a tiling, how do you understand/control the shapes of tiles? When can you realize the subdivisions so that the subtiles stay almost round? How do you proceed from combinatorial/topological information to analytic information?

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 11 / 29 Weight functions, combinatorial moduli (Cannon)

shingling (locally-ﬁnite covering by compact, connected sets) T on a surface S, ring (or quadrilateral) R ⊂ S

weight function ρ on T : ρ: T → R≥0 ρ-length of a curve, ρ-height Hρ of R, ρ-area Aρ of R, ρ-circumference Cρ of R 2 2 moduli Mρ = Hρ /Aρ and mρ = Aρ/Cρ 2 2 moduli M(R) = supρ Hρ /Aρ and m(R) = infρ Aρ/Cρ The sup and inf exist, and are unique up to scaling. (This follows from compactness and convexity.)

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 12 / 29 Optimal weight functions - an example

7 7

6 8

2 4 8 7

0 2 6 7

7

6

2 4 7 2 6 7 1 2

fat flows skinny cuts

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 13 / 29 Combinatorial Riemann Mapping Theorem

Now consider a sequence of shinglings of S. Axiom 1. Nondegeneration, comparability of asymptotic combinatorial moduli Axiom 2. Existence of local rings with large moduli conformal sequence of shinglings: Axioms 1 and 2, plus mesh locally approaching 0.

Theorem (C): If {Si } is a conformal sequence of shinglings on a topological surface S and R is a ring in S, then R has a metric which makes it a right-circular annulus such that analytic moduli and asymptotic combinatorial moduli on R are uniformly comparable. J. W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), 155–234.

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 14 / 29 Postcritically ﬁnite branched maps

An orientation-preserving branched map f : S2 → S2 is postcritically ﬁnite if the set Pf of postcritical points is ﬁnite. Two such maps f and g are equivalent if there is a 2 2 homeomorphism h: S → S such that h(Pf ) = Pg,

(h ◦ f ) = (g ◦ h) , and h ◦ f is isotopic, rel Pf , to g ◦ h. Pf Pf If a ﬁnite subdivision rule R is orientation-preserving and has subdivision complex a 2-sphere, then the subdivision map σ is postcritically ﬁnite. (The postcritical points are vertices of SR.)

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 15 / 29 Realizing subdivision maps by rational maps

When can the subdivision map of a ﬁnite subdivision rule be realized by a rational map? Theorem (C-F-Kenyon-P): Suppose R is an orientation-preserving ﬁnite subdivision rule which has bounded valence, mesh approaching zero, and subdivision complex a 2-sphere. If R is (combinatorially) conformal, then it is equivalent to a rational map. C-F-K-P, Constructing rational maps from ﬁnite subdivision rules, Conform. Geom. Dyn. 7 (2003), 76–102 (electronic). The converse follows from the expansion-complex machinery of C-F-P.

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 16 / 29 The pentagonal expansion complex P.L. Bowers and K. Stephenson, A “regular” pentagonal tiling of the plane, Conform. Geom. Dyn. 1 (1997), 58–68 (electronic).

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 17 / 29 The pentagonal subdivision rule

The pentagonal subdivision rule is closely associated with a fsr (with triangular tile types) which is realizable by the rational map

2z(z + 9/16)5 f (z) = . 27(z − 3/128)3(z − 1)2

We computed this at the Barrett Lectures in 1998. The expansion constant for the dilation of the pentagonal expansion complex is f 0(0)1/5 = (−324)1/5.

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 18 / 29 The pentagonal subdivision rule Here are subdivisions drawn by CirclePack and by preimages under the rational map (unfolding by z 7→ z5).

0.4

0.2

-0.4 -0.2 0.2 0.4

-0.2

-0.4

0.4

0.2

-0.4 -0.2 0.2 0.4

-0.2

-0.4

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 19 / 29 The barycentric subdivision rule The barycentric subdivision rule can be realized by the rational 2 3 ( ) = 4(z −z+1) map f z 27z2(z−1)2 . Here is the Julia set.

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 20 / 29 The ﬁrst two subdivisions

R ∪ ∞ is the 1-skeleton of the subdivision complex. Here are − − f 1(R ∪ ∞) and f 2(R ∪ ∞).

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 21 / 29 Realizing rational maps by fsr’s

Theorem (C-F-P): If f is a postcritically ﬁnite rational map without periodic critical points, then every sufﬁciently large iterate of f is equivalent to the subdivision map of a fsr. Idea: pick a simple closed curve containing the postcritical points. For a sufﬁciently large iterate, that curve can be approximated by a curve in its preimage. Now use the expansion-complex machinery. Do you need to pass to an iterate of the map? What about postcritically ﬁnite maps with periodic critical points? This corresponds to fsr’s with unbounded valence.

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 22 / 29 Realizing rational maps by fsr’s

Theorem (Bonk-Meyer, C-F-P-Pilgrim): If f is an expanding postcritically ﬁnite branched map, then every sufﬁciently large iterate of f is equivalent to the subdivision map of a fsr with mesh approaching zero. In both proofs, one contructs a fsr with 1-skeleton a circle.

Conjecture: Every postcritically ﬁnite branched map of S2 is the subdvision map of a fsr.

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 23 / 29 Thurston pullback map

Suppose f : S2 → S2 is a postcritically ﬁnite branched map. When is it equivalent to a rational map? 2 Let T (Of ) be the Teichmüller space of Of = S \ Pf .

f induces a pullback map τf : T (Of ) → T (Of ).

Theorem (Thurston): If τf has a ﬁxed point, then f is equivalent to a rational map.

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 24 / 29 Thurston obstructions

multicurve Γ: components are nontrivial, non-peripheral, and pairwise non-isotopic invariant multicurve Γ: each component of f −1(Γ) is trivial, peripheral, or isotopic to a component of Γ Thurston matrix AΓ for an invariant multicurve: Γ P 1 Aγδ = α deg(f : α→δ) Thurston obstruction: an invariant multicurve with spectral radius at least 1.

Theorem (Thurston): If Of is hyperbolic, then f is equivalent to a rational map if and only if there are no Thurston obstructions.

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 25 / 29 Levy obstructions

A diifﬁculty of working with Thurston’s characterization theorem is that there are inﬁnitely many possible multicurves to check for giving Thurston obstructions.

A Levy cycle is an invariant multicurve {γ1, . . . , γk } such that for −1 each i γi is homotopic rel Pf to a component of f (γi+1) (with indices mod k) which maps by degree one. A Levy obstruction is an essential, nonperipiheral simple closed curve γ such that there are a positive integer n and a component δ of f −n(γ) such that deg(f n : δ → γ) = 1.

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 26 / 29 Levy obstructions

Theorem (Levy): If a quadratic branched map has a Thurston obstruction, then it has a Levy cycle. Theorem (Bielefeld-Fisher-Hubbard): If a postcritically ﬁnite topological polynomial has a Thurston obstrution, then it has a Levy cycle. Theorem (C-F-P-Pilgrim): Suppose f is a postcritically ﬁnite branched map of S2 which is the subdivision map of a fsr. The tiling of the subdivision complex determines a ﬁnite set Γ of simple closed curves in S2 with the following property: If there is a Levy obstruction, then some element of Γ is a Levy obstruction (and the degree of the associated iterate is at most |Γ|).

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 27 / 29 Virtual endomorphisms

Suppose f : S2 → S2 is a postcritically ﬁnite branched map. Following Nekrashevych, one deﬁnes the virtual endomorphism φ of the orbifold fundamental group. The virtual endomorphism φ is contracting if

!1/n kφn(g)k lim sup lim sup < 1. n→∞ kgk→∞ kgk

The virtual endomorphism is contracting if f is expanding with respect to a complete length structure on S2. It is also contracting if f is equivalent to a rational map. Given f , how do you tell whether or not the virtual endomorphism is contracting?

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 28 / 29 Virtual endomorphisms

Theorem (C-F-P-Pilgrim): Suppose f : S2 → S2 is a postcritically ﬁnite branched map that is the subdivision map of a ﬁnite subdivision rule R. If R has bounded valence and mesh approaching zero combinatorially, then the associated virtual endomorphism is contracting.

A fsr R has mesh approaching zero combinatorially if edges of SR eventually subdivide and any tile of SR eventually has no subtile joining disjoint edges of SR. For “eventually”, one only needs to check the ﬁrst kl2 subdivisions, where k is the number of tiles in SR and l is the maximum number of edges in a tile type of SR.

Cannon-Floyd-Parry (BYU, VT, EMU) Finite subdivision rules and rational maps Barrett Lectures, May 2010 29 / 29