Non-Archimedean Dynamics andDegenerations of

Complex Dynamical Systems

Xander FaberUniversity of Hawaii at Manoa

Joint Mathematics Meetings in San Diego

Special Session on Complex Dynamics

January 9, 2013

Main Result — Joint with Laura De Marco

Theorem. Let ft : C → C be a family of degree d > 1 dynamical systemsthat varies holomorphically with t ∈ D∗

ε . Let µt be the measure of maximalentropy for ft. Then µ0 = limµt exists for the weak-∗ topology, and . . .

Main Result — Joint with Laura De Marco

Theorem. Let ft : C → C be a family of degree d > 1 dynamical systemsthat varies holomorphically with t ∈ D∗

ε . Let µt be the measure of maximalentropy for ft. Then µ0 = limµt exists for the weak-∗ topology, and . . .

1. (Mane ’86) If ft extends over Dε, then µ0 is the measure of maximalentropy for f0.

Main Result — Joint with Laura De Marco

Theorem. Let ft : C → C be a family of degree d > 1 dynamical systemsthat varies holomorphically with t ∈ D∗

ε . Let µt be the measure of maximalentropy for ft. Then µ0 = limµt exists for the weak-∗ topology, and . . .

1. (Mane ’86) If ft extends over Dε, then µ0 is the measure of maximalentropy for f0.

2. (DM / F) If ft does not extend over Dε, then µ0 is purely atomic. It maydescribed as the residual measure for the associated Berkovichdynamical system f : P1,an

L→ P

1,anL

. Here L is the minimal algebraicallyclosed and complete non-Archimedean field containing C((t)), andf = ft as a rational function with L-coefficients.

Main Result — Joint with Laura De Marco

Theorem. Let ft : C → C be a family of degree d > 1 dynamical systemsthat varies holomorphically with t ∈ D∗

ε . Let µt be the measure of maximalentropy for ft. Then µ0 = limµt exists for the weak-∗ topology, and . . .

1. (Mane ’86) If ft extends over Dε, then µ0 is the measure of maximalentropy for f0.

2. (DM / F) If ft does not extend over Dε, then µ0 is purely atomic. It maydescribed as the residual measure for the associated Berkovichdynamical system f : P1,an

L→ P

1,anL

. Here L is the minimal algebraicallyclosed and complete non-Archimedean field containing C((t)), andf = ft as a rational function with L-coefficients.

e.g., ft(z) = z2 +1

t, µ0 = δ∞

Main Result — Joint with Laura De Marco

Theorem. Let ft : C → C be a family of degree d > 1 dynamical systemsthat varies holomorphically with t ∈ D∗

ε . Let µt be the measure of maximalentropy for ft. Then µ0 = limµt exists for the weak-∗ topology, and . . .

1. (Mane ’86) If ft extends over Dε, then µ0 is the measure of maximalentropy for f0.

2. (DM / F) If ft does not extend over Dε, then µ0 is purely atomic. It maydescribed as the residual measure for the associated Berkovichdynamical system f : P1,an

L→ P

1,anL

. Here L is the minimal algebraicallyclosed and complete non-Archimedean field containing C((t)), andf = ft as a rational function with L-coefficients.

e.g., ft(z) = z2 +1

t, µ0 = δ∞

e.g., ft(z) =1√t(z2 + 1), µ0 =

1

2(δi + δ−i)

High Level Question

Given a family of objects over a small punctured disk,what reasonable ways are there to complete it over Dε?

X

��D∗ε

⊆

⊆

X

��Dε

High Level Question

Given a family of objects over a small punctured disk,what reasonable ways are there to complete it over Dε?

X

��D∗ε

⊆

⊆

X

��Dε

e.g., Smooth family of conics: xy = t

Does not extend to a smooth conic over t = 0.

D∗ε

High Level Question

Given a family of objects over a small punctured disk,what reasonable ways are there to complete it over Dε?

X

��D∗ε

⊆

⊆

X

��Dε

e.g., Smooth family of conics: xy = t

Does not extend to a smooth conic over t = 0.

Dε

Theorem. (Semistable reduction) Let X → D∗ε be a holomorphic family

of smooth projective curves. After shrinking ε if necessary, there existsa finite map δ : D∗

ε → D∗ε such that X ×δ D

∗ε extends over Dε to a family

X whose central fiber has at worst ordinary double points (snc).

Families of Dynamical Systems

f =adz

d + · · ·+ a0bdzd + · · ·+ b0

�→ (ad : · · · : a0 : bd : · · · : b0) ∈ P2d+1

Ratd(C) = P2d+1 � Res = “space of rational functions of degree d”Not projective ⇒ degenerate families

Families of Dynamical Systems

f =adz

d + · · ·+ a0bdzd + · · ·+ b0

�→ (ad : · · · : a0 : bd : · · · : b0) ∈ P2d+1

Ratd(C) = P2d+1 � Res = “space of rational functions of degree d”Not projective ⇒ degenerate families

If ft ∈ Ratd(C) is a 1-parameter family with t ∈ D∗ε , demand two things:

1. A limit object associated to “f0” should exist and be unique, and

2. The limit object should be dynamical: same answer if we replaceft by fn

t for any fixed n ≥ 1.

Families of Dynamical Systems

f =adz

d + · · ·+ a0bdzd + · · ·+ b0

�→ (ad : · · · : a0 : bd : · · · : b0) ∈ P2d+1

Ratd(C) = P2d+1 � Res = “space of rational functions of degree d”Not projective ⇒ degenerate families

If ft ∈ Ratd(C) is a 1-parameter family with t ∈ D∗ε , demand two things:

1. A limit object associated to “f0” should exist and be unique, and

2. The limit object should be dynamical: same answer if we replaceft by fn

t for any fixed n ≥ 1.

Key Idea. Pass from ft to its measure of maximal entropy µt.

1. Mane: µt varies continuously for t ∈ D∗ε

2. Characterization: f∗t µt = d ·µt, does not charge exceptional points

3. µt is invariant under ft �→ fnt

Examples

ft(z) = t

(z +

1

z

)

Julia set is very closed to theimaginary axis when t ≈ 0

t = 0.0001

Examples

ft(z) = t

(z +

1

z

)

Julia set is very closed to theimaginary axis when t ≈ 0

t = 0.0001

Theorem Redux — Complex Surface

Theorem. (DM / F) Let ft : C → C be a family of degree d > 1 dynamicalsystems that varies holomorphically with t ∈ D∗

ε , but that does not extend overDε. Let µt be the measure of maximal entropy for ft. Then µ0 = limµt existsfor the weak-∗ topology, and µ0 is purely atomic.

Theorem Redux — Complex Surface

Theorem. (DM / F) Let ft : C → C be a family of degree d > 1 dynamicalsystems that varies holomorphically with t ∈ D∗

ε , but that does not extend overDε. Let µt be the measure of maximal entropy for ft. Then µ0 = limµt existsfor the weak-∗ topology, and µ0 is purely atomic.

Step 1. Any weak-∗ limit µ0 must satisfy a “pullback formula” analogousto f∗

t µt = d · µt.

F

π

Dε × C Y

t = 0 t = 0

F ∗µ0 = d · π∗µ0

Theorem Redux — Transfer Principle

Theorem. (DM / F) Let ft : C → C be a family of degree d > 1 dynamicalsystems that varies holomorphically with t ∈ D∗

ε , but that does not extend overDε. Let µt be the measure of maximal entropy for ft. Then µ0 = limµt existsfor the weak-∗ topology, and µ0 is purely atomic.

Step 1. A weak-∗ limit µ0 satisfies a “pullback formula” F ∗µ0 = d · π∗µ0.

Step 2. ω0 = red∗µ0 is a “residual measure” on P1,anL

that satisfies a“pullback formula” f∗ω0 = d · π∗ω0. Here f = ft is a rational functionwith coefficients in L.

red

P1,anL Y0

Theorem Redux — Uniqueness

Theorem. (DM / F) Let ft : C → C be a family of degree d > 1 dynamicalsystems that varies holomorphically with t ∈ D∗

ε , but that does not extend overDε. Let µt be the measure of maximal entropy for ft. Then µ0 = limµt existsfor the weak-∗ topology, and µ0 is purely atomic.

Step 1. A weak-∗ limit µ0 satisfies a “pullback formula” F ∗µ0 = d · π∗µ0.

Step 2. ω0 = red∗µ0 is a “residual measure” on P1,anL

that satisfies a“pullback formula” f∗ω0 = d · π∗ω0. Here f = ft is a rational functionwith coefficients in L.

Step 3. There is a “unique residual measure” ω0 on P1,anL

that does notcharge exceptional points and satisfies all pullback formulas

(fn)∗ω0 = dn · πn∗ω0, n = 1, 2, 3, . . .

Moreover, ω0 does not charge type II points. Repeat Steps 1 and 2 forall n ≥ 1 to complete the proof.

Questions

Theorem. (DM / F) Let ft : C → C be a family of degree d > 1 dynamicalsystems that varies holomorphically with t ∈ D∗

ε , but that does not extend overDε. Let µt be the measure of maximal entropy for ft. Then µ0 = limµt existsfor the weak-∗ topology, and µ0 is purely atomic.

(i) What if ft varies in a higher dimensional family? Uniqueness of limµt

is lost, but it may be possible to extend our technique to show thatany limit is atomic.

(ii) A precise way to complete the family over Dε with a Berkovichdynamical system over t = 0? (Favre has made some progress.)

Questions

Theorem. (DM / F) Let ft : C → C be a family of degree d > 1 dynamicalsystems that varies holomorphically with t ∈ D∗

ε , but that does not extend overDε. Let µt be the measure of maximal entropy for ft. Then µ0 = limµt existsfor the weak-∗ topology, and µ0 is purely atomic.

(i) What if ft varies in a higher dimensional family? Uniqueness of limµt

is lost, but it may be possible to extend our technique to show thatany limit is atomic.

(ii) A precise way to complete the family over Dε with a Berkovichdynamical system over t = 0? (Favre has made some progress.)

Thanks for your attention.