Quantitative equidistribution in non-archimedean and complex - - PowerPoint PPT Presentation

Quantitative equidistribution in non-archimedean and complex dynamics

Yˆ usuke Okuyama (Kyoto Inst. Tech., okuyama@kit.ac.jp)

Complex and p-adic Dynamics; ICERN, Brown University 13 February, 2012

§Berkovich projective line: Notation

K:

algebraically closed field, complete WRT a non-trivial absolute value | · |. (either non-archimedean or archimedean. e.g. Cp, Cu, C)

P1 = P1(K): (classical) projective line [·, ·]: the normalized chordal distance on P1

P1 = P1(K): Berkovich projective line, compactifying P1 (Fact: For archimedean K, P1 P1) H1 := P1 \ P1: endowed with the hyperbolic metric ρ

δ(·, ·)can: the generalized Hsia kernel on P1 WRT Scan ∈ H1.

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§Gauss variational approach to dynamics

A rational function of degree d > 1

f : P1 → P1.

(Fact: this extends to P1 → P1, f(H1) = H1, conti, surj, open, discrete)

∃1(non-degenerate homogeneous polynomial) lift of f F : K2 → K2

(upto ×c ∈ K∗), i.e. for canonical projection π : K2 → P1(K),

π ◦ F = f ◦ π

and the homogeneous resultant Res F does not vanish.

2

Def . The dynamical Green function on P1

gF :=

∞

∑

j=0

1 d j(f j)∗ (1 d log |F| − log | · | )

(for ∀c ∈ K∗, gcF = gF + (log |c|)/(d − 1)).

An upper semicontinuous F-kernel on P1

ΦF(z, w) := log δ(z, w)can − gF(z) − gF(w).

The F-energy of a Radon measure µ on P1 (if exists)

IF(µ) := ∫

P1×P1 ΦF(z, w)d(µ × µ)(z, w).

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The F-equilibrium energy of (the whole) P1

VF := sup{IF(µ); µ is a prob. Radon measure on P1} > −∞.

A possible definition of the canonical measure µ f is Thm . There is the unique solution of Gauss variational problem WRT external field gF. Concretely, ∃1 probability Radon measure µ f on P1 s.t.

IF(µ f) = VF.

(Rem: µf is independent of choices of F)

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§Fekete configuration in dynamics

Now we can be more canonical: the f-kernel on P1

Φf(·, ·) := ΦF(·, ·) − VF,

independent of choices of F. (Rem:

−Φf is called the

Arakelov Green (kernel) function of f on P1) Def . A sequence (νn) of positive discrete measures on P1 is

f-asymptotically Fekete on P1 if as n → ∞, νn(P1) ր ∞, (νn × νn)(diagP1) = o(νn(P1)2), 1 νn(P1)2 ∫

P1×P1\diagP1

Φfd(νn × νn) → 0.

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(Rem: this is an analogue of Gauss variational problem for positive discrete measures. Φf(S, S) > 0 if S ∈ H1.) Def . The averaged pullback of a ∈ P1

( f n)∗(a) := ∑

w∈f−n(a)

degw( f n) · (a)

((a): the Dirac measure at a on P1). The algebraic exceptional set of f (Rem: this is in P1)

E( f) := {a ∈ P1; # ∪

n∈N

f −n(a) < ∞}. SAT( f): superattracting periodic points of f

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Def (main quantity). For each a ∈ P1 and each n ∈ N,

E f(n, a) := 1 d2n ∫

P1×P1\diag1

P

Φfd(( f n)∗(a) × ( f n)∗(a)) = − (( f n)∗(a) dn − µ f, ( f n)∗(a) dn − µ f )

f

(: the dyn version of Favre and Rivera-Letelier’s energy). (Fact) Then

• For ∀a ∈ P1 \ E( f),

((f n)∗(a)) is f-asymp Fekete on P1 ⇔ limn→∞ E f(n, a) = 0.

• For ∀a ∈ E( f), ((f n)∗(a)) is NEVER f-asymp Fekete on P1.

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Another fundamental quantity Def . For ∀a ∈ P1 \ E( f),

ηa,n = ηa,n( f) := max

w∈f−n(a) degw( f) ∈ N.

Rem: if K has characteristic 0, then

lim sup

j→∞

η1/ j

a,j

         ≤ (d3 − 1)1/3 (a ∈ P1 \ E( f)), = d (a ∈ E( f)),

(1)

sup

j∈N

ηa,j          ≤ d2d−2 (a ∈ P1 \ SAT( f)), = ∞ (a ∈ SAT( f)).

(2)

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§Main results: error estimates on Fekete

Let f be a rat function on P1 = P1(K) of degree d > 1. Put

C( f) := {c ∈ P1; f ′(c) = 0}, C( f)wan := {c ∈ C( f); ( f n(c)) is wandering under f}, CO(f)wan := {f n(c); c ∈ C( f)wan, n ∈ N}.

(Rem: if f has char 0, then ∑

c∈C( f)(degc f − 1) = 2d − 2.)

Thm 1 (principal estimates). For ∀a ∈ H1 and ∀n ∈ N,

|E f(n, a)| ≤ Cd−n

(3) for some C > 0 indep of n and loc bounded on a under ρ.

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(cont.)

If in addition K has char 0, then there is C′ > 0 s.t. for

∀a ∈ P1 and ∀n ∈ N, − 1 dn

n

∑

j=1

ηa,j ∑

c∈C(f)\ f− j(a)

1 d j log 1 [ f j(c), a] − C′ dn

n

∑

j=1

ηa, j − Ca dn ≤E f(n, a)

(4)

≤ − 1 dn

n

∑

j=1

∑

c∈C(f)\ f−j(a)

1 d j log 1 [ f j(c), a] + C′ dn

n

∑

j=1

ηa, j + Ca dn.

Here the constant Ca ≥ 0, which is independent of n, vanishes if a ∈ P1 \ CO( f)wan.

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Def . The classical omega limit set of each z0 ∈ P1

ω(z0) = ω(z0) := ∩

N∈N

{f n(z0); n ≥ N}

chordal

.

A point z0 ∈ P1 is pre-recurrent if ∃n0 ∈ N, f n0(z0) ∈ ω(z0).

(The chordal open ball with center w ∈ P1 and radius r > 0

B[w, r] := {z ∈ P1; [z, w] < r})

Thm 1 estimates the non-Fekete locus

EFekete( f) := {a ∈ P1; (( f n)∗(a)) is not f-asymp Fekete on P1}

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from above using

Ewan( f) := ∩

N∈N

∪

j≥N

∪

c∈C(f)wan

B[ f j(c), exp(−d j)].

Thm 2. Suppose K has characteristic 0. Then

E( f) ⊂ EFekete( f) ⊂ P1, EFekete( f) \ E(f) ⊂ Ewan( f) \ E(f),

and Ewan( f) is of capacity 0. (finite Hyllengren meas for (d j)). Moreover, EFekete( f) is Gδ-dense in ω(c) for every pre- recurrent c ∈ C( f)wan.

(so, possibly E( f) EFekete( f))

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§Application: quantitative equidistribution/K

Let f be a rational function on P1 = P1(K) of degree d > 1. Favre and Rivera-Letelier’s Cauchy-Schwarz inequality is Prop . For ∀a ∈ P1, C1-test function ∀φ on P1 and ∀n ∈ N,

• φ, ( f n)∗(a)

dn − µf

• ≤

         φ, φ1/2 √ |E f(n, a)| (a ∈ H1), C max{Lip(φ), φ, φ1/2} √ |E f(n, a)| + nd−nηa,n (a ∈ P1).

Here C > 0 is independent of a ∈ P1, φ and n.

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Theorem 1 establishes a quantitative equidistribution in terms of the proximity of wandering crit orbits to a ∈ P1. Thm 3 (Special case). Suppose that K has characteristic 0. Then there is C > 0 s.t. for ∀a ∈ P1 excluding Ewan( f) of capacity 0 and ∀n ∈ N large enough,

|E f(n, a)| ≤ Cnd−nηa,n,

(5) and there is C′ > 0 s.t. for C1-test function ∀φ on P1, ∀a ∈

P1 \ Ewan( f) and ∀n ∈ N large enough,

• φ, ( f n)∗(a)

dn − µ f

• ≤ C′ max{Lip(φ), φ, φ1/2}

√ nd−nηa,n

(Recall that supn∈N ηa,n ≤ d2d−2 if in addition a SAT(f)).

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(cont.)

On the other hand, for ∀a0 ∈ P1 excluding

PC(f) := { f n(c); c ∈ C( f), n ∈ N}

chordal

,

there are r0 > 0 and N = N(a0) s.t. for ∀a ∈ B[a0, r0], C1-test function ∀φ on P1, ∀k > N, the same (but locally uniform) estimate

• φ, ( f n)∗(a)

dn − µ f

• ≤ C′ max{Lip(φ), φ, φ1/2}

√ nd−n.

Rem .For K C, the better O(

√ d−n) estimate holds for ∀a ∈ P1 at which f is semihyp. (cf. D. Drasin and Ok, BLMS 2007).

15

§Arithmetic application/global fields

For a number field or a function field k, when f has its coefficients in k, the dynamics on algebraic points

f : P1(k) → P1(k),

is also interesting. Fix a non-trivial absolute value u on k, and set K = Cu. The dynamical Diophantine approximation (Silverman 1993, Szpiro and Tucker 2005): For ∀a ∈ P1(k) \ E( f) and wandering ∀z ∈ P1(k),

lim

n→∞

1 dn log[ f n(z), a]u = 0.

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Since (E( f) ⊂ SAT( f) ⊂)C( f) ⊂ P1(k), consequently

Ewan(f)u ∩ P1(k) ⊂ E( f),

(6) and Theorem 3 recovers (in a purely local manner) Favre and Rivera-Letelier’s arithmetic quantitative equidistribution: Under the above arithmetic setting, there is C > 0 s.t. for

∀a ∈ P1(k) \ E( f), C1-test function ∀φ on P1(Cu) and ∀n ∈ N

large enough,

• φ, ( f n)∗(a)

dn − µf,u

• ≤ C′ max{Lip(φ), φ, φ1/2}

√ nd−nηa,n.

Rem . By Thm 2 with (6), also EFekete( f)u ∩ P1(k) = E( f).

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§In complex dynamics/C

Let f be a rational function on P1(C) of degree > 1. Q . When E( f) = EFekete(f)? Cor 1. If ∃ Cremer periodic points, Siegel disks or Herman rings of f, then E( f) EFekete( f). If f is geometrically finite, then EFekete( f) = E( f). Rem . ∃ semihyperbolic real cubic polynomial f such that

EFekete( f) ∩ J( f) ∅, so E( f) EFekete( f).

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