Equidistribution of hyperbolic centers using Arakelov geometry

Transcription

1 Equidistribution of hyperbolic centers using Arakelov geometry July 8, 203

2 Number theory and dynamics Additive number theory: proof of Szemeredi s theorem by Furstenberg Diophantine approximation problems: proof of the Oppenheim conjecture by Margulis; proof of Khintchine theorem by Sullivan Arithmetic dynamics: set of conjectures promoted by Silverman concerning rational maps f : X K X K defined over a number field (Dynamical Lehmer s conjecture, Manin-Mumford, Mordell-Lang, Uniform Boundedness Conjecture, etc.)

3 The parameter space of quadratic polynomials Fix c C, and define P c (z) := z 2 + c Interested in the behaviour of z, P c (z), P 2 c (z) = P c (P c (z)),..., P n c (z),... K (c) := {z C, P n c (z) = O()} (the filled-in Julia set is compact) Dichotomy: either a Cantor set, or a connected set M := {c C, K (c) is connected } (the Mandelbrot set)

4 The parameter space of quadratic polynomials Fix c C, and define P c (z) := z 2 + c Interested in the behaviour of z, P c (z), P 2 c (z) = P c (P c (z)),..., P n c (z),... K (c) := {z C, P n c (z) = O()} (the filled-in Julia set is compact) Dichotomy: either a Cantor set, or a connected set M := {c C, K (c) is connected } (the Mandelbrot set)

5 The Mandelbrot set and some Julia sets

6 Special points in the Mandelbrot set Per(n) = {c, critical point has exact period n} = {c, P n c (0) = 0, P k c (0) 0 for k < n} Pc (0) = c Pc 2 (0) = c 2 + c = c(c + ) Pc 3 (0) = (c 2 + c) 2 + c = c( + c + 2c 2 + c 3 ) Pc 4 (0) = (c 2 + c)( + 2c 2 + 3c 3 + 3c 4 + 3c 5 + c 6 ) Theorem (Douady-Hubbard) #Per(m) = 2 n m n whence #Per(n) 2 n

7 Special points in the Mandelbrot set Per(n) = {c, critical point has exact period n} = {c, P n c (0) = 0, P k c (0) 0 for k < n} Pc (0) = c Pc 2 (0) = c 2 + c = c(c + ) Pc 3 (0) = (c 2 + c) 2 + c = c( + c + 2c 2 + c 3 ) Pc 4 (0) = (c 2 + c)( + 2c 2 + 3c 3 + 3c 4 + 3c 5 + c 6 ) Theorem (Douady-Hubbard) #Per(m) = 2 n m n whence #Per(n) 2 n

8 Special points in the Mandelbrot set Per(n) = {c, critical point has exact period n} = {c, P n c (0) = 0, P k c (0) 0 for k < n} Pc (0) = c Pc 2 (0) = c 2 + c = c(c + ) Pc 3 (0) = (c 2 + c) 2 + c = c( + c + 2c 2 + c 3 ) Pc 4 (0) = (c 2 + c)( + 2c 2 + 3c 3 + 3c 4 + 3c 5 + c 6 ) Theorem (Douady-Hubbard) #Per(m) = 2 n m n whence #Per(n) 2 n

9 Hyperbolic centers

10 Hyperbolic centers Theorem (Levine) µ n := 2 n c Per(n) δ c µ M := harmonic measure of M Basic tool: harmonic analysis µ M = g M = i π g M µ n = g n Aim: prove that g n g M in L loc.

11 Hyperbolic centers Theorem (Levine) µ n := 2 n c Per(n) δ c µ M := harmonic measure of M Basic tool: harmonic analysis µ M = g M = i π g M µ n = g n Aim: prove that g n g M in L loc.

12 Analytic proof log w c = δ c One may take g n (c) = 2 n log Pn c (0) Construction of g M : Green function: g c (z) := lim n log max{, P n 2 n c (z) } 0 K (c) = {g c = 0} g M (c) := g c (0) = lim n 2 n log max{, Pn c (0) }. g n g M everywhere 2. g n g M uniformly outside M 3. The trick: g n 0 on Int(M)

13 Analytic proof log w c = δ c One may take g n (c) = 2 n log Pn c (0) Construction of g M : Green function: g c (z) := lim n log max{, P n 2 n c (z) } 0 K (c) = {g c = 0} g M (c) := g c (0) = lim n 2 n log max{, Pn c (0) }. g n g M everywhere 2. g n g M uniformly outside M 3. The trick: g n 0 on Int(M)

14 Analytic proof log w c = δ c One may take g n (c) = 2 n log Pn c (0) Construction of g M : Green function: g c (z) := lim n log max{, P n 2 n c (z) } 0 K (c) = {g c = 0} g M (c) := g c (0) = lim n 2 n log max{, Pn c (0) }. g n g M everywhere 2. g n g M uniformly outside M 3. The trick: g n 0 on Int(M)

15 Adelic proof (Baker-H sia) Construction of a suitable height function such that Per(n) {h M = 0} Use of points of small height (Autissier) The latter result goes back to Bilu and Szpiro-Ullmo-Zhang in their work on the Bogomolov conjecture.

16 Adelic proof (Baker-H sia) Fix v a place in Q, and c C v. g c,v (z) := lim n 2 n log max{, Pn c (z) v } g M,v (c) = g c,v (0) g M, (c) = g M (c) and g M,p (c) = log + c p h M (c) := deg(c) c c v M Q g M,v (c ) h M differs from the standard height by a bounded function

17 Adelic proof (Baker-H sia) Theorem (Autissier) For any sequence of disjoint finite sets Z n Q that are invariant under Gal( Q/Q) and such that h M Zn = 0 then #Z n p Z n δ p µ M Apply this to Z n = Per(n)

18 Analytic vs Global method Theorem (F.- Rivera-Letelier, Okuyama) For any C function ϕ, 2 n ϕ(c) ϕµ M n C 2 n/2 ϕ C c Per(n) Theorem (Buff-Gauthier) 2 n where c Per(n,λ) δ c µ M := harmonic measure of M Per(n, λ) = {c, P n c (z) = z, (P n c ) (z) = λ}

19 Analytic vs Global method Theorem (F.- Rivera-Letelier, Okuyama) For any C function ϕ, 2 n ϕ(c) ϕµ M n C 2 n/2 ϕ C c Per(n) Theorem (Buff-Gauthier) 2 n where c Per(n,λ) δ c µ M := harmonic measure of M Per(n, λ) = {c, P n c (z) = z, (P n c ) (z) = λ}

20 The parameter space of polynomials of degree d = 3 P c,a (z) = 3 z3 c 2 z2 + a 3 Crit(P c,a ) = {P c,a = 0} = {c := c, c 0 := 0} Per(n 0, n ) := {(c, a) C 2, P n i c,a(c i ) = c i for i = 0, } Theorem (F.-Gauthier) If n (k) 0 n (k) and min n (k) i then 3 n(k) 0 +n(k) δ p µ M3 p Per(n (k) 0,n(k) ) where µ M3 is the equilibrium measure of the connectedness locus of cubic polynomials.

21 The parameter space of polynomials of degree d = 3 P c,a (z) = 3 z3 c 2 z2 + a 3 Crit(P c,a ) = {P c,a = 0} = {c := c, c 0 := 0} Per(n 0, n ) := {(c, a) C 2, P n i c,a(c i ) = c i for i = 0, } Theorem (F.-Gauthier) If n (k) 0 n (k) and min n (k) i then 3 n(k) 0 +n(k) δ p µ M3 p Per(n (k) 0,n(k) ) where µ M3 is the equilibrium measure of the connectedness locus of cubic polynomials.

22 Analytic method The Green function is well-defined: g c,a (z) := lim n 3 n log max{, Pn c,a(z) } g 0 = g c,a (c 0 ), g = g c,a (c ). Connectedness locus is {g 0 = g = 0} and is compact Equilibrium measure: µ M := (dd c ) 2 G(c, a) with G = max{g 0, g } Warning: the analytic method only applies when n (k) 0 n (k) (Dujardin -F.)

23 Analytic method The Green function is well-defined: g c,a (z) := lim n 3 n log max{, Pn c,a(z) } g 0 = g c,a (c 0 ), g = g c,a (c ). Connectedness locus is {g 0 = g = 0} and is compact Equilibrium measure: µ M := (dd c ) 2 G(c, a) with G = max{g 0, g } Warning: the analytic method only applies when n (k) 0 n (k) (Dujardin -F.)

24 Strategy Construction of a natural height where Per(n 0, n ) {h M3 = 0} Application of Yuan s theorem of of points of small heights Difficulties:. Height should be defined at finite places in a special way (semi-positive adelic metric) 2. Points should be generic

25 Strategy Construction of a natural height where Per(n 0, n ) {h M3 = 0} Application of Yuan s theorem of of points of small heights Difficulties:. Height should be defined at finite places in a special way (semi-positive adelic metric) 2. Points should be generic

26 Construction of the height The construction is similar to the quadratic case. g c,a,v (z) = lim n 3 n log max{, P n c,a(z) v } G v = max{g c,a,v (c 0 ), g c,a,v (c )} For p 5 then G v = log max{, c, a } h M3 (c, a) := deg(c, a) (c,a ) (c,a) v M Q g M,v (c, a ) It differs from the standard height by a bounded factor. Per(n 0, n ) {h M3 = 0}

27 Yuan s theorem

28 Yuan s theorem Theorem The line bundle: O() P 2 Q ; Metrization: σ v := e Gv on A 2 (with div(σ) the hyperplane at infinity) The associated height function is h M3. Suppose F n is a sequence of finite subsets of P 2 ( Q) that are defined over Q such that h M3 (F n ) 0; For any subvariety Z P 2, #(Fn Z ) #F n 0. Then #F n p F n δ p µ M3 in A 2 (C)

29 Yuan s theorem Theorem The line bundle: O() P 2 Q ; Metrization: σ v := e Gv on A 2 (with div(σ) the hyperplane at infinity) The associated height function is h M3. Suppose F n is a sequence of finite subsets of P 2 ( Q) that are defined over Q such that h M3 (F n ) 0; For any subvariety Z P 2, #(Fn Z ) #F n 0. Then #F n p F n δ p µ M3 in A 2 (C)

30 Genericity Theorem Fix a sequence n (k) 0 n (k) and min n (k) i, and pick any curve Z A 2. Then # Per(n (k) 0 lim, n(k) ) Z k # Per(n (k) 0, n(k) ) = 0 Proof. Per ε (n) = {(c, a), P n c,a(c ε ) = c ε } has degree 3 n ; Lower bound # Per(n 0, n ) = #Per 0 (n 0 ) Per (n ) = 3 n 0+n Upper bound Per(n 0, n ) Z (Per 0 (n 0 ) Z ) (Per (n ) Z ) #Per(n 0, n ) Z deg(z )3 max n 0,n

31 Genericity Theorem Fix a sequence n (k) 0 n (k) and min n (k) i, and pick any curve Z A 2. Then # Per(n (k) 0 lim, n(k) ) Z k # Per(n (k) 0, n(k) ) = 0 Proof. Per ε (n) = {(c, a), P n c,a(c ε ) = c ε } has degree 3 n ; Lower bound # Per(n 0, n ) = #Per 0 (n 0 ) Per (n ) = 3 n 0+n Upper bound Per(n 0, n ) Z (Per 0 (n 0 ) Z ) (Per (n ) Z ) #Per(n 0, n ) Z deg(z )3 max n 0,n

32 Transversality problems Theorem (Adam Epstein) Pick n 0 n. Then Per 0 (n 0 ) and Per (n ) are smooth at any of their intersection points, and intersect transversally there. Method inspired by Teichmüller theory. Relies on purely analytical tools (contraction properties of suitable operators in a complex Banach algebra).

33 Characterization of special subvarieties Special points: h M3 (c, a) = 0 both critical points have a finite orbit Question Describe irreducible curves in A 2 for which the set of special points is infinite. Chambert-Loir answered this for the standard height function (Bogomolov conjecture for semi-abelian varieties).

34 Characterization of special subvarieties Conjecture (Baker-DeMarco) Let V A 2 be an irreducible curve containing infinitely many (c, a) such that both critical points of P c,a have a finite orbit. Then either one of the two critical points has finite orbit for all v V ; or there exists a critical dynamically defined relation, i.e a closed subvariety Z V (A ) 2 invariant by the map (v, z, w) (v, P v (z), P v (w)) and containing (v, c 0, c ) for all v V.

35 : characterization of special subvarieties Example P c,a (c) = 0 defines a special curve {6a 3 = c 3 } Example The family P t (z) = z 3 3tz 2 + (2t 3 + t) is special. c 0 = 0, c = 2t h t (z) = z + 2t satisfies h t f t = f t h t Z = {(t, z, h t (z))}

36 : characterization of special subvarieties Example P c,a (c) = 0 defines a special curve {6a 3 = c 3 } Example The family P t (z) = z 3 3tz 2 + (2t 3 + t) is special. c 0 = 0, c = 2t h t (z) = z + 2t satisfies h t f t = f t h t Z = {(t, z, h t (z))}

37 Characterization of special subvarieties Theorem (Baker-DeMarco) In the space of cubic polynomials P a,b = z 3 + az + b. Consider the curve Per(λ) = {P a,b admits a fixed point with multiplier λ} Then Per(λ) contains infinitely many points for which both critical points are periodic iff λ = 0.