Distribution of PCF cubic polynomials Charles Favre - PowerPoint PPT Presentation

Distribution of PCF cubic polynomials Charles Favre charles.favre@polytechnique.edu June 28th, 2016

The parameter space of polynomials Poly d : degree d polynomials modulo conjugacy by affine transformations. ◮ { P = a d z d + · · · + a 0 , a d � = 0 } / { P ∼ φ ◦ P ◦ φ − 1 , φ ( z ) = az + b } ◮ C ∗ × C d / Aff ( 2 , C ) . Complex affine variety of dimension d − 1: finite quotient singularities.

Exploring the geometry of Poly 2 Interested in the geometry of the locus of polynomials with special dynamics in Poly d .

Exploring the geometry of Poly 2 Interested in the geometry of the locus of polynomials with special dynamics in Poly d . Work with a suitable ramified cover of Poly d by C d − 1 .

Exploring the geometry of Poly 2 Interested in the geometry of the locus of polynomials with special dynamics in Poly d . Work with a suitable ramified cover of Poly d by C d − 1 . ◮ d = 2: parameterization P c ( z ) = z 2 + c , c ∈ C . Critical point: 0. ◮ PCF maps: 0 is pre-periodic.

Exploring the geometry of Poly 2 Interested in the geometry of the locus of polynomials with special dynamics in Poly d . Work with a suitable ramified cover of Poly d by C d − 1 . ◮ d = 2: parameterization P c ( z ) = z 2 + c , c ∈ C . Critical point: 0. ◮ PCF maps: 0 is pre-periodic. PCF maps are defined over Q : c = 0 , c 2 + c = 0 , ( c 2 + c ) 2 + c = 0 , . . .

Distribution of hyperbolic quadratic PCF polynomials

Distribution of hyperbolic quadratic PCF polynomials

Distribution of quadratic PCF polynomials PCF ( n ) = { P c , the orbit of 0 has cardinality ≤ n } . Theorem (Levin, Baker-H’sia, F .-Rivera-Letelier, ...) The probability measures µ n equidistributed on PCF ( n ) converge weakly towards the harmonic measure of the Mandelbrot set.

Distribution of quadratic PCF polynomials PCF ( n ) = { P c , the orbit of 0 has cardinality ≤ n } . Theorem (Levin, Baker-H’sia, F .-Rivera-Letelier, ...) The probability measures µ n equidistributed on PCF ( n ) converge weakly towards the harmonic measure of the Mandelbrot set. ◮ Levin: potential theoretic arguments ◮ Baker-H’sia: exploit adelic arguments, and work over all completions of Q both Archimedean and non-Archimedean: C , C p for p prime.

The cubic case Poly 3 3 z 3 − c 2 z 2 + a 3 , a , c ∈ C . ◮ Parameterization: P c , a ( z ) = 1 ◮ Crit ( P c , a ) = { 0 , c } ◮ P c , a ( 0 ) = a 3 , P c , a ( c ) = a 3 − c 3 6 . PCF ( n , m ) = { orbit of 0 has cardinality ≤ n } & { orbit of c has cardinality ≤ m }

Finiteness of PCF maps 3 z 3 − c 2 z 2 + a 3 P c , a ( z ) = 1 Proposition (Branner-Hubbard) The set PCF ( n , m ) is finite and defined over Q .

Finiteness of PCF maps 3 z 3 − c 2 z 2 + a 3 P c , a ( z ) = 1 Proposition (Branner-Hubbard) The set PCF ( n , m ) is finite and defined over Q . ◮ PCF ( n , m ) is bounded in C 2 by Koebe distortion estimates;

Finiteness of PCF maps 3 z 3 − c 2 z 2 + a 3 P c , a ( z ) = 1 Proposition (Branner-Hubbard) The set PCF ( n , m ) is finite and defined over Q . ◮ in C p with p ≥ 5:

Finiteness of PCF maps 3 z 3 − c 2 z 2 + a 3 P c , a ( z ) = 1 Proposition (Branner-Hubbard) The set PCF ( n , m ) is finite and defined over Q . ◮ in C p with p ≥ 5: ◮ if | a | > max { 1 , | c |} , then | P c , a ( 0 ) | = | a | 3 , | P 2 c , a ( 0 ) | = | a | 9 c , a ( 0 ) | = | a | 3 n → ∞ ; and, | P n

Finiteness of PCF maps 3 z 3 − c 2 z 2 + a 3 P c , a ( z ) = 1 Proposition (Branner-Hubbard) The set PCF ( n , m ) is finite and defined over Q . ◮ in C p with p ≥ 5: ◮ if | a | > max { 1 , | c |} , then | P c , a ( 0 ) | = | a | 3 , | P 2 c , a ( 0 ) | = | a | 9 c , a ( 0 ) | = | a | 3 n → ∞ ; and, | P n ◮ PCF ( n , m ) ⊂ {| c | , | a | ≤ 1 } .

Finiteness of PCF maps 3 z 3 − c 2 z 2 + a 3 P c , a ( z ) = 1 Proposition (Branner-Hubbard) The set PCF ( n , m ) is finite and defined over Q . Proposition (Ingram) For any finite extension K / Q the set PCF ∩ K 2 is finite.

Distribution of PCF maps 3 z 3 − c 2 z 2 + a 3 P c , a ( z ) = 1 Theorem (F.-Gauthier) The probability measures µ n , m equidistributed on PCF ( n , m ) converge weakly towards the equilibrium measure µ of the connectedness locus C as n , m → ∞ .

Distribution of PCF maps 3 z 3 − c 2 z 2 + a 3 P c , a ( z ) = 1 Theorem (F.-Gauthier) The probability measures µ n , m equidistributed on PCF ( n , m ) converge weakly towards the equilibrium measure µ of the connectedness locus C as n , m → ∞ . ◮ C = { ( c , a ) , Julia set of P c , a is connected } = { ( c , a ) , 0 and c have bounded orbit } ;

Distribution of PCF maps 3 z 3 − c 2 z 2 + a 3 P c , a ( z ) = 1 Theorem (F.-Gauthier) The probability measures µ n , m equidistributed on PCF ( n , m ) converge weakly towards the equilibrium measure µ of the connectedness locus C as n , m → ∞ . ◮ C = { ( c , a ) , Julia set of P c , a is connected } = { ( c , a ) , 0 and c have bounded orbit } ; ◮ Green function G C : C 0 psh ≥ 0 function, C = { G C = 0 } , G ( c , a ) = log max { 1 , | c | , | a |} + O ( 1 ) ;

Distribution of PCF maps 3 z 3 − c 2 z 2 + a 3 P c , a ( z ) = 1 Theorem (F.-Gauthier) The probability measures µ n , m equidistributed on PCF ( n , m ) converge weakly towards the equilibrium measure µ of the connectedness locus C as n , m → ∞ . ◮ C = { ( c , a ) , Julia set of P c , a is connected } = { ( c , a ) , 0 and c have bounded orbit } ; ◮ Green function G C : C 0 psh ≥ 0 function, C = { G C = 0 } , G ( c , a ) = log max { 1 , | c | , | a |} + O ( 1 ) ; µ = Monge-Ampère ( G C ) = ( dd c ) 2 G C .

The adelic approach Interpretation of the Green function in the parameter space: G C = max { G c , a ( c ) , G c , a ( 0 ) } ◮ Dynamical Green function: G c , a = lim 1 3 n log max { 1 , | P n c , a |}

The adelic approach Interpretation of the Green function in the parameter space: G C = max { G c , a ( c ) , G c , a ( 0 ) } ◮ Dynamical Green function: G c , a = lim 1 3 n log max { 1 , | P n c , a |} ◮ Same construction for any norm | · | p on Q : G C , p

The adelic approach Interpretation of the Green function in the parameter space: G C = max { G c , a ( c ) , G c , a ( 0 ) } ◮ Dynamical Green function: G c , a = lim 1 3 n log max { 1 , | P n c , a |} ◮ Same construction for any norm | · | p on Q : G C , p Key observation: P c , a ∈ PCF iff 1 � � G C , p ( c ′ , a ′ ) = 0 Height ( c , a ) := deg ( c , a ) p c ′ , a ′ − → Apply Yuan’s theorem!

Special curves Problem (Baker-DeMarco) Describe all irreducible algebraic curves C in Poly 3 such that PCF ∩ C is infinite. ◮ motivated by the André-Oort conjecture in arithmetic geometry

Special curves Problem (Baker-DeMarco) Describe all irreducible algebraic curves C in Poly 3 such that PCF ∩ C is infinite. ◮ motivated by the André-Oort conjecture in arithmetic geometry Theorem (Baker-DeMarco, Ghioca-Ye, F .-Gauthier) Suppose C is an irreducible algebraic curve in Poly 3 such that PCF ∩ C is infinite. Then there exists a persistent critical relation.

The curves Per n ( λ ) : DeMarco’s conjecture Per n ( λ ) = { P c , a admitting a periodic point of period n and multiplier λ }

The curves Per n ( λ ) : DeMarco’s conjecture Per n ( λ ) = { P c , a admitting a periodic point of period n and multiplier λ } ◮ Geometry of Per n ( 0 ) : Milnor, DeMarco-Schiff, irreducibility by Arfeux-Kiwi ( n prime);

The curves Per n ( λ ) : DeMarco’s conjecture Per n ( λ ) = { P c , a admitting a periodic point of period n and multiplier λ } ◮ Geometry of Per n ( 0 ) : Milnor, DeMarco-Schiff, irreducibility by Arfeux-Kiwi ( n prime); ◮ Distribution of Per n ( λ ) when n → ∞ described by Bassaneli-Berteloot.

The curves Per n ( λ ) : DeMarco’s conjecture Per n ( λ ) = { P c , a admitting a periodic point of period n and multiplier λ } ◮ Geometry of Per n ( 0 ) : Milnor, DeMarco-Schiff, irreducibility by Arfeux-Kiwi; ◮ Distribution of Per n ( λ ) when n → ∞ described by Bassaneli-Berteloot. Theorem (F.-Gauthier) The set PCF ∩ Per n ( λ ) is infinite iff λ = 0 . n = 1 by Baker-DeMarco

Scheme of proof C irreducible component of Per n ( λ ) containing infinitely many PCF: λ ∈ ¯ Q . 1. One of the critical point is persistently preperiodic on C . 2. C contains a unicritical PCF polynomial = ⇒ | λ | 3 < 1. 3. There exists a quadratic PCF polynomial having λ as a multiplier = ⇒ | λ | 3 = 1.

Step 2 3 z 3 − c 2 z 2 + a 3 P c , a ( z ) = 1 C contains a unicritical PCF polynomial, and | λ | 3 < 1. a) Existence of the PCF unicritical polynomial by Bezout and Step 1.

Step 2 3 z 3 − c 2 z 2 + a 3 P c , a ( z ) = 1 C contains a unicritical PCF polynomial, and | λ | 3 < 1. a) Existence of the PCF unicritical polynomial by Bezout and Step 1. b) Suppose P ( z ) = z 3 + b is PCF ◮ | b | 3 ≤ 1 ◮ periodic orbits are included in | z | 3 ≤ 1 ◮ | P ′ ( z ) | 3 = | 3 z 2 | 3 < 1 on the unit ball hence | λ | 3 < 1

Step 1 One of the critical point is persistently preperiodic on C .

Step 1 One of the critical point is persistently preperiodic on C . 1. Apply the previous theorem: there exists a persistent critical relation.