Critical points of the multiplier map Igors Gorbovickis Jacobs - PowerPoint PPT Presentation

Critical points of the multiplier map Igors Gorbovickis Jacobs University Bremen March 25, 2019 Igors Gorbovickis Critical points of the multiplier map

Independence of multipliers Theorem (G. 2014): The multipliers of any n − 1 distinct periodic orbits, considered as algebraic maps on the space of degree n polynomials, are locally independent at a generic polynomial f . Igors Gorbovickis Critical points of the multiplier map

The multiplier map on the space Poly 2 = { z 2 + c | c ∈ C } For any k ∈ N , ◮ let Poly k 2 be the set of all pairs ( f c , O ), such that f c ∈ Poly 2 and O is a periodic orbit of f c of period k . ◮ The multiplier map ρ k : Poly k 2 → C is defined by ρ k ( f c , O ) := the multiplier of the periodic orbit O . Question: What can we say about the critical points of the maps ρ k ? Igors Gorbovickis Critical points of the multiplier map

The multiplier map on the space Poly 2 = { z 2 + c | c ∈ C } For any k ∈ N , ◮ let Poly k 2 be the set of all pairs ( f c , O ), such that f c ∈ Poly 2 and O is a periodic orbit of f c of period k . ◮ The multiplier map ρ k : Poly k 2 → C is defined by ρ k ( f c , O ) := the multiplier of the periodic orbit O . Question: What can we say about the critical points of the maps ρ k ? When c = 0, k − 1 d ρ k z − 2 j +1 � dc (0 , � z 0 � ) = − 2 k . 0 j =0 6 12 18 20 21 24 30 k e 2 π i / 9 e 2 π i / 45 e 2 π i / 27 e 2 π i / 25 e 2 π i / 49 e 2 π i / 153 e 2 π i / 99 z 0 Table: The list of all k ≤ 30, for which ρ k has a critical point at c = 0. ( z 0 is a corresponding periodic point.) Igors Gorbovickis Critical points of the multiplier map

Critical points of the multiplier maps ρ k For any k ∈ N , define ◮ σ k ( f c , O ) := d ρ k dc ( f c , O ); ◮ X k := { c ∈ C | σ k ( f c , O ) = 0 , for some periodic orbit O} . (Points in X k are counted with multiplicity.) 1 � ν k := δ c . # X k c ∈ X k Theorem (Firsova, G.): The sequence of measures { ν k } k ∈ N converges to µ bif in the weak sense of measures on C . Theorem (Firsova, G.): For every k 0 ∈ N and c ∈ X k 0 \ M , there exists a sequence { c k } ∞ k =3 , such that each c k ∈ X k and k →∞ c k = c . lim Igors Gorbovickis Critical points of the multiplier map

Related results for quadratic polynomials µ bif = ∆ G M , where G M : C → [0 , + ∞ ) is the Green’s function of the Mandelbrot set and ∆ is the generalized Laplacian. n → + ∞ max { 2 − n log | f ◦ n G c ( z ) = lim c ( z ) | , 0 } , G M ( c ) = G c ( c ) . Igors Gorbovickis Critical points of the multiplier map

Related results for quadratic polynomials µ bif = ∆ G M , where G M : C → [0 , + ∞ ) is the Green’s function of the Mandelbrot set and ∆ is the generalized Laplacian. n → + ∞ max { 2 − n log | f ◦ n G c ( z ) = lim c ( z ) | , 0 } , G M ( c ) = G c ( c ) . Theorem (Brolin 1965): For any z 0 ∈ C (possibly avoiding two exceptional values), the points f − k ( z 0 ) (counted with multiplicity) c equidistribute on the Julia set J c , as k → ∞ . Theorem (Levin 1989, Bassanelli-Berteloot 2011, Buff-Gauthier 2015): For any ρ 0 ∈ C , the set of parameters c (counted with multiplicity), such that ρ k ( f c , O ) = ρ 0 , for some ( f c , O ) ∈ P k , equidistributes on the boundary of M , as k → ∞ . Igors Gorbovickis Critical points of the multiplier map

Critical points of the multiplier maps ρ k For any s ∈ C and any k ∈ N , ◮ define X s , k := { c ∈ C | σ ( f c , O ) = s , for some periodic orbit O} . (Points in X s , k are counted with multiplicity.) 1 � ν s , k := δ c . # X s , k c ∈ X s , k Theorem (Firsova, G.): For every sequence of complex numbers { s k } k ∈ N , such that 1 lim sup k log | s k | ≤ log 2 , k → + ∞ the sequence of measures { ν s k , k } k ∈ N converges to µ bif in the weak sense of measures on C . Igors Gorbovickis Critical points of the multiplier map

Idea of the proof: Potentials! Step 1: For each measure ν k , construct a potential (a subharmonic function) u k : C → [ −∞ , + ∞ ), such that ∆ u k = ν k . Step 2: Then convergence u k → G M in L 1 loc as k → ∞ implies weak convergence of measures ν k → µ bif . Igors Gorbovickis Critical points of the multiplier map

Step 1: Potentials ˜ � S k ( c , s ) := ( s − σ k ( f c , O )) O| ( c , O ) ∈ P k ˜ S k is a rational map in c with simple poles at primitive parabolic c . � C k ( c ) := ( c − ˜ c ) . c ∈ ˜ ˜ P k S k ( c , s ) = C k ( c ) ˜ S k ( c , s ) – polynomials in c and s . Lemma: S k ( c , 0) = 0, iff c is a critical point of the multiplier map ρ k . Igors Gorbovickis Critical points of the multiplier map

Step 1: Potentials ˜ � S k ( c , s ) := ( s − σ k ( f c , O )) O| ( c , O ) ∈ P k ˜ S k is a rational map in c with simple poles at primitive parabolic c . � C k ( c ) := ( c − ˜ c ) . c ∈ ˜ ˜ P k S k ( c , s ) = C k ( c ) ˜ S k ( c , s ) – polynomials in c and s . Lemma: S k ( c , 0) = 0, iff c is a critical point of the multiplier map ρ k . For all c ∈ C , define 1 1 � � log | ˜ u k ( c ) := log | S k ( c , 0) | = S k ( c , 0) | + log | C k ( c ) | . deg c S k deg c S k Then ν k = ∆ u k . Igors Gorbovickis Critical points of the multiplier map

Step 2: L 1 loc convergence of potentials Lemma (Buff, Gauthier): Any subharmonic function u : C → [ −∞ , + ∞ ) which coincides with G M outside M , coincides with G M everywhere. Lemma (Buff, Gauthier): Let K ⊂ C be a compact set such that C \ K is connected. Let v be a subharmonic function on C such that ∆ v is supported on ∂ K and does not charge the boundary of the connected components of the interior of K . Then, any subharmonic function u on C which coincides with v outside K , coincides with v everywhere. Igors Gorbovickis Critical points of the multiplier map

Step 2: L 1 loc convergence of potentials Lemma (Buff, Gauthier): Any subharmonic function u : C → [ −∞ , + ∞ ) which coincides with G M outside M , coincides with G M everywhere. Lemma (Buff, Gauthier): Let K ⊂ C be a compact set such that C \ K is connected. Let v be a subharmonic function on C such that ∆ v is supported on ∂ K and does not charge the boundary of the connected components of the interior of K . Then, any subharmonic function u on C which coincides with v outside K , coincides with v everywhere. Corollary: We need to prove u k → G M only outside M . Igors Gorbovickis Critical points of the multiplier map

Roots of the multiplier maps c ( λ ) := φ − 1 M ( λ 2 ) c : C \ D → C \ M – conformal double covering, Ω := { 0 , 1 } N , σ : Ω → Ω is the left shift. Igors Gorbovickis Critical points of the multiplier map

Roots of the multiplier maps c ( λ ) := φ − 1 M ( λ 2 ) c : C \ D → C \ M – conformal double covering, Ω := { 0 , 1 } N , σ : Ω → Ω is the left shift. For any λ ∈ C \ D , the map ψ λ : Ω → C is ◮ a homeomorphism between Ω and J c ( λ ) , conjugating σ to f c ( λ ) : ψ λ ◦ σ = f c ( λ ) ◦ ψ λ ; (1) ◮ ψ λ ( w ) depends analytically on λ ∈ C \ D ; Igors Gorbovickis Critical points of the multiplier map

Roots of the multiplier maps c ( λ ) := φ − 1 M ( λ 2 ) c : C \ D → C \ M – conformal double covering, Ω := { 0 , 1 } N , σ : Ω → Ω is the left shift. For any λ ∈ C \ D , the map ψ λ : Ω → C is ◮ a homeomorphism between Ω and J c ( λ ) , conjugating σ to f c ( λ ) : ψ λ ◦ σ = f c ( λ ) ◦ ψ λ ; (1) ◮ ψ λ ( w ) depends analytically on λ ∈ C \ D ; For λ ∈ C \ D , define 1 / k   k − 1 �  2 k ψ λ ( σ j w ) g k , w ( λ ) := .  j =0 Motivation: If w is k -periodic, then g k , w ( λ ) is the k -th degree root of the multiplier. Igors Gorbovickis Critical points of the multiplier map

Roots of the multiplier maps Ergodic Theorem: For a.e. w ∈ Ω, the sequence of maps { g k , w } k ∈ N converges to 2 λ on compact subsets of C \ D , as k → ∞ . Igors Gorbovickis Critical points of the multiplier map

Roots of the multiplier maps Ergodic Theorem: For a.e. w ∈ Ω, the sequence of maps { g k , w } k ∈ N converges to 2 λ on compact subsets of C \ D , as k → ∞ . Ω k ⊂ Ω – periodic itineraries of period k . For any w ∈ Ω k , define g w ( λ ) := g k , w ( λ ) – the k -th degree root of the multiplier. Theorem: For any ε, δ > 0 and a compact subset K ⊂ C \ D , there exists k 0 ∈ N , such that for any k ≥ k 0 , the following holds: # { w ∈ Ω k : � g w − 2 · id � K < δ } > 1 − ε. #Ω k Igors Gorbovickis Critical points of the multiplier map

Potentials outside of M 1 � � log | ˜ Recall: u k ( c ) := S k ( c , 0) | + log | C k ( c ) | . deg c S k According to Buff-Gauthier, for any c ∈ C \ M , 1 log | C k ( c ) | → log | λ ( c ) | , pointwise as k → ∞ . deg c S k Next � � 1 S k ( c , 0) | ∼ 1 1 d log | ˜ � � dc ([ g w ( λ )] k ) � k · log � = � � deg c S k 2 k � w ∈ Ω k 1 1 � � � � d λ � log k + ( k − 1) log | g w ( λ ) | + log | g ′ � � w ( λ ) | + log → � � 2 k k dc � � w ∈ Ω k → log | g w ( λ ) | (for “nice” w ) = log | 2 λ | = log | λ ( c ) | + log 2 . Igors Gorbovickis Critical points of the multiplier map