On accessibility of hyperbolic components of the tricorn Hiroyuki - PowerPoint PPT Presentation

On accessibility of hyperbolic components of the tricorn Hiroyuki Inou (Joint work in progress with Sabyasachi Mukherjee) Department of Mathematics, Kyoto University Inperial College London Parameter Problems in Analytic Dynamics June 30, 2016 1 / 30

The tricorn family z 2 + c . ◮ f c ( z ) = ¯ c ( z ) = ( z 2 + ¯ c ) 2 + c : real-analytic 2-parameter family of ◮ f 2 biquadratic (quartic) polynomials. ◮ K c = { z ∈ C ; f n c ( z ) �→ ∞} : filled Julia set. ◮ J c = ∂ K c : Julia set. ◮ M ∗ = { c ∈ C ; K c : connected } : The tricorn . ◮ Periodic points: ◮ x : p -periodic point. def ◮ λ : multiplier of x ⇔ multiplier for f 2 c . ◮ k : odd ⇒ λ = ( ∂ f k z ( x ))( ∂ f k z ( x )) ≥ 0. c c ∂ ¯ ∂ ¯ 2 / 30

Hyperbolic components ◮ Hyperbolic component = (bounded) connected component of the hyperbolicity locus Hyp ∗ � int M ∗ . ◮ Remark: � = component of int M ∗ . ◮ e.g., period 1 and 2 hyperbolic components are contained in the same component of int M ∗ (Crowe et al.). ◮ H : hyperbolic component, p : period. ◮ c ∈ ∂ H ⇒ f c has an indifferent fixed point of f p c . ◮ p : odd ⇒ ∂ H consists of 3 parabolic arcs and 3 cusps . ◮ parabolic arc ⇔ ∃ simple 1-parabolic p -periodic point (1 attracting petal). ◮ cusp ⇔ ∃ double 1-parabolic p -periodic point (2 invariant attracting petals). 3 / 30

It is a “1.5-dim family”! ◮ The tricorn is connected (Nakane). ◮ ∃ Φ : C \ M ∗ → C \ D : real-analytic diffeomorphism. ◮ Therefore, we can define external rays (parameter rays) and do some combinatorics with them as in the case of the Mandelbrot set. ◮ Parameter rays are stretching rays. ◮ Even iterate is holomorphic � 1D phenomena. ◮ discrete parabolic maps, ◮ baby Mandelbrot sets. ◮ Odd iterate is anti-holomorphic � 2D phenomena. ◮ parabolic arcs, ◮ baby tricorn-like sets, ◮ wiggly features, ◮ discontinuous straightening maps (baby tricorn-like sets are not homeomorphic to the tricorn). 4 / 30

2D-phenomena: Wiggly features The existence of parabolic arcs (arcs consisting parabolic parameters) induces many wiggly features: ◮ Non-landing umbilical cords (Hubbard-Schleicher, I, I-Mukherjee). ◮ Non-landing parameter rays (I-Mukherjee). ◮ baby tricorn-like sets are NOT (dynamically) homeomorphic to the tricorn (I-Mukherjee). Conjecture No pair of baby tricorn-like sets are dynamically homeomorphic unless they are symmetric (in which case they are trivially affinely homeomorphic). 5 / 30

2D-phenomena: Wiggly features The existence of parabolic arcs (arcs consisting parabolic parameters) induces many wiggly features: ◮ Non-landing umbilical cords (Hubbard-Schleicher, I, I-Mukherjee). ◮ Non-landing parameter rays (I-Mukherjee). ◮ baby tricorn-like sets are NOT (dynamically) homeomorphic to the tricorn (I-Mukherjee). Conjecture No pair of baby tricorn-like sets are dynamically homeomorphic unless they are symmetric (in which case they are trivially affinely homeomorphic). 5 / 30

2D-phenomena: Wiggly features The existence of parabolic arcs (arcs consisting parabolic parameters) induces many wiggly features: ◮ Non-landing umbilical cords (Hubbard-Schleicher, I, I-Mukherjee). ◮ Non-landing parameter rays (I-Mukherjee). ◮ baby tricorn-like sets are NOT (dynamically) homeomorphic to the tricorn (I-Mukherjee). Conjecture No pair of baby tricorn-like sets are dynamically homeomorphic unless they are symmetric (in which case they are trivially affinely homeomorphic). 5 / 30

2D-phenomena: Wiggly features The existence of parabolic arcs (arcs consisting parabolic parameters) induces many wiggly features: ◮ Non-landing umbilical cords (Hubbard-Schleicher, I, I-Mukherjee). ◮ Non-landing parameter rays (I-Mukherjee). ◮ baby tricorn-like sets are NOT (dynamically) homeomorphic to the tricorn (I-Mukherjee). Conjecture No pair of baby tricorn-like sets are dynamically homeomorphic unless they are symmetric (in which case they are trivially affinely homeomorphic). 5 / 30

Accessible/inaccessible hyperbolic components We say a hyperbolic component H is accessible if there is a path γ : ( 0 , 1 ] → C \ M ∗ such that γ ( t ) converges to a point in ∂ H as t ց 0. Theorem 1 (I-Mukherjee) Any hyperbolic component of period 1 and 3 in M ∗ are accessible. ◮ Seems reasonable to conjecture that “most” hyperbolic components are inaccessible. ◮ An attempt to find infinitely many accessible hyperbolic component converging to the Chebyshev map f − 2 (I-Kawahira, in progress). 6 / 30

Accessible/inaccessible hyperbolic components 7 / 30

Fatou coordinates and Lavaurs maps ◮ H 0 : a hyperbolic component in M ∗ . ◮ C 0 ⊂ ∂ H 0 : parabolic arc of a hyperbolic component of odd period p . ◮ c ∈ C 0 . ◮ φ c , ∗ ( ∗ = attr , rep): normalized attracting/repelling Fatou coordinate, i.e., φ c , ∗ ( f c ( z )) = φ c , ∗ ( z ) + 1 2 . Re φ c , attr ( 0 ) = 0. z + 1 ◮ Remark: R is invariant by z �→ ¯ 2 , hence it follows that φ c , ∗ is unique up to real translation. ◮ Therefore, Im φ c , ∗ is well-defined ( Ecalle height ). ◮ Fact: C 0 is analytically parametrized by Im φ c , attr ( c ) (the critical Ecalle height ). 8 / 30

◮ T τ ( z ) = z + τ . ◮ g c ,τ = φ − 1 c , rep ◦ T τ ◦ φ c , attr : int K c → C : Lavaurs map with phase τ . ◮ We only consider the case τ ∈ R (reason explained later). ◮ Thus g c ,τ ◦ f c = f c ◦ g c ,τ . ◮ K c ,τ = K ( f c , g c ,τ ) = { z ∈ C ; ( f c , g c ,τ ) -orbit of z is bounded } : filled Julia-Lavaurs set. ◮ J c ,τ = J ( f c , g c ,τ ) = ∂ K ( f c , g c ,τ ) : Julia-Lavaurs set. 9 / 30

Julia-Lavaurs set 10 / 30

Julia-Lavaurs set 10 / 30

Geometric limits and Lavaurs maps ◮ Let c n �∈ H 0 → c 0 ∈ C 0 . ◮ φ c n : normalized Fatou coordinate s.t. φ c n → φ c , rep . ◮ Assume ∃ k n → ∞ s.t. φ c n ( f 2 pk n ( 0 )) → τ . c n ◮ Then we have f 2 pk n → g c ,τ ( n → ∞ ) . c n ◮ Notice: τ ∈ R ! ◮ φ c n − α n → φ c , attr for some α n ∈ R . ◮ Therefore, τ ← Im φ c n ( f 2 k n c n ( 0 )) = Im ( φ c n ( f 2 k n c n ( 0 )) − α n ) → Im φ c , attr ( 0 ) . ◮ Hubbard-Schleicher (implicitly proved): { c n } �→ ( c 0 , τ ) ∈ C 0 × R / Z is surjective. 11 / 30

Parameter space of Julia-Lavaurs sets Julia-Lavaurs family normalized tricorn family The vertical direction is a parametrization of C 0 , and the horizontal direction is (an approximation of) the phase. 12 / 30

Parameter space of Julia-Lavaurs sets 13 / 30

Parameter space of Julia-Lavaurs sets 14 / 30

Inaccessibility for the family of geometric limits ˜ H ⊂ C 0 × S 1 : “primitive” hyperbolic component. Lemma 2 1. The attractive basins are inaccessible from the escape region for ( c , τ ) ∈ ˜ H . 2. ˜ H is inaccessible from the escape locus. Remark ◮ Indeed, there is no path accumulating to the boundary. ◮ The first statement still holds for the parabolic basin of ( c , τ ) ∈ ∂ ˜ H . 15 / 30

Lemma 2 1. The attractive basins are inaccessible from the escape region for ( c , τ ) ∈ ˜ H . 2. ˜ H is inaccessible from the escape locus. Proof of 1. ◮ Assume f l c ◦ g m c ,τ has an attracting fixed point. ◮ a polynomial-like restriction h c ,τ := f l c ◦ g m c ,τ : U ′ c ,τ → U c ,τ exists. ◮ Let K n = { z ∈ int K c ; g k c ,τ ( z ) ∈ int K c ( k = 1 , . . . , n ) } . ◮ K n ⊃ h − n c ,τ ( U c ,τ ) is a neighborhood of K c ,τ . ◮ ∂ K n is contained in J c ,τ . ◮ Therefore, the attractive basin is inaccessible from C \ K n . ◮ Escape region for ( f c , g c ,τ ) = � ( C \ K n ) . 16 / 30

Lemma 2 1. The attractive basins are inaccessible from the escape region for ( c , τ ) ∈ ˜ H . 2. ˜ H is inaccessible from the escape locus. Proof of 2. ◮ Assume f l c ◦ g m c ,τ has an attracting fixed point. ◮ ∃U : nbd of ˜ H where polynomial-like restriction h c ,τ := f l c ◦ g m c ,τ : U ′ c ,τ → U c ,τ exists. ◮ Let A n = { ( c , τ ); g k c ,τ ( 0 ) ∈ int K c ,τ ( k = 1 , . . . , n ) } and ◮ C n = { ( c , τ ) ∈ U ; g k c ,τ ( 0 ) ∈ U c ,τ ( k = 1 , . . . , n ) } . ◮ C n is a neighborhood of C ( ˜ k C k ⊃ ˜ H ) = � H . ◮ ∂ A n is contained in the bifurcation locus. ◮ Therefore, ˜ H is inaccessible from ( C 0 × S 1 ) \ A n . ◮ Escape locus = � (( C 0 × S 1 ) \ A n ) . 17 / 30

Criterion for inaccessible hyperbolic components Let H ⊂ Hyp ∗ be a hyperbolic component of odd period. Lemma 3 Assume for any non-cusp c ∈ ∂ H the following holds: ◮ Let E 1 , . . . , E K be the connected components of K c ∩ Dom ( φ c , rep ) such that the parabolic periodic point is in ∂ E k . ◮ I k := int Im φ c , rep ( E k ) ⊂ R (Im φ c , rep : Ecalle height). ◮ { I k } K k = 1 is an open cover of R . Then ∂ H is inaccessible from the escape locus. 18 / 30

Criterion for inaccessible hyperbolic components Let H ⊂ Hyp ∗ be a hyperbolic component of odd period. Lemma 3 Assume for any non-cusp c ∈ ∂ H the following holds: ◮ Let E 1 , . . . , E K be the connected components of K c ∩ Dom ( φ c , rep ) such that the parabolic periodic point is in ∂ E k . ◮ I k := int Im φ c , rep ( E k ) ⊂ R (Im φ c , rep : Ecalle height). ◮ { I k } K k = 1 is an open cover of R . Then ∂ H is inaccessible from the escape locus. 18 / 30