On accessibility of hyperbolic components of the tricorn

1 / 30 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

The tricorn family 2 / 30 f c (z) = z 2 + c. fc 2 (z) = (z 2 + c) 2 + c: real-analytic 2-parameter family of biquadratic (quartic) polynomials. K c = {z C; fc n (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 fc 2. f k: odd λ = ( k c z (x))( f c k z (x)) 0.

Hyperbolic components 3 / 30 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 fc p. 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).

It is a 1.5-dim family! 4 / 30 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).

2D-phenomena: Wiggly features 5 / 30 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).

6 / 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 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).

Accessible/inaccessible hyperbolic components 7 / 30

8 / 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. Remark: R is invariant by z z + 1 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).

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

Geometric limits and Lavaurs maps 11 / 30 Let c n H 0 c 0 C 0. φ cn : normalized Fatou coordinate s.t. φ cn φ c,rep. Assume k n s.t. φ cn (fc 2pkn n (0)) τ. Then we have fc 2pkn n g c,τ (n ). Notice: τ R! φcn α n φ c,attr for some α n R. Therefore, τ Im φ cn (f 2kn c n (0)) = Im(φ cn (f 2kn 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.

Parameter space of Julia-Lavaurs sets 12 / 30 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.

Parameter space of Julia-Lavaurs sets 13 / 30

Parameter space of Julia-Lavaurs sets 14 / 30

Inaccessibility for the family of geometric limits 15 / 30 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.

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 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 ).

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( H) = k C k 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

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 {I k } K k=1 is an open cover of R. (Im φ c,rep : Ecalle height). Then H is inaccessible from the escape locus.

19 / 30 The bifurcation locus near a parabolic arc looks like the blow-up of the Julia set at the parabolic periodic point w.r.t. the Ecalle height. Since any parabolic-attracting (virtually attracting) c C (or cusp) lies in the interior of M (in the common boundary arc with another hyperbolic components of double period), we need only check the assumption for c in the compact subarc of C consisting of non-parabolic-attracting parameters.

Perturbations Assume Hn Hyp satisfy Hn H geometrically. 20 / 30

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Inaccessible hyperbolic components for M 24 / 30 Theorem 4 (I-Mukherjee, in progress) H 0 Hyp : real hyperbolic component of odd period p > 1. C H 0 : the root arc (i.e., the umbilical cord converges to it). H C S 1 : real hyperbolic component in the space of geometric limits. H n Hyp converges to H geometrically. Then for sufficiently large n, the root arc of H n is inaccessible from the escape locus.

Outline of proof 25 / 30 For any c n H n, there are three connected components E = E (c n ) ( = ±, 0) of K c Dom(φ c,rep ) in Lemma 3. Let I = I (c n ) = int Im φ c,rep (E ). I + is unbounded above, I0 is bounded, I is unbounded below. Consider a geometric limit of a sequence H n c n (c 0, τ). (f c0, g c0,τ ) has an inaccessible parabolic basin. Therefore, int Im φ c0,τ,rep(k c0,τ ) = R.

Outline of proof (continued) 26 / 30 Moreover, by assumption that H 0 and H is real, E + and E must touch in the limit, so we have lim inf I + lim sup I. n n I = I + and I 0 = I 0 by symmetry w.r.t. z z + 1 2, so lim inf I + 0 < lim sup I 0, n n lim inf I 0 < 0 lim sup I. n n Hence for sufficiently large n, {I } =±,0 is an open covering of R, and we can apply Lemma 3.

Remarks The assumptions that I the period p > 1, I hyperbolic components are real, and I the parabolic arc is the root arc seems unnecessary, but we do not know how to exclude degenarate case: On root arcs: lim inf I+ = lim sup I0, lim inf I0 = lim sup I. On co-root arcs: lim inf I+ = lim sup I. (E0 = I0 = in this case.) 27 / 30

On root arc 28 / 30

On co-root arcs 29 / 30

On co-root arcs 30 / 30