The topological differences between the Mandelbrot set and the tricorn

Transcription

1 The topological differences between the Mandelbrot set and the tricorn Sabyasachi Mukherjee Jacobs University Bremen Warwick, November 2014

2 Basic definitions We consider the iteration of quadratic anti-polynomials f c = z 2 + c, c C.

3 Basic definitions We consider the iteration of quadratic anti-polynomials f c = z 2 + c, c C. Here are some of the dynamically interesting sets that we would be interested in:

4 Basic definitions We consider the iteration of quadratic anti-polynomials f c = z 2 + c, c C. Here are some of the dynamically interesting sets that we would be interested in: The set of all points which remain bounded under all iterations of f c is called the Filled-in Julia set K(f c ).

5 Basic definitions We consider the iteration of quadratic anti-polynomials f c = z 2 + c, c C. Here are some of the dynamically interesting sets that we would be interested in: The set of all points which remain bounded under all iterations of f c is called the Filled-in Julia set K(f c ). The boundary of the Filled-in Julia set is defined to be the Julia set J(f c )

6 Basic definitions We consider the iteration of quadratic anti-polynomials f c = z 2 + c, c C. Here are some of the dynamically interesting sets that we would be interested in: The set of all points which remain bounded under all iterations of f c is called the Filled-in Julia set K(f c ). The boundary of the Filled-in Julia set is defined to be the Julia set J(f c ) The set of all points whose orbits converge to is called the basin of infinity A (f c ).

7 This leads, as in the holomorphic case, to the notion of the connectedness locus of quadratic anti-polynomials:

8 This leads, as in the holomorphic case, to the notion of the connectedness locus of quadratic anti-polynomials: Definition The tricorn is defined as T = {c C : K(f c ) is connected}

9 This leads, as in the holomorphic case, to the notion of the connectedness locus of quadratic anti-polynomials: Definition The tricorn is defined as T = {c C : K(f c ) is connected} Since the second iterate of f c is holomorphic, the dynamics of anti-polynomials is similar to that of ordinary polynomials in many respects. Here are some subtle differences that occur at odd-periodic cycles:

10 This leads, as in the holomorphic case, to the notion of the connectedness locus of quadratic anti-polynomials: Definition The tricorn is defined as T = {c C : K(f c ) is connected} Since the second iterate of f c is holomorphic, the dynamics of anti-polynomials is similar to that of ordinary polynomials in many respects. Here are some subtle differences that occur at odd-periodic cycles: The first return map of an odd-periodic cycle is an orientation reversing map.

11 This leads, as in the holomorphic case, to the notion of the connectedness locus of quadratic anti-polynomials: Definition The tricorn is defined as T = {c C : K(f c ) is connected} Since the second iterate of f c is holomorphic, the dynamics of anti-polynomials is similar to that of ordinary polynomials in many respects. Here are some subtle differences that occur at odd-periodic cycles: The first return map of an odd-periodic cycle is an orientation reversing map. Every indifferent periodic point of odd period is parabolic.

12 This leads, as in the holomorphic case, to the notion of the connectedness locus of quadratic anti-polynomials: Definition The tricorn is defined as T = {c C : K(f c ) is connected} Since the second iterate of f c is holomorphic, the dynamics of anti-polynomials is similar to that of ordinary polynomials in many respects. Here are some subtle differences that occur at odd-periodic cycles: The first return map of an odd-periodic cycle is an orientation reversing map. Every indifferent periodic point of odd period is parabolic. There are, however, striking differences between the topological features of the tricorn and those of the Mandelbrot set.

13 The tricorn and the Mandelbrot set Left: The tricorn. Right: The Mandelbrot set.

14 Parameter rays Theorem (Nakane) The map Φ : C \ T C \ D, defined by c φ c (c) (where φ c is the Böttcher coordinate near ) is a real-analytic diffeomorphism.

15 Parameter rays Theorem (Nakane) The map Φ : C \ T C \ D, defined by c φ c (c) (where φ c is the Böttcher coordinate near ) is a real-analytic diffeomorphism. The previous theorem allows us to define parameter rays of the tricorn as pre-images of radial lines in C \ D under the map Φ.

16 Anti-holomorphic Fatou Coordinate, Equator and Ecalle height Lemma Suppose z 0 is a parabolic periodic point of odd period of f c with only one petal and U is an immediate basin of attraction. Then there exists a petal V U and a univalent map Φ: V C conjugating the first return map to z This map Φ is unique up to horizontal translation.

17 Anti-holomorphic Fatou Coordinate, Equator and Ecalle height Lemma Suppose z 0 is a parabolic periodic point of odd period of f c with only one petal and U is an immediate basin of attraction. Then there exists a petal V U and a univalent map Φ: V C conjugating the first return map to z This map Φ is unique up to horizontal translation. The anti-holomorphic iterate interchanges both ends of the Ecalle cylinder, so it must fix one horizontal line around this cylinder (the equator).

18 Anti-holomorphic Fatou Coordinate, Equator and Ecalle height Lemma Suppose z 0 is a parabolic periodic point of odd period of f c with only one petal and U is an immediate basin of attraction. Then there exists a petal V U and a univalent map Φ: V C conjugating the first return map to z This map Φ is unique up to horizontal translation. The anti-holomorphic iterate interchanges both ends of the Ecalle cylinder, so it must fix one horizontal line around this cylinder (the equator). The change of coordinate has been so chosen that the equator is the projection of the real axis.

19 Anti-holomorphic Fatou Coordinate, Equator and Ecalle height Lemma Suppose z 0 is a parabolic periodic point of odd period of f c with only one petal and U is an immediate basin of attraction. Then there exists a petal V U and a univalent map Φ: V C conjugating the first return map to z This map Φ is unique up to horizontal translation. The anti-holomorphic iterate interchanges both ends of the Ecalle cylinder, so it must fix one horizontal line around this cylinder (the equator). The change of coordinate has been so chosen that the equator is the projection of the real axis. We will call the vertical Fatou coordinate the Ecalle height. Its origin is the equator.

20 Changing the Ecalle height gives parabolic arcs Theorem Every parameter with a simple odd-periodic parabolic orbit lies in the interior of a real-analytic arc consisting of parabolic parameters with quasiconformally equivalent but conformally nonequivalent dynamics. These arcs are called parabolic arcs.

21 Changing the Ecalle height gives parabolic arcs Theorem Every parameter with a simple odd-periodic parabolic orbit lies in the interior of a real-analytic arc consisting of parabolic parameters with quasiconformally equivalent but conformally nonequivalent dynamics. These arcs are called parabolic arcs.

22 The eggbeater

23 Perturbation of parabolics and Ecalle heights

24 Perturbation of parabolics and Ecalle heights Cc in := Vc in /fc 2k and Cc out := Vc out /fc 2k are bi-infinite cylinders. The disjoin unions C in = Cc in and C out = Cc out are topologically trivial bundles with fibers isomorphic to C/Z.

25 Perturbation of parabolics and Ecalle heights Cc in := Vc in /fc 2k and Cc out := Vc out /fc 2k are bi-infinite cylinders. The disjoin unions C in = Cc in and C out = Cc out are topologically trivial bundles with fibers isomorphic to C/Z. Ecalle heights make sense in the incoming and outgoing cylinders even after perturbation.

26 Perturbation of parabolics and Ecalle heights Cc in := Vc in /fc 2k and Cc out := Vc out /fc 2k are bi-infinite cylinders. The disjoin unions C in = Cc in and C out = Cc out are topologically trivial bundles with fibers isomorphic to C/Z. Ecalle heights make sense in the incoming and outgoing cylinders even after perturbation. In the perturbed situation, the holomorphic first return map induces a conformal isomorphism (called the transit map ) between the incoming and outgoing cylinders and it preserves Ecalle heights.

27 Most odd-periodic rays wiggle Theorem (Inou, M) The accumulation set of every parameter ray accumulating on the boundary of a hyperbolic component of odd period (except period one) of M d contains an arc of positive length.

28 Most odd-periodic rays wiggle Theorem (Inou, M) The accumulation set of every parameter ray accumulating on the boundary of a hyperbolic component of odd period (except period one) of M d contains an arc of positive length.

29 Landing implies horizontal projection Lemma Suppose R t lands at a single parabolic parameter c on C. Then the dynamical ray R c t must project to a horizontal line under the repelling Fatou coordinate.

30 Landing implies horizontal projection Lemma Suppose R t lands at a single parabolic parameter c on C. Then the dynamical ray R c t must project to a horizontal line under the repelling Fatou coordinate.

31 Choose any smooth path γ : [0, δ] C with γ(0) = c.

32 Choose any smooth path γ : [0, δ] C with γ(0) = c.

33 Choose any smooth path γ : [0, δ] C with γ(0) = c. As s 0, the critical orbit takes longer to transit from the incoming Ecalle cylinder to the outgoing cylinder.

34 Choose any smooth path γ : [0, δ] C with γ(0) = c. As s 0, the critical orbit takes longer to transit from the incoming Ecalle cylinder to the outgoing cylinder. The projection of the curve spirals around the cylinder while the height converges to 0.

35 Choose any smooth path γ : [0, δ] C with γ(0) = c. As s 0, the critical orbit takes longer to transit from the incoming Ecalle cylinder to the outgoing cylinder. The projection of the curve spirals around the cylinder while the height converges to 0. This implies that the curve γ intersects the parameter ray infinitely often.

36 Symmetry of rational laminations Lemma Let c C and the dynamical ray R c t projects to a horizontal line under the repelling Fatou coordinate. Then the rational lamination RL (f c ) is invariant under the transformation s 2t s.

37 Symmetry of rational laminations Hence, ψ 1 c ι 1 ψ c = φ 1 c ι 2 φ c.

38 Symmetric lamination only for the period 1 component Lemma Let c H and the rational lamination RL (f c ) be invariant under the transformation s 2t s. Then the period of H must be 1.

39 Symmetric lamination only for the period 1 component Lemma Let c H and the rational lamination RL (f c ) be invariant under the transformation s 2t s. Then the period of H must be 1.

40 The invariance would force four different rays to land at a common point, which is not allowed by the combinatorics of unicritical anti-polynomials (at most three dynamical rays can land at a periodic point of odd period). Symmetric lamination only for the period 1 component Lemma Let c H and the rational lamination RL (f c ) be invariant under the transformation s 2t s. Then the period of H must be 1.

41 Centers are not equidistributed w.r.t. the harmonic measure Theorem (M) The centers of the hyperbolic components of the tricorn are not equidistributed w.r.t. the harmonic measure.

42 Centers are not equidistributed w.r.t. the harmonic measure Theorem (M) The centers of the hyperbolic components of the tricorn are not equidistributed w.r.t. the harmonic measure. Consider the period 1 parabolic arcs.

43 Centers are not equidistributed w.r.t. the harmonic measure Theorem (M) The centers of the hyperbolic components of the tricorn are not equidistributed w.r.t. the harmonic measure. Consider the period 1 parabolic arcs.

44 Centers are not equidistributed w.r.t. the harmonic measure Theorem (M) The centers of the hyperbolic components of the tricorn are not equidistributed w.r.t. the harmonic measure. Consider the period 1 parabolic arcs. The projection of the basin of infinity onto the repelling Ecalle cylinder contains a round cylinder.

45 The projection of the basin of infinity onto the repelling Ecalle cylinder contains a round cylinder. Transfer this round cylinder to the parameter plane (using parabolic perturbation techniques) to obtain an undecorated parabolic arc. Centers are not equidistributed w.r.t. the harmonic measure Theorem (M) The centers of the hyperbolic components of the tricorn are not equidistributed w.r.t. the harmonic measure. Consider the period 1 parabolic arcs.

46 Real-analyticity of HD Theorem (M, Urbanski) Let C be a parabolic arc of M d and let c : R C, h c(h) be its critical Ecalle height parametrization. Then the function is real-analytic. R h HD(J(f c(h) )) Proof. Complexify the parabolic arc to obatin an analytic family of q.c.-conjugate maps. Work with the radial Julia sets to associate an analytic family of conformal (infinite) IFS to the complexified family. Thermodynamic formalism (technical) + good control over the infinite IFS => R h HD(J r (f c(h) )) is real-analytic.

47 Thank you!