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Dynamics of Complex Maps in Parameter Space with McMullen Domains and Julia Sets

Explore the intricate behaviors of complex maps in the parameter space when the case n > 2. Discover McMullen domains, Mandelpinski necklaces, and Julia sets with symmetrical patterns and critical points. Learn about the Escape Trichotomy and Cantor sets, including Sierpinski curves, in different scenarios. From fundamental definitions to topological characterizations, delve into the captivating world of complex dynamics.

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Dynamics of Complex Maps in Parameter Space with McMullen Domains and Julia Sets

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  1. Dynamics of the family of complex maps C Y A (why the case n = 2 is ) R Z with: Mark Morabito Monica Moreno Rocha Kevin Pilgrim Elizabeth Russell Yakov Shapiro David Uminsky Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta

  2. The case n > 2 is great because:

  3. The case n > 2 is great because: There exists a McMullen domain around = 0 .... Parameter plane for n=3

  4. The case n > 2 is great because: There exists a McMullen domain around = 0 .... ... surrounded by infinitely many “Mandelpinski” necklaces...

  5. The case n > 2 is great because: There exists a McMullen domain around = 0 .... ... surrounded by infinitely many “Mandelpinski” necklaces... ... and the Julia sets behave nicely as

  6. The case n = 2 is crazy because: There is no McMullen domain....

  7. The case n = 2 is crazy because: There is no McMullen domain.... ... and no “Mandelpinski” necklaces...

  8. The case n = 2 is crazy because: There is no McMullen domain.... ... and no “Mandelpinski” necklaces... ... and the Julia sets “go crazy” as

  9. Some definitions: Julia set of J = boundary of {orbits that escape to } = closure {repelling periodic orbits} = {chaotic set} Fatou set = complement of J = predictable set

  10. Computation of J: Color points that escape to infinity shades of red orange yellow green blue violet Black points do not escape. J = boundary of the black region.

  11. Easy computations: is superattracting, so have immediate basin B mapped n-to-1 to itself. B

  12. Easy computations: is superattracting, so have immediate basin B mapped n-to-1 to itself. B T 0 is a pole, so have trap door T mapped n-to-1 to B.

  13. Easy computations: is superattracting, so have immediate basin B mapped n-to-1 to itself. B T 0 is a pole, so have trap door T mapped n-to-1 to B. The Julia set has 2n-fold symmetry.

  14. Easy computations: 2n free critical points

  15. Easy computations: 2n free critical points

  16. Easy computations: 2n free critical points Only 2 critical values

  17. Easy computations: 2n free critical points Only 2 critical values

  18. Easy computations: 2n free critical points Only 2 critical values

  19. Easy computations: 2n free critical points Only 2 critical values But really only 1 free critical orbit

  20. Easy computations: 2n free critical points Only 2 critical values But really only 1 free critical orbit And 2n prepoles

  21. The Escape Trichotomy (with D. Look & D Uminsky) There are three possible ways that the critical orbits can escape to infinity, and each yields a different type of Julia set.

  22. The Escape Trichotomy (with D. Look & D Uminsky) B is a Cantor set

  23. The Escape Trichotomy (with D. Look & D Uminsky) B is a Cantor set T is a Cantor set of simple closed curves (n > 2) (McMullen)

  24. The Escape Trichotomy (with D. Look & D Uminsky) B is a Cantor set T is a Cantor set of simple closed curves (n > 2) (McMullen) In all other cases is a connected set, and if T is a Sierpinski curve

  25. B Case 1: is a Cantor set parameter plane when n = 3

  26. B Case 1: is a Cantor set parameter plane when n = 3

  27. B Case 1: is a Cantor set parameter plane when n = 3 J is a Cantor set

  28. B Case 1: is a Cantor set parameter plane when n = 3 J is a Cantor set

  29. T Case 2: is a Cantor set of simple closed curves parameter plane when n = 3

  30. T Case 2: is a Cantor set of simple closed curves parameter plane when n = 3

  31. T Case 2: is a Cantor set of simple closed curves The central disk is the McMullen domain

  32. T Case 2: is a Cantor set of simple closed curves B T parameter plane when n = 3 J is a Cantor set of simple closed curves

  33. T Case 2: is a Cantor set of simple closed curves parameter plane when n = 3 J is a Cantor set of simple closed curves

  34. Case 3: T is a Sierpinski curve parameter plane when n = 3

  35. Case 3: T is a Sierpinski curve A Sierpinski curve is a planar set homeomorphic to the Sierpinski carpet fractal parameter plane when n = 3

  36. Sierpinski curves are important for two reasons: There is a “topological characterization” of the carpet 2. A Sierpinski curve is a “universal plane continuum”

  37. Topological Characterization Any planar set that is: 1. compact 2. connected 3. locally connected 4. nowhere dense 5. any two complementary domains are bounded by simple closed curves that are pairwise disjoint is a Sierpinski curve. The Sierpinski Carpet

  38. Universal Plane Continuum Any planar, one-dimensional, compact, connected set can be homeomorphically embedded in a Sierpinski curve. For example....

  39. This set

  40. This set can be embedded inside

  41. Case 3: T is a Sierpinski curve parameter plane when n = 3

  42. Case 3: T is a Sierpinski curve A Sierpinski “hole”

  43. Case 3: T is a Sierpinski curve A Sierpinski “hole” A Sierpinski curve

  44. Case 3: T is a Sierpinski curve A Sierpinski “hole” A Sierpinski curve Escape time 3

  45. Case 3: T is a Sierpinski curve Another Sierpinski “hole” A Sierpinski curve

  46. Case 3: T is a Sierpinski curve Another Sierpinski “hole” A Sierpinski curve Escape time 4

  47. Case 3: T is a Sierpinski curve Another Sierpinski “hole” A Sierpinski curve Escape time 7

  48. Case 3: T is a Sierpinski curve Another Sierpinski “hole” A Sierpinski curve Escape time 5

  49. is homeomorphic to So to show that

  50. Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s

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