Dynamics of the family of complex maps

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