Exponential Dynamics and (Crazy) Topology

1. Exponential Dynamics and (Crazy) Topology Cantor bouquets Indecomposable continua

2. Exponential Dynamics and (Crazy) Topology Cantor bouquets Indecomposable continua These are Julia sets of

3. Example 1: Cantor Bouquets with Clara Bodelon Michael Hayes Gareth Roberts Ranjit Bhattacharjee Lee DeVille Monica Moreno Rocha Kreso Josic Alex Frumosu Eileen Lee

4. Orbit of z: Question: What is the fate of orbits?

5. Julia set of * J = closure of {orbits that escape to } = closure {repelling periodic orbits} = {chaotic set} Fatou set = complement of J = predictable set * not the boundary of {orbits that escape to }

6. For polynomials, it was the orbit of the critical points that determined everything. But has no critical points.

7. For polynomials, it was the orbit of the critical points that determined everything. But has no critical points. But 0 is an asymptotic value; any far left half-plane is wrapped infinitely often around 0, just like a critical value. So the orbit of 0 for the exponential plays a similar role as in the quadratic family (only what happens to the Julia sets is very different in this case).

8. Example 1: is a “Cantor bouquet”

9. Example 1: is a “Cantor bouquet”

10. Example 1: is a “Cantor bouquet” attracting fixed point q

11. Example 1: is a “Cantor bouquet” The orbit of 0 always tends this attracting fixed point attracting fixed point q

12. Example 1: is a “Cantor bouquet” q p repelling fixed point

13. Example 1: is a “Cantor bouquet” q x0 p

14. Example 1: is a “Cantor bouquet” And for all q x0 p

15. So where is J?

16. So where is J?

17. So where is J? in this half plane

18. So where is J? Green points lie in the Fatou set

19. So where is J? Green points lie in the Fatou set

20. So where is J? Green points lie in the Fatou set

21. So where is J? Green points lie in the Fatou set

22. So where is J? Green points lie in the Fatou set

23. The Julia set is a collection of curves (hairs) in the right half plane, each with an endpoint and a stem. hairs endpoints stems

24. A “Cantor bouquet” q p

25. Colored points escape to and so now are in the Julia set. q p

30. One such hair lies on the real axis. repelling fixed point stem

31. Orbits of points on the stems all tend to . hairs

32. So bounded orbits lie in the set of endpoints. hairs

33. So bounded orbits lie in the set of endpoints. Repelling cycles lie in the set of endpoints. hairs

34. So bounded orbits lie in the set of endpoints. Repelling cycles lie in the set of endpoints. hairs So the endpoints are dense in the bouquet.

35. So bounded orbits lie in the set of endpoints. Repelling cycles lie in the set of endpoints. hairs So the endpoints are dense in the bouquet.

36. S Some Facts:

37. S Some Facts: The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stems

38. S Some Crazy Facts: The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stems The set of endpoints together with the point at infinity is connected ...

39. S Some Crazy Facts: The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stems The set of endpoints together with the point at infinity is connected ... but the set of endpoints is totally disconnected (Mayer)

40. S Some Crazy Facts: The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stems The set of endpoints together with the point at infinity is connected ... but the set of endpoints is totally disconnected (Mayer) Hausdorff dimension of {stems} = 1...

41. S Some Crazy Facts: The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stems The set of endpoints together with the point at infinity is connected ... but the set of endpoints is totally disconnected (Mayer) Hausdorff dimension of {stems} = 1... but the Hausdorff dimension of {endpoints} = 2! (Karpinska)

42. Another example: Looks a little different, but still a pair of Cantor bouquets

43. Another example: The interval [-, ] is contracted inside itself, and all these orbits tend to 0 (so are in the Fatou set)

44. Another example: The real line is contracted onto the interval , and all these orbits tend to 0 (so are in the Fatou set)

45. Another example: -/2 /2 The vertical lines x = n + /2 are mapped to either [, ∞) or (-∞, - ], so these lines are in the Fatou set....

46. Another example: c -c The lines y = c are both wrapped around an ellipse with foci at

47. Another example: c -c The lines y = c are both wrapped around an ellipse with foci at , and all orbits in the ellipse tend to 0 if c is small enough

48. Another example: c -c So all points in the ellipse lie in the Fatou set

49. Another example: c -c So do all points in the strip

50. Another example: c -c The vertical lines given by x = n + /2 are also in the Fatou set.