1 Computability Julia sets Mark Braverman Microsoft Research January 6, 2009

2 Julia Sets Focus on the quadratic family. Let c be a complex parameter. Consider the following discrete-time dynamical system on the complex plane: z ↦ z 2 + c The simplest non-trivial polynomial dynamical system on the complex plane.

3 The Julia set The filled Julia set K c is the set of initial z’s for which the orbit does not escape to ∞. The Julia set J c is the boundary of K c : J c = ∂ K c. J c is also the set of points around which the dynamics is unstable.

4 Example: c = i z ↦ z 2 + c

5 Iterating z ↦ z i z ↦ z 2 + c

6 Iterating z ↦ z i z ↦ z 2 + c

7 Iterating z ↦ z i z ↦ z 2 + c

8 Iterating z ↦ z i z ↦ z 2 + c

9 Iterating z ↦ z i z ↦ z 2 + c

10 Iterating z ↦ z i z ↦ z 2 + c

11 The Julia set The filled Julia set K c is the set of initial z’s for which the orbit does not escape to ∞. The Julia set J c is the boundary of K c : J c = ∂ K c. J c is also the set of points around which the dynamics is unstable.

12 Example: c ≈ i

13 Iterating z ↦ z i z ↦ z 2 + c

14 Iterating z ↦ z i “rotation” by an angle θ z ↦ z 2 + c

15 Computing K c and J c Given the parameter c as an input, compute K c and J c. The parameter c describes the rule of the dynamics – “its world”.

16 Computability model We use the “bit-computability” notion, which accounts for the cost of the operations on a Turing Machine. Dates back to [Turing36], [BanachMazur37]. TM c = − … …i JcJc

17 Input – giving c to TM The input c is given by an oracle φ(m). –On query m the oracle outputs a rational approximation of c within an error of 2 -m. TM is allowed to query c with any finite precision. TM c = − … …i

Input – giving c to TM The input c is given by an oracle φ(m). –On query m the oracle outputs a rational approximation of c within an error of 2 -m. TM is allowed to query c with any finite precision. TM c = − … …i φ(m) m=2 φ (2) = − i

Input – giving c to TM The input c is given by an oracle φ(m). –On query m the oracle outputs a rational approximation of c within an error of 2 -m. TM is allowed to query c with any finite precision. TM c = − … …i φ(m) m=10 φ(10) = − i

Input – giving c to TM The input c is given by an oracle φ(m). –On query m the oracle outputs a rational approximation of c within an error of 2 -m. TM is allowed to query c with any finite precision. TM c = − … …i φ(m) m=7 φ(7) = − i

21 Output Given a precision parameter n, TM needs to output a 2 -n -approximation of J c, which is a “picture” of the set. TM φ(m) JcJc

22 Output A 2 -n -approximation of J c, is made of pixels of size ≈ 2 -n. For each pixel, need to decide whether to paint it white or black. JcJc

23 JcJc Coloring a Pixel We use round pixels – equivalent up to a constant. A pixel is a circle of radius 2 -n with a rational center. No ??? Yes Put it in if it intersects J c. If twice the pixel does not intersect J c, leave it out. Otherwise, don’t care.

Summary TypeEmpirical and prior work New Hyperbolicempirically easy; some shown in poly-time poly-time computable Parabolicempirically computable (exp-time) poly-time computable Siegelempirically computable in many cases some are computable some are not Cremerno useful pictures to datecomputable Filled Julia set K c thought to be tightly linked to J c always computable ?

25 Theorem [BY07]: There is an algorithm A that computes a number c such that no machine with access to c can compute J c. In fact, computing J c is as hard as solving the Halting Problem. Under a reasonable conjecture from Complex Dynamics, c can be made poly-time computable. TM c JcJc A ?

26 Proving the non-computability theorem Consider the family z ↦ z 2 + e 2 π i θ z. When is there a Siegel disc? Theorem [Brjuno’65]: When the function converges.

27 Geometric Meaning of Φ ( θ ) r( θ ) is a measure on the size of the Siegel disc. r( θ ) can be computed from J c and vice versa. Theorem [Yoccoz’88], [Buff,Cheritat’03]: The function Φ ( θ ) + log r( θ ) is continuous. In particular, when Φ ( θ )=∞, r( θ )=0. Theorem [BY’07]: There is an explicit poly-time algorithm that generates a θ such that Φ ( θ ) is as hard to compute as the Halting Problem.

c such that J c has a Siegel disc JcJc ? A the rotation angle 2πθ of the Siegel disc r( θ ) – the “size” of the Siegel Disc Continuous: heavy machinery [Yoccoz’88], [Buff,Cheritat’03]. AlgebraicGeometric

Summary TypeEmpirical and prior work New Hyperbolicempirically easy; some shown in poly-time poly-time computable Parabolicempirically computable (exp-time) poly-time computable Siegelempirically computable in many cases some are computable some are not Cremerno useful pictures to datecomputable Filled Julia set K c thought to be tightly linked to J c always computable ?

30 The map of parameters c disconnected; poly-time connected; poly-time poly- time Non- computable

Thank You Computability of Julia Sets, M. Braverman and M. Yampolsky. Springer, 2008.