Graph Mandelbrot Problems

See the code page and the Mathematica notebook.
  1. [Mandelbrot problem 1] Given necessary and sufficient conditions so that the graph Zn,T) with T(x) = ax+b is connected.
  2. [Mandelbrot problem 2] Are all graphs on Zn generated by T(x)=3x+1, S(x)=2x connected? We have checked this until n=200'000.
  3. [Mandelbrot problem 3] Find necessary and sufficient conditions that all graphs on Zn* generated by (x)=x^a S(x)=xb on n* are connected.
  4. [Mandelbrot problem 4] Which 2 x 2 matrices A in R=M(2,Zn) have the property that T(x)=x2 + A produces a connected graph?
  5. [Mandelbrot problem 5] Which of the 256x256 pairs of elementary cellular automata on Z2n produce connected graphs eventually for large enough n.

Statistical problems

  1. [Length-Cluster Problem I] On the probability space of all permutations T,S on Zn, the expectation E[ -mu(G)/log(nu(G) ] converges for n to infinity.
  2. [Length-Cluster Problem II] For 2 or more quadratic maps, the expectation E[ -mu(G)/log(nu(G) ] converges on Zp, if p runs along primes.

Quadratic graph problems

See the code page and the Mathematica notebook.
  1. [One dimensional quadratic graph] The probability to have a connected graph on Zp goes to zero for primes p to infinity.
  2. [Two dimensional quadratic graphs] The probability to have a connected graph on Zp goes to one for primes p to infinity.
  3. [Three dimensional quadratic graphs] For all large enough primes, and three different quadratic maps, the graph is connected.