## Graph Mandelbrot Problems

See the code page and the

Mathematica notebook.

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

## Statistical problems

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

## Quadratic graph problems

See the code page and the

Mathematica notebook.

- [One dimensional quadratic graph] The probability to have a connected graph on Z
_{p} goes to
zero for primes p to infinity.
- [Two dimensional quadratic graphs] The probability to have a connected graph on Z
_{p} goes to
one for primes p to infinity.
- [Three dimensional quadratic graphs] For all large enough primes, and three different quadratic maps,
the graph is connected.