The famous recursion relation $$z_{n+1} := z_n^2 + c$$
defining the Mandelbrot set can be viewed as a 2D Poincaré map of larger dimensional dynamical system, withe initial value $z_0=c$ of the recurrence determined by the initial conditions of the said dynamical system.
- Does such a dynamical system exist?
- Can it be phase-volume preserving (Hamiltonian)?
- How to construct an explicit example of the corresponding flow?
My motivation is to put the Mandelbrot recursion in the context of an introductory course on analytical mechanics where we discuss fractal boundaries of phase space regions with chaotic dynamics.
Best Answer
Yes you can construct such flow.
So you have your function $f_c : \hat{\mathbb{C}} \to \hat{\mathbb{C}}, z \mapsto c + z^2$ ($\hat{\mathbb{C}}$ is the complex sphere, that is the complexe space plus a point at infinity).
Now take any fonction $h: \hat{\mathbb{C}} \to \mathbb{R_{>0}}$, and define the following space $\hat{\mathbb{C}} \times \mathbb{R}/(x,h(t))\sim(f_c(x),0)$, then you have a natural flow $$ T_t:(x,s)=\left \{ \begin{matrix} (x,t+s) \text{ if }t+s\leq h(x) \\ (f_c(x),t+s-h(x)) \text{ otherwise} \end{matrix} \right . $$ Then your function is the return map to the subset $\hat{\mathbb{C}} \times \{0\}$. As this is happening in a compact set there is at least one invariant measure, but I am not sure it will be continuous with respect to Lesbegue measure (I suspect ergodic measure to be Dirac times a line or fractal,...). So for your second question it is mostly no.
If things are not clear you can look at Wikipedia ( https://en.wikipedia.org/wiki/Suspension_(dynamical_systems) ) or this master thesis https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=9389&context=etd