The simplest double pendulum are just two rods attached to each other. The main rod may or may not flip depending on the initial conditions.
If it flips, the path that the end of pendulum passes covers the space between two concentric circles. Is it true that all points are covered if we let pendulum spinning to infinity?
If it does not ever flip, there is the max height the path will reach on both sides, but apart from that it is still a space between two concentric circles. The same question in this case, all points that are bellow this height and between the two circles will be visited?
The simulations are strongly confirming this, making the path if we take it as a fractal having dimension $2$.
Any proof of this in the literature?
If you take it like a probability of the end of pendulum being somewhere like in this post
I need only the entire space of probability being different from $0$. I am somewhat sure it has a higher dimension than $1$, but not certain if it has dimension $2$ most of the time, all of the time, sometimes or never.
Regarding the dynamics and complexity it would be very natural to assume that, at least, the end of pendulum reaches almost every point meaning the number of points it does not reach has measure $0$.
But it is just me. (I am not sure if this question is that difficult that I cannot find even a hint about it. It is still just a question about fractal dimension.)
This is an illustration of day or two the simulation running:
