TITLE: Lorenz attractor path-connected?
QUESTION [9 upvotes]: Can we tell if the Lorenz attractor is path-connected?  By the attractor I do not mean only the line weaving around, but rather its closure.

EDIT:  The answer below is unsatisfactory, and possibly incorrect. It does not account for the path-component of the main orbit being enlarged when the the closure is taken.  In the forking regions there will be a Cantor set times "T", with one leg of "T" butting into the main orbit.

REPLY [14 votes]: The question has been answered here: https://mathoverflow.net/a/297836/121665. It is connected but not path connected.
The situation is somewhat similar to the topologist's sine curve: the graph of
$$
f(x)=\sin\frac{1}{x},
\quad x\in (0,1]
$$
is path connected, but its closure (as a subset of $\mathbb{R}^2$) is connected, but not path connected.