Epistemic Status: Almost certainly wrong, but fun to think about.
A fixed point theorem says that, as long as certain conditions are satisfied, a function that has the same domain and range will have at least one point that gets mapped to itself. The best example of this is Brouwer's fixed point theorem, which proves the existence of fixed points for continuous functions on a convex and compact set. There are other fixed point theorems that apply in other cases.
These would be mildy interesting factoids if it weren't possible to represent an enormous number of common tasks in life as functions on a set. In fact, thinking itself could be represented as a function. Specifically, you could represent a thought as a function that maps one point in mind-space to another (nearby) point in mind-space.
If your mind when you wake up is at one point, then when you think about breakfast your mind is now at a different point.
In that case, we can ask if there is a fixed point in such a circumstance. If there is, we can ask what that fixed point might be.
I certainly don't know enough about neuroscience yet to figure out what properties the set of minds has, or what properties the function of thought has. But I'm more interested in the second question anyway: assuming a fixed point of mind exists, what is it?
A fixed point in mind-state is a point where, once you reach it, the act of thinking doesn't take you away from it. Since thinking is a function implemented by the mind, a fixed point in mind-space endorses its own existence.
One of the interesting fixed points that may exist for mind-space is probably enlightment. In fact, meditation as a search for enlightenment seems to be a search function implemented on the mind. You start with your mind as it is, and then successively apply the meditation function until you get to the fixed point.
In that case, you could ask if such a search always succeeds. It seems clear the the answer is no. In fact, people with certain mental or emotional disorders are often advised not to start meditating. You probably want to search for the fixed point of meditation only when you're within a topological basin of attraction for it. So it may be worth e.g. getting therapy to put yourself into the basin of attraction for enlightenment before beginning meditation.
Furthermore, doing some kind of iterated search through mind-space isn't guaranteed to ever converge. I know I'll often cycle on some subject "I should call so-and-so. But what if she's mad about the thing I said last week? I wonder if she is. I should call her." And then those thoughts go around a bunch more times. In this case, the thought-function doesn't converge. It seems likely that there are many cycles of this type, perhaps much longer than can be readily noticed by introspection.
This is why just closing your eyes and letting your mind drift is insufficient as meditation. Proper meditation must be a though-function that, for a large set of mind-states, does converge to some fixed point.
It also becomes clear that closing your eyes, and in general just avoiding distractions, is also important for seeking a fixed point in mind-space. The more inputs you have, the more complex a search function would need to be. This implies that enlightenment (if it is a fixed point) may actually be more of a moving target. As you interact with the world and learn things, your mind-state will necessarily change. Perhaps it changes in a way that's easy to adjust to a new fixed-point, and perhaps not.
Finally, it seems likely that fixed points in mind-space aren't necessarily good. Wireheading, for instance, seems like it could be represented as a fixed point. Just because a point in mind-space is stable doesn't mean it satisfies your goals right now.