If you are consistently right more than 50% of the time in the stock market, and you have an infinite lifetime, then (assuming a plausible nature of the stock market), your wealth from this will be unbounded.
This seems like it is a special case of some result within the theory of dynamical systems. So, assuming I am right about that, my question is: what canonical result in the theory of dynamical system is that stock market speculation result a special case of? Is this one of those situations involving an ‘attractor’ or a ‘basin’?
I would like to mention that this is actually not strictly true. The problem lies with the understanding of probability and wealth. If you are right more than 50% of the time, which presumably means that you gain money more than 50% of the time, you aren't guaranteed to get infinite wealth provided you started with a finite amount of wealth. This is because you are missing two key points:
Consider the possibility of buying 1000 dollars worth of stock with your last thousand dollars, then the company goes out of business - your stock is worth 0.
The main issue is that probability doesn't work like you think it does. Here's an example. Say we bought stocks 10 times. Then there are multiple ways in which I can have a 50 percent success rate. For instance, I might lose 5 times in a row, then gain 5 times. I might gain 5 times then lose 5 times. I might have some mixture. Furthermore, probability doesn't work like this anyway! With only 10 trades I might just lose on all of them and run out of money. Probabilities only work when the numbers are "large", so you'd have to be able to make a lot of trades before your 50 percent average works, and you'd have to make those trades without running out of money.
Furthermore, I don't accept your claim that your wealth will be unbounded even with infinite time. This is where the magnitude of the loss matters. You might, for instance, always gain 50 dollars and then always lose it with your next stock purchase. Similarly, you might be right 50 percent of the time in that you make a positive return but, when you're wrong about getting a positive return you lose much more.
To answer the original question, though, all you're looking for is an "unstable" dynamical system. The simplest one that (vaguely) captures what you want is this:
Presume that, on average, I will have a 2 percent return on my money for every transaction. This is an aggregate amount. That is, I might lose money on some trades, but on others I make back the loss and then some. When that's happening I can treat my trades as always giving me 2 percent. Let $M_{k+1}$ be the amount of money I have when I complete trade $k+1$, and assume I go all or nothing - I put all my money into my next trade. The modification for when that isn't true isn't really important or difficult here. Then:
$M_{k+1} = 1.02M_k$
is the dynamical system. However, your original premise needs some work!