A given point has current wealth as it's level and moves randomly up and down in that level according to a process that's random.
https://en.wikipedia.org/wiki/Dynamical_system https://en.wikipedia.org/wiki/Random_dynamical_system
I suppose you can make it fit the definition for the random dynamical system, though in that case the underlying dynamical system is ergodic.
This's got nothing to do with ergodic theory though, but feel free to use that word in whatever weird distorted way you like, I literally can't seem to stop you.
1. A state space X
2. An evolution map f: X -> X
3. A measure \mu on X, preserved by f (such that f_*\mu = \mu) and where \mu(X) = 1.
To represent a series of i.i.d. random variables you can typically just represent them as a sequence of values with a joint probability distribution as the invariant measure.
If you've got a space X of such sequence you can also add the current value to it to get IR x X, with a map f: (x,t) -> x * 1.5 if t[0] is heads & x * 0.6 otherwise. But you'll have to tell me it's invariant measure because I can't come up with one.
The process of simply flipping coins, even a biased coin, does correspond to a dynamic system where the space is all infinite sequences of heads and tails, the invariant measure is the joint probability, and the operator is the left-shift (i.e. if you sample 1 series of coin flips and throw away the first result you still end up with a sample of the same distribution).
But this doesn't translate at all to their proposed scenario of starting everyone at 1 and letting the results diverge from there.