Consider a dynamical system and let $v(t)=(v^1(t),\dots,v^n(t))$ be an orbit.
Suppose we are able to prove, for some $k=1,\dots,n$:
$$\lim_{t\to\infty} v^k(t)=0$$
Of course, we cannot be sure that $\exists T\ v^k(T)=0$. In any arbitrary large, but finite time, the limit point can only be approximated. Intuitively, it takes $\omega$ steps to reach the limit point. Then, we can imagine of "placing the system in its limit point" and study its behavior iterating from there. In a sense, study the $\omega+\omega$-limit of the system. My first question is:
- Is there something as the "transfinite limit" of a dynamical system? Can we formalize the previous idea, maybe using nets associated to the orbits of the system?
I am asking this because in (evolutionary) game theory an interesting result hold, in words: considering a symmetric normal form game, we can associate to it a dynamical system describing how it would be played by a population of players interacting by imitation. This is a replicator dynamics, and the orbits of the system describe the evolution of of the frequency with which an action is played by the population.
Then, one can prove that in the limit, the frequency with which a dominated action is played tend to $0$. An interesting set of actions is the set of rationalizable or equivalently iteratively undominated actions. Is there a "limit" in which the population plays only rationalizable actions? If the previous idea can be formalized, I would say that the answer is positive: in the first $\omega$ steps dominated actions are played with frequency $0$, hence eliminated. Then, $\omega+\omega$ steps the dominated actions of the resulting subgames are played with frequency $0$ and hence deleted, and so on until in some $\omega+\dots+\omega$ steps only rationalizable actions are played with positive probability.
My second curiosity is the:
- If the "transfinite limit" of a dynamical system is studied, are there general results linking the behavior at various ordinal levels?
Since everything in the dynamical system that ever happens, happens already in finite time, there is no point in looking at anything transfinite. Luckily, this is enough here.
The relative frequency of pure strategies that do not survive iterated elimination of strictly dominated strategies goes to zero under the replicator dynamics starting from any interior point. This follows from Theorem 1 in [Hofbauer, Josef, and Jörgen W. Weibull. "Evolutionary selection against dominated strategies." Journal of Economic Theory 71.2 (1996): 558-573.]