Trace is something like a list of execution steps of a program, a list of what happens at each step, for all steps. When a program runs indefinitely, it'll be a potentially infinite list (or actually infinite if we know the program won't terminate). Finite dynamical system is something like a program (together with its current state) that is itself finite, and allows to compute data of the same kind (e.g. program + state) for the next step: this transition from the current step to the next step is the dynamic. Infinity appears in this process when we consider all the (future) steps, not just one, even though one step is enough to determine them all. Objectification as I used it is a concept from mathematics, when you are trying to capture some phenomenon as a certain kind of single mathematical object (as opposed to a thing with whistles, processes and hand-waving).