interesting patterns about the time-evolution of that system can be found in each of those state spaces—has tremendous implications. Each of these patterns, of course, represents a constraint on the behavior of the system in question; if some system’s state is evolving in a way that is described by some pattern, then (by definition) its future states are constrained by that pattern. As long as the pattern continues to describe the time-evolution of the system, then states that it can transition into are limited by the presence of the constraints that constitute the pattern. To put the point another way: patterns in the time-evolution of systems just are constraints on the system’s evolution over time.
It’s worth emphasizing that all these constraints can (and to some degree must) apply to all the state spaces in which a particular system can be represented. After all, the choice of a state space in which to represent a system is just a choice of how to describe that system, and so to notice that a system’s behavior is constrained in one space is just to notice that the system’s behavior is constrained period. Of course, it’s not always the case that the introduction of a new constraint at a particular level will result in a new relevant constraint in every other space in which the system can be described. For a basic example, visualize the following scenario.
Suppose we have three parallel Euclidean planes stacked on top of one another, with a rigid rod passing through the three planes perpendicularly (think of three sheets of printer paper stacked, with a pencil poking through the middle of them). If we move the rod along the axis that’s parallel to the planes, we can think of this as representing a toy multi-level system: the rod represents the system’s state; the planes represent the different state-spaces we could use to describe the system’s position (i.e. by specifying its location along each plane). Of course, if the paper is intact, we’d rip the sheets as we dragged the pencil around. Suppose, then, that the rod