Pugh's closing lemma

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Pugh's closing lemma is a key idea in dynamical systems. It says that if a point in a system keeps returning arbitrarily close to its starting position (it's nonwandering), you can adjust the rules of the system by a very small amount to make that point truly periodic (it repeats its path after a fixed number of steps). In other words, almost-repeating behavior can be turned into exact repeating behavior with a tiny tweak. This helps explain the connection between chaotic dynamics and regular, repeating motion. For example, if a system is constrained so that periodic orbits cannot occur, then it cannot exhibit the kind of chaotic behavior that relies on those orbits. The lemma underpins some results about convergence in autonomous systems.