A **periodic point** is a concept from dynamical systems and mathematical analysis. Specifically, a point \( x \) in a dynamical system is said to be periodic with period \( n \) if, when the system iteratively applies a function \( f \), the point eventually returns to its original position after \( n \) iterations.
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A point is periodic for an iterated map if for some positive integer . Its exact period is the least such , and the distinct points form its periodic orbit. For a holomorphic map its periodic-orbit multiplier is , computed in a local coordinate; at infinity use a reciprocal coordinate. The chain rule makes this the cyclic product of derivatives along the orbit, so it is the same at every point of that orbit.