Normal family

A family of meromorphic functions on a domain is normal if every sequence in it has a subsequence converging locally uniformly in the spherical metric to a meromorphic function or infinity.

Montel's theorem: A theorem giving normality criteria for families of holomorphic or meromorphic functions, including local boundedness for holomorphic families. It provides a central criterion for recognizing normal families from local boundedness.
Fatou set: The set of points where the iterates of a rational map form a normal family in a neighborhood. Normality of iterates defines the region of stable dynamics.
Meromorphic function: A function holomorphic except at isolated poles on a complex domain. Normal families commonly consist of meromorphic functions, whose poles the spherical metric accommodates.
Uniform boundedness: A property that one common finite bound controls the values of every function in a family throughout a set. Meromorphic normality does not require bounded values when spherical convergence handles poles.
Arzelà–Ascoli theorem: A compactness theorem characterizing when families of functions have uniformly convergent subsequences on compact domains. Normality is its locally uniform, spherical-metric counterpart for meromorphic functions.
Julia set: The boundary between stable and unstable behavior under iteration of a rational map. Failure of normality for the iterates characterizes this chaotic boundary.
Holomorphic function: A complex function that is complex differentiable at every point of an open domain. Holomorphic families are the basic setting for classical normality criteria.
Pointwise convergence: Convergence of a sequence of functions separately at each point of its domain. Normality demands subsequential convergence uniformly on compact sets, not merely pointwise.
Spherical metric: A distance on the Riemann sphere that treats infinity as an ordinary point. It permits convergence across poles and to the constant function infinity.
Montel's theorem in complex dynamics: Applications of Montel's normality criterion to prove properties of iterated holomorphic maps. Omitting enough values can force normality and constrain an orbit.

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