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 Incoming links: Postcritical set

A critical orbit is the forward sequence

p, R (p), R^{2} (p), \dots

starting at a critical point of a rational map. The postcritical set records the forward images, usually from the first iterate onward, of all critical points; a single critical orbit and the union of all such images are distinct notions. For quadratic polynomials there is one finite critical point, zero, and its orbit controls connectedness of the filled Julia set and membership in the Mandelbrot set. A bounded orbit can be preperiodic to a repelling cycle, as

0 \mapsto i \mapsto - 1 + i \mapsto - i \mapsto - 1 + i

for

z^{2} + i

; boundedness then does not imply Fatou membership.

An immediate attracting or superattracting basin contains a critical point whose forward orbit converges to the attracting cycle.
An immediate parabolic basin contains a critical point whose forward orbit converges to the parabolic cycle along an attracting direction.
A Siegel disc is conformally conjugate to an irrational rotation of a disc. It contains no critical point, and its boundary is contained in the closure of the postcritical set.
A Herman ring is conformally conjugate to an irrational rotation of an annulus. It contains no critical point, and both boundary components are contained in the closure of the postcritical set.