the automorphisms of the disk

Take the unit disk and ask: what are all the ways to map it conformally onto itself? That is, which holomorphic maps send the disk bijectively back to the disk, with a holomorphic inverse? Such a self-map is called an automorphism of the disk, and the wonderful surprise is that there are exactly enough of them to be interesting but few enough to write down completely. They form a group (compose two, get a third; each has an inverse), and that group has a clean, explicit description.

Every automorphism of the unit disk has the form f(z) = e^(i theta) * (z - a)/(1 - a-bar * z), where a is any point inside the disk (|a| < 1) and theta is any real angle. Read the two pieces: the factor (z - a)/(1 - a-bar * z), a Blaschke-type factor, is the basic move that slides the chosen interior point a to the center 0 while keeping the disk mapped onto itself; the factor e^(i theta) then rotates the disk about its center. So every disk symmetry is 'move some point to the center, then spin'. You can check (1 - a-bar * z) never vanishes inside the disk, and that |f(z)| = 1 exactly when |z| = 1, so the boundary circle goes to the boundary circle and the interior to the interior.

These maps are the symmetries of the disk in the deepest sense: they are exactly the rigid motions of the disk's natural hyperbolic geometry, the distance-preserving maps of the Poincare metric. Counting parameters, there are three real degrees of freedom (two for the point a, one for the angle theta) — the same count as Mobius maps, which is no accident, since disk automorphisms are precisely the Mobius transformations that happen to preserve the disk. A common misconception worth dispelling: the only automorphisms that fix the center 0 are the plain rotations f(z) = e^(i theta) z (a fact that drops straight out of the Schwarz lemma) — every other automorphism necessarily moves the center somewhere else.