the fixed points of a Mobius transformation
When a map shuffles the plane around, the points it leaves exactly where they are tell you almost everything about how the map behaves. A fixed point of a transformation is a point z with f(z) = z — the map holds it still. For Mobius transformations these anchor points are easy to find and remarkably informative: locate them, and you can read off the entire dynamical character of the map at a glance.
To find the fixed points of w = (a z + b)/(c z + d), set the output equal to the input: (a z + b)/(c z + d) = z. Clearing the denominator gives c z^2 + (d - a) z - b = 0, a quadratic. A quadratic has at most two roots, so a Mobius map that is not the identity has at most two fixed points (when c = 0, infinity is automatically fixed, and you solve a linear equation for the other). This is a startling rigidity result: if a Mobius map fixes three distinct points, it must be the identity map that fixes everything. Three is one too many for any nontrivial map.
The number and type of fixed points classify the map. Two distinct fixed points: the map is loxodromic, hyperbolic, or elliptic, depending on a number called the multiplier — points flow from one fixed point (a repeller) toward the other (an attractor), possibly spiraling. One repeated fixed point: the map is parabolic, a kind of shear that slides points along circles tangent at that single point. This 'fixed-point picture' is the fastest way to understand and even sketch the action of a Mobius map, and it underlies how the map carries the geometry of the disk or the half-plane. One caution: always work on the extended plane — infinity counts as a possible fixed point, and forgetting it makes you miscount.
The map w = (z - 1)/(z + 1) has fixed points where (z - 1)/(z + 1) = z, i.e. z^2 + z = z - 1, giving z^2 = -1, so z = i and z = -i. Two distinct fixed points (here both finite) — so this map flows points from one of them toward the other.
Solving f(z) = z reduces to a quadratic, so a non-identity Mobius map has at most two fixed points.
Having at most two fixed points is exactly why three points determine a Mobius map: if two maps agreed on three points, their composite (one followed by the other's inverse) would fix three points and so be the identity, forcing the maps to be equal.