Properties of Möbius Maps

Complex Analysis · Axiom Academy

LESSON Properties of Möbius Maps Every Möbius map has at most two fixed points — and those two points decide everything the map does. A fixed point is a z_0 the map leaves alone: f(z_0)=z_0 . Setting and clearing the denominator turns the search into one quadratic. The fixed-point equation (when ) A quadratic has at most two roots, so a Möbius map has at most two fixed points — unless it is the identity, which fixes everything. Below, the point joins the plane as one more candidate (it is fixed exactly when c=0 ). Conjugate the map so its two fixed points sit at 0 and . Then f becomes a multiplication , and the single number — the multiplier — names the behavior. Each panel below shows the orbit of one starting point. . Points circle the fixed points; nothing is attracted or repelled. real, . Points stream along arcs from the repelling fixed point to the attracting one. and . Rotation and flow at once: orbits spiral out of one anchor into the other. , the two anchors merged into one. Conjugate to : every orbit drifts past the single fixed point. rotates (elliptic); f(z)=2z dilates outward (hyperbolic); spirals (loxodromic); f(z)=z+1 translates and fixes only (parabolic). 3. Reading the Type Off the Trace You never have to plot an orbit. Normalize the matrix so ad-bc=1 , then look at the single number , the trace . Where lands in the complex plane is the type.

This is the written version of the interactive lesson above. See the full Complex Analysis course.