Complex Branching of Inverse Monomial Mappings from Lissajous Figures
This Demonstration shows the branches of inverted monomial mappings from various Lissajous figures parametrized by . The symmetry is a direct consequence of the fundamental theorem of algebra: each point on the Lissajous figure maps via the inverse of to precisely radially symmetric points given by the roots of . Since each branch preserves continuity, the Lissajous figure in its entirety is mapped to distinct radially symmetric branches. Each colored curve is the image of the Lissajous figure under a different branch of the inverse of .
For , two such branches exist: and . Some of the curves are closed, while others are not; this is not a property of the inverse mapping, but of the Lissajous figures. The inverse monomial mapping preserves compactness on each branch.