Fractal sets attached to homogeneous quadratic maps in two variables

Abstract

The fractal set attached to the iteration of a “generic” homogeneous quadratic map from the plane to itself is studied and depicted, by using a two-parametric family of normal forms obtained from the theory of invariants of symmetric bilinear maps F:R×R→R under the full linear group of the plane. While invariant theory classifies maps F:C×C→C on the complex plane, we confine ourselves to consider maps on the real plane, in order to include the results obtained from the theory of two-dimensional discrete dynamical systems. A discrete number of “topological types” for such fractals is conjectured to exist.