Fractal sets attached to homogeneous quadratic maps in two variables

• Published in: Physica. D Vol. 245; no. 1; pp. 8 - 18
• Main Authors: Durán Díaz, R., Hernández Encinas, L., Muñoz Masqué, J.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15-02-2013
• Subjects: Bilinear symmetric map; Discrete quadratic dynamic systems; Fractal set; Invariant functions; Fractal set; Bilinear symmetric map; Discrete quadratic dynamic systems; Invariant functions
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2012.11.002

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.
► Fractal sets attached to “generic” homogeneous quadratic maps are studied. ► Using invariant theory, two invariants give rise to only six “normal forms”. ► Normal forms partition the plane of invariants into eleven connected regions. ► Comparison of “topological types” for pairs of fractals, per region. ► Conjecture: only a finite and small number of distinct topological types exist.