On self-similarity in homogeneous quadratic transformations
Takeshi Yoshikawa · AIP conference proceedings · 2000
In this paper, we derive invariants for discriminating the existence of self-similar parts in the shape of a divergence-convergence boundary of two-dimensional real homogeneous quadratic transformations. A self-similar part in this context includes an infinite number of its own contracted images ranging closely. To explain the properties of this shape, we analyze the self-similarity in the portrait of the behavior of directions in the transformation process. For two-dimensional real homogeneous quadratic transformations, Da-te and Iri gave the invariant series in Ref. (4). We found an additional invariant to discriminate the existence of self-similar parts.