Geometric Characterizations of Symmetric Maps in the Complex Plane

In this paper, we establish several characterizations of planar symmetric maps, such as preserving Euclidean circles, preserving k-Apollonius circles for some \(k\in (0,\infty )\) $k\in (0,\infty )$ , and preserving geometric moduli of all pairs of disjoint continua in the extended complex plane.