Structural Properties of Isosymmetric Transformations on Metric Spaces

In this paper, we introduce and study m,n;q-isosymmetric transformations on metric spaces, generalizing the notions of m;q-isometries and n;q-symmetries that have recently been investigated in the literature. Our approach combines these two metric structures within a common framework governed by the parameters m, n, and q>0. We establish a collection of structural properties for m,n;q-isosymmetric transformations, including their behavior under iteration and composition, and derive several characterizations and equivalent conditions. We further examine their relationships with Lipschitz, nonexpansive, and contractive mappings, revealing conditions under which the corresponding metric properties are inherited or recovered. In particular, we show how the new class connects the previously studied m;q-isometric and n;q-symmetric settings within a single metric framework. These results provide a foundation for the systematic study of the structure and behavior of isosymmetric transformations on metric spaces.