Symmetry Transfer to Basins and Enhancement at Basin Boundaries

Abstract

For an equivariant dynamical system, we prove that a compact Lyapunov-stable attractor realized as an $\omega$-limit set has the same setwise symmetry group as its basin. Admissible attractor and basin symmetries therefore coincide within each fixed dynamical category. For a finite symmetry group, the index of the attractor stabilizer in the boundary stabilizer counts the symmetry-related basins sharing that full boundary. This yields a criterion for boundary symmetry enhancement, a dichotomy for attractor orbits of prime size, and a connection with Wada basins. A common boundary of $m$ disjoint open basins has at least $m$ local complementary components, which excludes smooth hypersurface patches when $m\ge 3$. Smooth flows and Newton maps illustrate the different possibilities.

Substantially revised version with a new theoretical structure and streamlined results. The manuscript now focuses on attractor-basin symmetry transfer and common basin boundaries. 7 pages, 1 figure