in this work, different types of chaotic 1-manifolds which lie on the chaotic spheres or on a torus are introduced.
some types of retractions of the chaotic spheres affect on the 1-chaotic systems, and other types of retractions
occur to the geometric manifold but make the 1-chaotic manifold invariant. the existed retractions are discussed
through new proved theorems. also we construct diffe...
