Generalized snap-back repeller and semi-conjugacy to shift operators of piecewise continuous transformations

In this paper, we attempt to clarify an open problem related to a generalization of the snap-back repeller. Constructing a semi-conjugacy from the finite product of a transformation $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ on an invariant set $\Lambda$ to a sub-shift of the finite type on a $w$-symbolic space, we show that the corresponding transformation associated with the generalized snap-back repeller on $\mathbb{R}^{n}$ exhibits chaotic dynamics in the sense of having a positive topological entropy. The argument leading to this conclusion also shows that a certain kind of degenerate transformations, admitting a point in the unstable manifold of a repeller mapping back to the repeller, have positive topological entropies on the orbits of their invariant sets. Furthermore, we present two feasible sufficient conditions for obtaining an unstable manifold. Finally, we provide two illustrative examples to show that chaotic degenerate transformations are omnipresent.