Stability of orbit spaces of endomorphisms

In this paper it is first proved that, for a hyperbolic set of aC1 (non-invertible) endomorphism of a compact manifold, the dynamical structure of its orbit space (inverse limit space) is stable underC1-small perturbations and is semi-stable underC0-small perturbations. It is then proved that if an Axiom A endomorphism satisfies no-cycle condition then its orbit space is Θ-stable andR-stable underC1-small perturbations and is semi-Θ-stable and semi-R-stable underC0-small perturbations.