Fibrations of the Persistent Invariant Manifolds

We continue to work in $${{\tilde{D}}_{k}}$$, where the bumped perturbed flow (2.6.27) is defined. From Proposition 3.1.1, we know the existence of the Cn codimension 1 center-unstable manifold $$W_{{{{\delta }_{1}},\delta }}^{{cu}}$$, the Cn codimension 1 center-stable manifold $$W_{{{{\delta }_{1}},\delta }}^{{cs}}$$, and the Cn codimension 2 center manifold $${{W}_{{{{\delta }_{1}},\delta }}}$$, under the bumped perturbed flow (2.6.27). More specifically, $$W_{{{{\delta }_{1}},\delta }}^{{cu}}$$ exists in $$\tilde{D}_{k}^{{(1)}}$$; moreover, it is overflowing invariant. $$W_{{{{\delta }_{1}},\delta }}^{{cs}}$$ exists in $$\tilde{D}_{k}^{{(2)}}$$; moreover, it is inflowing invariant. Then $${{W}_{{{{\delta }_{1}},\delta }}} \equiv W_{{{{\delta }_{1}},\delta }}^{{cu}} \cap W_{{{{\delta }_{1}},\delta }}^{{cs}}$$ exists in $$\tilde{D}_{k}^{{(2)}}$$, and it is inflowing invariant. Since the fibration theorem is concerned with the fiber representations of $$W_{{{{\delta }_{1}},\delta }}^{{cu}}$$ and $$W_{{{{\delta }_{1}},\delta }}^{{cs}}$$ with respect to $${{W}_{{{{\delta }_{1}},\delta }}}$$ as the base, we have to work in a region where $${{W}_{{{{\delta }_{1}},\delta }}}$$ exists. Therefore, we can work only inside $$\tilde{D}_{k}^{{(2)}}$$. We know that $${{W}_{{{{\delta }_{1}},\delta }}}$$ is inflowing invariant in $$\tilde{D}_{k}^{{(2)}}$$. Next, we will prove the following lemma: