Fibrations of the Persistent Invariant Manifolds

Abstract

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 C ⁿ 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: