On asymptotic phase of dynamical system hyperbolic along attracting invariant manifold

We consider a dynamical system which has the hyperbolic structure along an attracting invariant manifold $M$. The problem is whether every motion starting in a neighborhood of $M$ possesses an asymptotic phase, i.e. eventually approaches a particular motion on $M$. Earlier, positive solutions to the problem were obtained under the condition that the decay rate of solutions toward the manifold exceeds the decay rate of the solutions within the manifold. We show that in our case the above condition is not necessary. To prove that a neighborhood of $M$ is filled with motions for each of which there exists an asymptotic phase we apply the Brouwer fixed point theorem. An invariant foliation structure which appears in the neighborhood of $M$ is discussed.