Geometry in the neighborhood of invariant manifolds of maps and flows in linearization.

*(English)*Zbl 0746.58008
Pitman Research Notes in Mathematics. 233. Harlow, New York: Longman Scientific & Technical, John Wiley & Sons, Inc. 89 p. (1990).

The book consists of two parts of approximately equal lengths each written by one of the authors. In the first part homomorphisms \(P: \mathbb{R}^ r\times \mathbb{R}^ t\to\mathbb{R}^ r\times\mathbb{R}^ t\) given by \(P(z,y)=(f(z,y),L(y)+Y(z,y))\) are considered where \(L: \mathbb{R}^ t\to\mathbb{R}^ t\) is linear and expanding, and \(P\) is (by Lipschitz constraints) sufficiently close to a mapping \(P_ z\times L\) with \(P_ z: \mathbb{R}^ r\to\mathbb{R}^ r\) less expanding than \(L\). Under these assumptions a horizontal and a vertical foliation \(H\), \(V\), respectively of \(\mathbb{R}^ r\times\mathbb{R}^ t\) are constructed such that \(P\) maps leaves to leaves. There is an invariant leaf of \(H\) which is the graph of a function \(G: \mathbb{R}^ r\to \mathbb{R}^ t\). Using these foliations new coordinates \(\bar z,\;\bar y\) can be introduced. (The leaves correspond to the manifolds given by \(\bar y=\text{const.}\) or \(\bar z=\text{const.}\), respectively.) With respect to the new coordinates \(P\) splits as
\[
P(\bar z,\bar y)=(f(\bar z,G(\bar z)),L(\bar y))
\]
and is therefore linearized with respect to \(y\). This is applied to get a corresponding linearization for flows given by autonomous differential equations on \(\mathbb{R}^ r\times\mathbb{R}^ t\). In the second part this linearization for the solution of differential equations is constructed directly (also using a horizontal and a vertical foliation).

Reviewer: H.G.Bothe (Berlin)