Invariant manifold theory via generating maps (Q957560)

A correspondence of a set \(Z\) into itself is a map \(h\) of \(Z\) into the set \(\mathcal{P}( Z) \) of subsets of \(Z.\) This map is determined by its graph \(\text{graph}( h) :=\{(z,z^{\prime}) \in Z^{2}:z^{\prime}\in h(z)\}\). If \(Z\) is a product \(X\times Y,\) the correspondence \(h\) admits a generating map \(H=( F,G) :Z\to Z\) when the graph of \(h\) is the set of those \(( x,y,x^{\prime},y^{\prime}) \in Z^{2}\) which satisfy \[ x^{\prime}=F(x,y^{\prime}) \qquad\text{and}\quad y=G(x,y^{\prime}). \] Consequently, for each \(( x,y^{\prime}) \in Z,\) there exists a unique orbit \((z_{0},z_{1})\) of \(h\) with the length one such that the first component of \(z_{0}\) is \(x\) and the second component of \(z_{1}\) is \(y^{\prime}.\) In this note, the author presents a synthetic approach to invariant manifold theorems in terms of generating maps. Two sets of results are briefly outlined for the cases when (a) \(F\) and \(G\) are Lipschitzian and (b) \(F\) and \(G\) are \(C^{r}\) with \(X\) and \(Y\) being \(C^{r}\) Finsler manifolds with corners. Further details and explanations will appear elsewhere.