A qualitative theory of similarity pseudogroups and an analogy of Sacksteder's theorem (Q1190712)

An important theorem of Sacksteder implies that, in a \(C^ 2\)-foliation of codimension one on a closed manifold, an exceptional minimal set contains a leaf with contracting linear holonomy. Sacksteder's theorem was stated for finitely generated, \(C^ 2\) pseudogroups \(\Gamma\) on \(\mathbb{R}\), asserting that an exceptional, semiproper orbit \(\Gamma(x_ 0)\) closes on a point \(x\in \mathbb{R}\) whose stabilizer \(\Gamma_ x\) contains an element \(g\) with \(0 < g'(x) < 1\). The paper under review extends this result to certain foliations of codimension \(q>1\). Not surprisingly, this requires a quite restrictive hypothesis on the foliation, namely, that the holonomy pseudogroup \(\Gamma\) is generated by finitely many local similarity transforms \(\{g_ i\}^ n_{i=1}\) in \(\mathbb{R}^ q\). It is assumed that a given holonomy orbit \(\Gamma(x_ 0)\) is nonproper, is bounded away from the boundary of the domains of the generators \(g_ i\), and has ``bubbles''. Taken together, these conditions are analogous in higher codimension, to the notion of an ``exceptional, semiproper orbit'' in codimension one. The author proves that, under these conditions, there is \(x\in\overline{\Gamma(x_ 0)}\) and a contracting element \(g\in \Gamma_ x\).