Motions in G-vector bundles (linear extensions) that are locally bounded and locally separated from zero (Q914191)

Let p: \(X\to Y\) be a metric G-vector bundle. Let \(x\in X\), \(s\in S\). We say that the motion of the point x is bounded (or separated from zero) on the set S, if sup\(\{| sx|:\) \(s\in S\}<\infty\) (or, respectively, inf\(\{| sx|:\) \(s\in S\}>0)\). The motion of x is called locally bounded (locally separated from zero) if every point \(y\in Y\) has a neighbourhood U such that the motion of x is bounded (respectively, separated from zero) on the set \(\{\) \(g\in G:\) p(gx)\(\in U\}\). Let LB (or LD) denote the set of points with locally bounded motions (or, respectively, locally separated from zero). Let APL denote the set of Levitan almost periodic points. The results of the paper are the following: if LB\(\subset LD\) then: 1) LB is a G-invariant subbundle of X with a G-invariant metric. 2) LB is a direct sum of irreducible G-invariant bundles. 3) If for every \(x\in APL\) the closure of its orbit is in APL then APL is a G-invariant subbundle of X, and, for a commutative G, APL is a direct sum of one-dimensional trivial G-invariant subbundles.