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A criterion for the existence of a continuous invariant section of an affine extension. (Russian) Zbl 0706.55012

The authors’ main result can be stated as follows: Let (X,p,B) be a vector bundle of finite rank with compact base space B. Suppose that p: (X,\({\mathbb{R}},\pi)\to (B,{\mathbb{R}},\rho)\) is a linear extension which has no nontrivial bounded motions. Let f: \(B\to X\) be a continuous section of (X,p,B). Then an affine extension p: (X,\({\mathbb{R}},\pi_ f)\to (B,{\mathbb{R}},\rho)\) has a continuous invariant section iff \[ \int^{\infty}_{-\infty}\sigma_*(\rho^ t(b))(f(\rho^ t(b))dt=0 \] (b\(\in B\); \(\pi \circ \rho^ t=\rho^ t\circ p)\) for each continuous section \(\sigma_*: B\to X^*\) which is invariant w.r.t. the linear extension \(p^*: (X^*,{\mathbb{R}},\pi^*)\to (B,{\mathbb{R}},\rho)\).
Reviewer: J.Szilasi

MSC:

55R99 Fiber spaces and bundles in algebraic topology
37-XX Dynamical systems and ergodic theory
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