Parallelizability in Banach spaces: Applications of negligibility theory. (English) Zbl 0699.54018
Using the negligibility theory of infinite-dimensional topology, the author modifies the classical Beputov example to give a dynamical system \(\rho\) on an infinite-dimensional Banach space E for which the restriction of \(\rho\) to \(E\setminus \{0_ E\}\) is dispersive but not parallelizable. This example shows that the infinite-dimensional analogue of the famous Nemyckii-Stepanov theorem (which states that a dynamical system \(\pi\) on a finite-dimensional normed space X or more generally, on a locally compact metric space X is parallelizable if and only if it is dispersive) fails. Also is given another example of a dynamical system \(\eta\) on \(E\times F\), where E, F are Banach spaces with E infinite- dimensional, showing that all aspects of the infinite-dimensional counterpart of Coleman’s conjecture (which states that a suitable topological notion of hyperbolicity for an equilibrium point \(x_ 0\) of a dynamical system \(\pi\) in \({\mathbb{R}}^{n+m}\) should guarantee that near \(x_ 0\), \(\pi\) is conjugate to a dynamical system induced by a linear hyperbolic vector field) are false.
Reviewer: T.Panchapagesan
MSC:
| 54H20 | Topological dynamics (MSC2010) |
| 46B99 | Normed linear spaces and Banach spaces; Banach lattices |
| 37-XX | Dynamical systems and ergodic theory |
Keywords:
globally locally uniformly asymptotically stable; hyperbolic; equilibria; negligibility theory of infinite-dimensional topology; Beputov example; dynamical system; dispersive but not parallelizable; infinite-dimensional analogue of the famous Nemyckii-Stepanov; theorem; infinite-dimensional counterpart of Coleman’s conjecture; infinite-dimensional analogue of the famous Nemyckii-Stepanov theoremReferences:
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