Minimal Flows and their Extensions
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Different definitions begin with different interpretations of this divergence. We present here some popular ideas and relations between them for minimal dynamical systems see Fig. The following concept from [AK] links the Li-Yorke versions of chaos with the notion of sensitivity to initial conditions.
A dynamical system is scattering if and only if its cartesian product with any minimal dynamical system is transitive. Then the definitions of minimal sets and minimality for a ttg are analogous to those for maps.
Minimal Flows and Their Extensions - J. Auslander - Google книги
For a homeomorphism, minimality in the sense of the density of all full orbits is in general not equivalent to minimality in the sense of the density of all forward orbits. There are locally compact but not compact metric spaces which admit minimal homeomorphisms in the former sense but do not admit any minimal map in the latter one. For the structure of minimal sets of a ttg the same alternative holds as the one discussed above in the case of a homeomorphism -- a minimal set is either nowhere dense or clopen.
For connected groups the following general result holds. For more information on minimality in the setting of topological transformation groups see the books [GH], [Ell1], [Br], [Au] and [Vri]. The converse is not true the Klein bottle does not admit a minimal continuous flow though it admits a minimal homeomorphism. In this section we partially follow the article Topological dynamics.
A pair of points is called distal if it is not proximal. An equicontinuous flow on a compact Hausdorff space is distal, but the converse is not true in general. If a flow is obtained by an equicontinuous extension of an equicontinuous flow then it need not be equicontinuous but it is necessarily distal. In fact more is true: An equicontinuous extension of a distal flow and even a distal extension of a distal flow is distal.
Therefore if we start with a distal flow, say the trivial one point flow, and extend it equicontinuously again and again, possibly transfinitely many times by passing to inverse limits of flows at limit ordinals see e. The following deep Furstenberg structure theorem says that the converse is also true:.
The Furstenberg structure theorem was extended by several authors. In particular, there are structure theorems for so called point distal minimal flows, prodal minimal flows, normal minimal flows. There is also a structure theorem for general minimal flows equicontinuous, proximal and so-called weakly mixing extensions appear in it. For more details see [Gl], [Au], [Vri] and references which can be found there.
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Various classes of minimal flows equicontinuous, distal, point distal are intensively studied, and a general structure theorem is obtained. Another theme is the universal'' approach - entire classes of minimal flows are studied, rather than flows in isolation. This leads to the consideration of disjointness of flows, which is a kind of independence condition.
Among the topics unique to this book are a proof of the Ellisjoint continuity theorem'', a characterization of the equicontinuous structure relation, and the aforementioned structure theorem for minimal flows. Ellis qu:The book under review is a pleasant, leisurely account of the topics discussed. In places it reads like a novel. We are always looking for ways to improve customer experience on Elsevier. We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit.
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