Automorphisms of compact groups
Automorphisms of compact groups
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紧群的自同构
DOI：
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发表时间：
1989
影响因子：
0.9
通讯作者：
K. Schmidt
中科院分区：
文献类型：
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作者：
B. Kitchens;K. Schmidt
研究了紧可制群X的连续自同构的有限生成阿贝尔群Γ，并引入了这类群对的降链条件（X， Γ）。如果Γ扩展作用于X，则（X， Γ）满足降链条件，且（X， Γ）满足降链条件当且仅当它在代数和拓扑上同构于GΓ的闭平移不变子群，其中G是紧李群。更进一步，GΓ的每一个这样的子群都是一个（高维）马尔可夫移位，其字母表是紧李群。例如，利用降链条件证明了当Γ作用于X时，在X中Γ-periodic点的集合是密集的，并且，如果X是紧群且（X， Γ）满足降链条件，则Γ的每一个遍历元素都有一个密集的周期点集合。最后，在假设X是阿贝尔的前提下，给出了满足降链条件的对（X， Γ）的代数描述。
Abstract We study finitely generated, abelian groups Γ of continuous automorphisms of a compact, metrizable group X and introduce the descending chain condition for such pairs (X, Γ). If Γ acts expansively on X then (X, Γ) satisfies the descending chain condition, and (X, Γ) satisfies the descending chain condition if and only if it is algebraically and topologically isomorphic to a closed, shift-invariant subgroup of GΓ, where G is a compact Lie group. Furthermore every such subgroup of GΓ is a (higher dimensional) Markov shift whose alphabet is a compact Lie group. By using the descending chain condition we prove, for example, that the set of Γ-periodic points is dense in X whenever Γ acts expansively on X. Furthermore, if X is a compact group and (X, Γ) satisfies the descending chain condition, then every ergodic element of Γ has a dense set of periodic points. Finally we give an algebraic description of pairs (X, Γ) satisfying the descending chain condition under the assumption that X is abelian.