初值敏感依赖与混沌群作用
Sensitive dependence on initial conditions and chaotic group actions
摘要：
紧度量空间上群G的连续作用对初始条件有敏感依赖性，如果存在一个数e>0，使得对任意开集U，我们可以在G中找到g，使得g.U的直径大于e.证明了如果G作用保持全支持度的概率测度，则系统要么是极小的等度连续的，要么对初始条件具有敏感依赖性。这推广了可逆的情况下的一个定理的Reinner和Weiss。证明了当一个可解生成的可解群、作用和某些循环子作用具有稠密的极小点集时，系统对初始条件具有敏感依赖性。此外，我们展示了如何构建非紧一元论群，传递，非极小，几乎等度连续，经常性，群作用的例子。
英文摘要：
A continuous action of a group G on a compact metric space has sensitive dependence on initial conditions if there is a number e>0 such that for any open set U we can find g in G such that g.U has diameter greater than e. We prove that if a G action preserves a probability measure of full support, then the system is either minimal and equicontinuous, or has sensitive dependence on initial conditions. This generalizes the invertible case of a theorem of Glasner and Weiss. We prove that when a finitely generated, solvable group, acts and certain cyclic subactions have dense sets of minimal points, then the system has sensitive dependence on initial conditions. Additionally, we show how to construct examples of non-compact monothetic groups, and transitive, non-minimal, almost-equicontinuous, recurrent, group actions.
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