We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact quantum metric space in a natural way. Moreover, we provide a flexible set of assumptions ensuring that a continuous family of $\ast$-automorphisms of a compact quantum metric space yields a field of crossed product algebras which varies continuously in Rieffel’s quantum Gromov–Hausdorff distance. Finally, we show how our results apply to continuous families of Lip-isometric actions on compact quantum metric spaces and to families of diffeomorphisms of compact Riemannian manifolds which vary continuously in the Whitney $C^{1}$-topology.

中文翻译：

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紧致量子度量空间的动力学

我们详细研究了整数在紧致量子度量空间上的作用，其中包括确保相关叉积代数以自然方式再次成为紧致量子度量空间的一般标准。此外，我们提供了一套灵活的假设，以确保一个连续的家庭$\ast$- 紧致量子度量空间的自同构产生一个叉积代数域，该域在 Rieffel 的量子 Gromov-Hausdorff 距离中连续变化。最后，我们展示了我们的结果如何应用于紧致量子度量空间上的唇等距作用的连续族以及惠特尼中连续变化的紧致黎曼流形的微分同胚族$C^{1}$-拓扑。