In this paper, we consider a family of the hypercomplex rings \({\mathscr {H}}=\left\{ {\mathbb {H}}_{t}\right\} _{t\in {\mathbb {R}}}\) scaled by \({\mathbb {R}}\), and the dynamical system of \({\mathbb {R}}\) acting on \({\mathscr {H}}\) via a certain action \(\theta \) of \({\mathbb {R}}\). i.e., we study an analysis on dynamical system induced by \({\mathscr {H}}\). In particular, we are interested in free-probabilistic information on the dynamical system dictated by our hypercomplex analysis.

中文翻译：

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缩放超复环诱发的算子动力系统

在本文中，我们考虑一族超复环\({\mathscr {H}}=\left\{ {\mathbb {H}}_{t}\right\} _{t\in {\mathbb { R}}}\)按\({\mathbb {R}}\)缩放，而\({\mathbb {R}}\)的动力系统作用于\({\mathscr {H}}\)通过\({\mathbb {R}}\)的某个动作\(\theta \)。即，我们研究由\({\mathscr {H}}\)引起的动力系统的分析。特别是，我们对由超复杂分析决定的动力系统的自由概率信息感兴趣。