Dynamical Systems of Operators Induced by Scaled Hypercomplex Rings

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.