Ergodicity of continuous and discrete quaternionic semigroups and Tauberian theorems

In this paper, the ergodic theory of continuous and discrete quaternionic semigroups and Tauberian theory are developed under the quaternionic setting. The notions of Cesàro and Abel convergence for the quaternionic Banach-valued locally integrable functions are introduced and the corresponding Tauberian conditions are established. Moreover, the Cesàro convergence is studied based on the slice hyperholomorphic extension and the relations between Cesàro and Abel convergence are formulated. We weaken the boundary condition of the slice hypercomplex domain of Cauchy integral via the geometric techniques of cutting rectifiable Jordan curves by finite circles and prove the quaternionic Cesàro mean ergodic theorem. Furthermore, the Tauberian theorems for the quaternionic power series are deduced. On the other hand, the notions of Cesàro and Abel ergodicity of the continuous and discrete quaternionic semigroups are proposed and the S-resolvent and spectral conditions for Cesàro ergodicity of these semigroups are obtained. In addition, some basic properties of Cesàro and Abel-ergodic projections are derived and Abel ergodic and Cesàro mean ergodic theorems are established. Besides, some equivalent characterizations of the quaternionic operator matrix as the semigroup generator are formulated and proved and the ergodic theorems of the quaternionic semigroup generated by quaternionic operator matrix are demonstrated. Finally, the ergodicity of solutions for the inhomogeneous Cauchy problem with periodic inhomogeneity in quaternionic setting is achieved.