Multiparameter ratio ergodic theorems for semigroups

Abstract

After one-parameter treatment of ratio ergodic theorems for semigroups, we formulate the Sucheston a.e. convergence principle of continuous parameter type. This principle plays an effective role in proving some multiparameter generalizations of Chaconʼs type continuous ratio ergodic theorems for semigroups and of Jacobsʼ type continuous random ratio ergodic theorems for quasi-semigroups. In addition, a continuous analogue of the Brunel–Dunford–Schwartz ergodic theorem is given of sectorially restricted averages for a commutative family of semigroups. We also formulate a local a.e. convergence principle of Suchestonʼs type. The local convergence principle is effective in proving multiparameter local ergodic theorems. In fact, a multiparameter generalization of Akcoglu–Chaconʼs local ratio ergodic theorem for semigroups of positive linear contractions on L1 is proved. Moreover, some multiparameter martingale theorems are obtained as applications of convergence principles.