Orbit equivalence is a part of Ergodic theory, studying infinite countable groups of actions on standard measure spaces. These actions depend on the orbit partitions of the space. In fact, orbit equivalence determine the isomorphic actions of equivalent orbit. Ergodic theory researches into the feature of certain group actions on certain spaces. Nowadays, this theory has been enlarged to the extend that is applicable in various parts of mathematics. The possibility Ergodic theory is considered as the main part Ergodic theory, indicating the existing limitations. The other part is called infinite Ergodic which is more pervasive. This study, in addition to the investigating possibility Ergodic theory, explore the infinite Ergodic theory; furthermore , it surveys the Poincar´e theorem, a principle of Ergodic theory, from two perspectives. In possibility Ergodic theory that embraces orbit equivalence, major groups of orbit equivalence actions will be determind through the groups’ amenability, thus it represents the low capacity spaces and introduces spaces with higher capacity. This thesis attempted to represent these kind of actions and to uncover the spaces with more appropriate feature for studying Ergodic theory.