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Techniques from Descriptive Set Theory are applied in order to study theTopological Complexity of families of operators naturally connected to ergodicoperators in infinite dimensional Banach Spaces. The families of ergodic,uniform-ergodic,Cesaro-bounded and power-bounded operators are shown to beBorel sets, while the family of super-ergodic operators is shown to be eithercoanalytic or Borel according to specific structures of the Space. Moreover,trees and coanalytic ranks are introduced to characterize super-ergodicoperators as well as spaces where the above classes of operators do notcoincide.

A Coanalytic Rank on Super-Ergodic Operators