  Techniques from Descriptive Set Theory are applied in order to study the
Topological Complexity of families of operators naturally connected to ergodic
operators in infinite dimensional Banach Spaces. The families of ergodic,
uniform-ergodic,Cesaro-bounded and power-bounded operators are shown to be
Borel sets, while the family of super-ergodic operators is shown to be either
coanalytic or Borel according to specific structures of the Space. Moreover,
trees and coanalytic ranks are introduced to characterize super-ergodic
operators as well as spaces where the above classes of operators do not
coincide.
