Example of a diffeomorphism for which the special ergodic theorem   doesn't hold
  In this work we present an example of C^\infty-diffeomorphism of a compact 4-manifold such that it admits a global SRB measure \mu but for which the special ergodic theorem doesn't hold. Namely, for this transformation there exist a continuous function \phi and a positive constant \alpha such that the following holds: the set of the initial points for which the Birkhoff time averages of the function \phi differ from its \mu--space average by at least \alpha has zero Lebesgue measure but full Hausdorff dimension. 
Dmitry Ryzhov
math.DS