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.