  We study the multifractal analysis for smooth dynamical systems in dimension
one. It is characterized the Hausdorff dimension of the level set obtained from
the Birkhoff averages of a continuous function by the local dimensions of
hyperbolic measures for a topologically mixing $C^2$ map modelled by an
abstract dynamical system. A characterization which corresponds to above is
also given for the ergodic basins of invariant probability measures. And it is
shown that the complement of the set of quasi-regular points carries full
Hausdorff dimension.
