  We discuss on an example a general mechanism of apparition of anomalous
scaling in scale invariant systems via zero modes of a scale invariant
operator. We discuss the relevance of such mechanism in turbulence, and point
out a peculiarity of turbulent flows, due to the existence of both forcing and
dissipation. Following these considerations, we show that if this mechanism of
anomalous scaling is operating in turbulence, the structure functions can be
constructed by simple symmetry considerations. We find that the generical scale
behavior of structure functions in the inertial range is not self-similar
$S_n(\ell)\propto \ell^{\zeta_n}$ but includes an "exponential self-similar"
behavior $S_n(\ell) \propto \exp[\zeta_n\alpha^{-1} \ell^{\alpha}]$ where
$\alpha$ is a parameter proportional to the inverse of the logarithm of the
Reynolds number. The solution also follows exact General Scaling and
approximate Extended Self-Similarity.
