Multifractal structure of Ly$\alpha$ clouds: An example with the spectrum of QSO 0055--26

Ly$\alpha$ forests are usually associated with intergalactic gas clouds intercepting the quasar sight--line. Using as an example the Ly$\alpha$ forest of QSO 0055--26 ($z_{em} = 3.66$), we show that the probability of observing a line at the velocity difference $\Delta v$ has an intermittent structure in the redshift space. On small scales (70 km~s$^{-1} \leq \Delta v \leq 1580$ km~s$^{-1}$) the signature of intermittency appears as a self--similar structure of spikes systematically visible at the same redshift on all scales. This behaviour can be interpreted as due to the presence of clustering of the Ly$\alpha$ lines, which appear as singularities of the density of the probability measure. From a direct measurement of the generalized dimensions, we show that the intermittency can be described by a multifractal structure which is due to a Fourier phase correlation of the signal. The multifractal structure disappears at scales larger than $\Delta v \simeq 1580$ km~s$^{-1}$.