Broadband ultrasonic attenuation spectra (100 kHz to 2 GHz) of aqueous solutions of vesicles from 1,2-dimyristoyl-L-3-phosphatidylcholine, with different amounts of cholesterol admixed, have been measured at temperatures between 20 and 28 degrees C. The spectra have been evaluated in terms of suitable relaxation functions. They are discussed in view of the effect of cholesterol on the membrane behavior around the gel-fluid phase transition temperature T-m. In addition to a frequency-independent asymptotic high-frequency term, all spectra reveal a critical term and a Debye-type relaxation term with relaxation time around 0.5 ns. The former is evaluated in the light of the Bhattacharjee-Ferrell dynamic scaling theory. It is,assigned to the critical domain structure fluctuations of the membranes. Critical slowing of fluctuations is demonstrated Also shown are relations of the critical amplitudes to thermodynamic parameters. The Debye term reflects the rotational isomerization of the phospholipid alkyl chains. The relaxation time of isomerization reveals a significant steplike change at T-m. At moderate cholesterol content an additional Debye relaxation term exists. It is assigned to the axial diffusion of the membrane molecules. Because it likewise shows effects of slowing near T-m, the diffusion appears to be coupled to the domain structure fluctuations. A further relaxation term at small cholesterol concentration is assumed to be due to small-range shape fluctuations of vesicles near the phase transition temperature.