  Owing to the frequency and reproducibility of its outbursts, the black-hole
candidate GX 339-4 has become the standard against which the outbursts of other
black-hole candidate are matched up. Here we present the first systematic study
of the evolution of the X-ray lags of the broad-band variability component
(0.008-5 Hz) in GX 339-4 as a function of the position of the source in the
hardness-intensity diagram. The hard photons always lag the soft ones,
consistent with previous results. In the low-hard state the lags correlate with
X-ray intensity, and as the source starts the transition to the
intermediate/soft states, the lags first increase faster, and then appear to
reach a maximum, although the exact evolution depends on the outburst and the
energy band used to calculate the lags. The time of the maximum of the lags
appears to coincide with a sudden drop of the Optical/NIR flux, the fractional
RMS amplitude of the broadband component in the power spectrum, and the
appearance of a thermal component in the X-ray spectra, strongly suggesting
that the lags can be very useful to understand the physical changes that GX
339-4 undergoes during an outburst. We find strong evidence for a connection
between the evolution of the cut-off energy of the hard component in the energy
spectrum and the phase lags, suggesting that the average magnitude of the lags
is correlated with the properties of the corona/jet rather than those of the
disc. Finally, we show that the lags in GX 339-4 evolve in a similar manner to
those of the black-hole candidate Cygnus X-1, suggesting similar phenomena
could be observable in other black-hole systems.
