  The study of young Sun-like stars is of fundamental importance to understand
the magnetic activity and rotational evolution of the Sun. Space-borne
photometry by the Kepler telescope provides unprecedented datasets to
investigate these phenomena in Sun-like stars. We present a new analysis of the
entire Kepler photometric time series of the moderately young Sun-like star
Kepler-17 that is accompanied by a transiting hot Jupiter. We applied a
maximum-entropy spot model to the long-cadence out-of-transit photometry of the
target to derive maps of the starspot filling factor versus the longitude and
the time. These maps are compared to the spots occulted during transits to
validate our reconstruction and derive information on the latitudes of the
starspots. We find two main active longitudes on the photosphere of Kepler-17,
one of which has a lifetime of at least $\sim 1400$ days, although with a
varying level of activity. The latitudinal differential rotation is of solar
type, that is, with the equator rotating faster than the poles. We estimate a
minimum relative amplitude $\Delta \Omega/ \Omega$ between $\sim 0.08 \pm 0.05$
and $0.14 \pm 0.05$, our determination being affected by the finite lifetime of
individual starspots and depending on the adopted spot model parameters. We
find marginal evidence of a short-term intermittent activity cycle of $\sim 48$
days and an indication of a longer cycle of $400-600$ days characterized by an
equatorward migration of the mean latitude of the spots as in the Sun. The
rotation of Kepler-17 is likely to be significantly affected by the tides
raised by its massive close-by planet. We confirm the reliability of
maximum-entropy spot models to map starspots in young active stars and
characterize the activity and differential rotation of this young Sun-like
planetary host.
