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The study of young Sun-like stars is of fundamental importance to understandthe magnetic activity and rotational evolution of the Sun. Space-bornephotometry by the Kepler telescope provides unprecedented datasets toinvestigate these phenomena in Sun-like stars. We present a new analysis of theentire Kepler photometric time series of the moderately young Sun-like starKepler-17 that is accompanied by a transiting hot Jupiter. We applied amaximum-entropy spot model to the long-cadence out-of-transit photometry of thetarget to derive maps of the starspot filling factor versus the longitude andthe time. These maps are compared to the spots occulted during transits tovalidate our reconstruction and derive information on the latitudes of thestarspots. 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 avarying level of activity. The latitudinal differential rotation is of solartype, that is, with the equator rotating faster than the poles. We estimate aminimum 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 ofindividual starspots and depending on the adopted spot model parameters. Wefind 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 anequatorward migration of the mean latitude of the spots as in the Sun. Therotation of Kepler-17 is likely to be significantly affected by the tidesraised by its massive close-by planet. We confirm the reliability ofmaximum-entropy spot models to map starspots in young active stars andcharacterize the activity and differential rotation of this young Sun-likeplanetary host.

Stellar activity and rotation of the planet host Kepler-17 from long-term space-borne photometry