  Kepler-13b (KOI-13.01) is a most intriguing exoplanet system due to the rapid
precession rate, exhibiting several exotic phenomena. We analyzed $Kepler$
Short Cadence data up to Quarter 14, with a total time-span of 928 days, to
reveal changes in transit duration, depth, asymmetry, and identify the possible
signals of stellar rotation and low-level activity. We investigated long-term
variations of transit light curves, testing for duration, peak depth and
asymmetry. We also performed cluster analysis on $Kepler$ quarters. We computed
the autocorrelation function of the out-of-transit light variations. Transit
duration, peak depth, and asymmetry evolve slowly, due to the slowly drifting
transit path through the stellar disk. The detected transit shapes will map the
stellar surface on the time scale of decades. We found a very significant
clustering pattern with 3-orbit period. Its source is very probably the
rotating stellar surface, in the 5:3 spin-orbit resonance reported in a
previous study. The autocorrelation function of the out-of-transit light
variations, filtered to 25.4 hours and harmonics, shows slow variations and a
peak around 300--360 day period, which could be related to the activity cycle
of the host star.
