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Using recently proposed measures for non-Markovianity [H. P. Breuer, E. M.Laine, and J. Piilo, Phys. Rev. Lett. {\bf 103}, 210401 (2009)], we study thedynamics of a qubit coupled to a spin environment via an energy-exchangemechanism. We show the existence of a point, in the parameter space of thesystem, where the qubit dynamics is effectively Markovian and that such a pointseparates two regions with completely different dynamical behaviors. Indeed,our study demonstrates that the qubit evolution can in principle be tuned froma perfectly forgetful one to a deep non-Markovian regime where the qubit isstrongly affected by the dynamical back-action of the environmental spins. Bymeans of quantum process tomography, we provide a complete and intuitivecharacterization of the qubit channel.

Memory-keeping effects and forgetfulness in the dynamics of a qubit coupled to a spin chain