  We consider a generic one dimensional spin system of length L, arbitrarily
large, with strictly local interactions, for example nearest neighbor, and
prove that the dynamical alpha-R\'enyi entropies, 0 < alpha < 1, of an initial
product state grow at most linearly in time. We extend our bound on the
dynamical generation of entropy to systems with exponential decay of
interactions, for values of alpha close enough to 1. We provide a non rigorous
argument to extend our results to initial low-entangled, meaning O(log L)
states. This class of states includes many examples of spin systems ground
states, and also critical states. This implies that low entanglement states
have an efficient MPS representation that persists at least up to times of
order log L. The main technical tools are the Lieb-Robinson bounds, to locally
approximate the dynamics of the spin chain, a strict upper bound of Audenaert
on alpha-R\'enyi entropies and a bound on their concavity. Such a bound, that
we provide in an appendix, can be of independent interest.
