  We use character sums to confirm several recent conjectures of V. I. Arnold
on the uniformity of distribution properties of a certain dynamical system in a
finite field. On the other hand, we show that some conjectures are wrong. We
also analyze several other conjectures of V. I. Arnold related to the orbit
length of similar dynamical systems in residue rings and outline possible ways
to prove them. We also show that some of them require further tuning.
