Unimodular eigenvalues, uniformly distributed sequences and linear dynamics

We study increasing sequences of positive integers (nk)k⩾1 with the following property: every bounded linear operator T acting on a separable Banach (or Hilbert) space with supk⩾1‖Tnk‖<∞ has a countable set of unimodular eigenvalues. Whether this property holds or not depends on the distribution (modulo one) of sequences (nkα)k⩾1, α∈R, or on the growth of nk+1/nk. Counterexamples to some conjectures in linear dynamics are given. For instance, a Hilbert space operator which is frequently hypercyclic, chaotic, but not topologically mixing is constructed. The situation of C0-semigroups is also discussed.