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 sup(k >=)1 parallel to T-nk parallel to < infinity has a countable set of unimodular eigenvalues. Whether this property holds or not depends on the distribution (modulo one) of sequences (n(k alpha))k >= 1, alpha epsilon R, or on the growth of n(k+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 C-0-semigroups is also discussed. (C) 2006 Elsevier Inc. All rights reserved.