Hypercyclicity and unimodular point spectrum

We show that an operator on a separable complex Banach space with sufficiently many eigenvectors associated to eigenvalues of modulus 1 is hypercyclic. We apply this result to construct hypercyclic operators with prescribed K_σ unimodular point spectrum. We show how eigenvectors associated to unimodular eigenvalues can be used to exhibit common hypercyclic vectors for uncountable families of operators, and prove that the family of composition operators C_? on H²(D), where ? is a disk automorphism having +1 as attractive fixed point, has a residual set of common hypercyclic vectors.