  It is shown that Jamison sequences, introduced in 2007 by Badea and Grivaux
([C. Badea and S. Grivaux, Unimodular eigenvalues, uniformly distributed
sequences and linear dynamics, Adv. Math. 211 (2007), no. 2, 766--793]), arise
naturally in the study of topological groups with no small subgroups, of Banach
or normed algebra elements whose powers are close to identity along
subsequences, and in characterizations of (self-adjoint) positive operators by
the accretiveness of some of their powers. The common core of these results is
a description of those sequences for which non-identity elements in Lie groups
or normed algebras escape an arbitrary small neighborhood of the identity in a
number of steps belonging to the given sequence. Several spectral
characterizations of Jamison sequences are given and other related results are
proved.
