  Dynamical systems that are contracting on a subspace are said to be
semicontracting. Semicontraction theory is a useful tool in the study of
consensus algorithms and dynamical flow systems such as Markov chains. To
develop a comprehensive theory of semicontracting systems, we investigate
seminorms on vector spaces and define two canonical notions: projection and
distance semi-norms. We show that the well-known lp ergodic coefficients are
induced matrix seminorms and play a central role in stability problems. In
particular, we formulate a duality theorem that explains why the
Markov-Dobrushin coefficient is the rate of contraction for both averaging and
conservation flows in discrete time. Moreover, we obtain parallel results for
induced matrix log seminorms. Finally, we propose comprehensive theorems for
strong semicontractivity of linear and non-linear time-varying dynamical
systems with invariance and conservation properties both in discrete and
continuous time.
