  We introduce invariants, called shifting numbers, that measure the asymptotic
amount by which an autoequivalence of a triangulated category translates inside
the category. The invariants are analogous to Poincare translation numbers that
are widely used in dynamical systems. We additionally establish that in some
examples the shifting numbers provide a quasimorphism on the group of
autoequivalences. Additionally, we relate our shifting numbers to the entropy
function introduced by Dimitrov, Haiden, Katzarkov, and Kontsevich.
