  It follows from Oseledec Multiplicative Ergodic Theorem (or Kingmans
Subadditional Ergodic Theorem) that the Lyapunov-irregular set of points for
which the Oseledec averages of a given continuous cocycle diverge has zero
measure with respect to any invariant probability measure. In strong contrast,
for any dynamical system f with exponential specification property and a Holder
continuous matrix cocycle A, we show here that if there exist ergodic measures
with different Lyapunov spectrum, then the Lyapunov-irregular set of A displays
distributional chaos of type 1.
