  Using the techniques in topological dynamics, we give a uniform treatment of
Li-Yorke chaos, mean Li-Yorke chaos and distributional chaos for continuous
endomorphisms of completely metrizable groups, and characterize three kinds of
chaos (resp. extreme chaos) in term of the existence of corresponding
semi-irregular points (resp. irregular points). We demonstrate some inner
automorphisms of Polish groups to illustrate the results. We also apply the
results to the chaos theory of continuous linear operators on Fr\'echet spaces,
which improves some results in the literature.
