Etale and motivic variations on Smith's theory on action of cyclic groups

In this talk, after recalling some facts on Smith's theory in classical topology, I will present some analogues in algebraic geometry using \'etale cohomology first, and I will also give analogues in \A^1-homotopy theory and motives of Smith's theory concerning the multiplicative group \G_m, which behaves like a  "cyclic of order 2".