Wreath products along the period-doubling route to chaos

The wreath-product construction is used to give a complete combinatorial de scription of the dynamics of period-doubling quadratic maps leading to the Feigenbaum map. An explicit description of the action on periodic points us es the Thue-Morse sequence. in particular, a wreath-product construction of this sequence is given. The combinatorial renormalization operator on the period-doubling family of maps is invertible.