On an infinite permutation similar to the Thue–Morse word

Infinite permutations (in our sense) were introduced in 2005 by Fon-Der-Flaass and Frid. We introduce an infinite permutation τ in some sense similar to the Thue–Morse word w (where w is the fixed point of the morphism φ w : 0 ↦ 01 , 1 ↦ 10 ). We suggest two essentially different definitions of τ and show their equivalence.