On the infinite permutation generated by the period doubling word

The paper is devoted to the properties of infinite permutations. We introduce the infinite permutation ξξ, in some sense similar to the period doubling word ww (which is the fixed point of the morphism φw:0↦0100,1↦0101φw:0↦0100,1↦0101). We investigate combinatorial properties of ξξ and find the complexity of ξξ.