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 ξ , in some sense similar to the period doubling word w (which is the fixed point of the morphism φ w : 0 ↦ 0100 , 1 ↦ 0101 ). We investigate combinatorial properties of ξ and find the complexity of ξ .