The measure preserving and nonsingular transformations of the jump Lévy processes

Let $\xi(t), t\in[0, 1],$ be a jump Lévy process. By ${\mathcal P}_\xi,$ we denote the law of $\xi$ in the Skorokhod space ${\mathbb D}[0, 1].$ Under some conditions on the Lévy measure of the process, we construct the group of ${\mathcal P}_\xi$ – preserving transformations of ${\mathbb D}[0, 1].$ For the Lévy process that has only positive (or only negative) jumps, we construct the semigroup of nonsingular transformations.