Infinite ergodic index ${\displaystyle {\mathbb{Z}}^d}$ -actions in infinite measure

We construct infinite measure preserving and nonsingular rank one ${\displaystyle {\mathbb{Z}}^d}$-actions. The first example is ergodic infinite measure preserving but with nonergodic, infinite conservative index, basis transformations; in this case we exhibit sets of increasing finite and infinite measure which are properly exhaustive and weakly wandering. The next examples are staircase rank one infinite measure preserving ${\displaystyle {\mathbb{Z}}^d}$-actions; for these we show that the individual basis transformations have conservative ergodic Cartesian products of all orders, hence infinite ergodic index. We generalize this example to obtain a stronger condition called power weakly mixing. The last examples are nonsingular ${\displaystyle {\mathbb{Z}}^d}$-actions for each Krieger ratio set type with individual basis transformations with similar properties.