Infinite symmetric ergodic index and related examples in infinite measure
For infinite-measure-preserving rank-one transformations, we give a condition guaranteeing that all finite Cartesian products of the transformation with its inverse are ergodic. We show that the infinite Chacón transformation satisfies this condition. We then explore the relationship between product conservativity and product ergodicity and answer a question of Danilenko. Finally, we define a class of infinite Chacón type transformations and show they do not have products of all powers conservative and are therefore not power weakly mixing.