Ergodicity and Conservativity of products of infinite transformations   and their inverses
  We construct a class of rank-one infinite measure-preserving transformations such that for each transformation $T$ in the class, the cartesian product $T\times T$ of the transformation with itself is ergodic, but the product $T\times T^{-1}$ of the transformation with its inverse is not ergodic. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative. 
Julien Clancy, Rina Friedberg, Indraneel Kasmalkar, Isaac Loh, Tudor   P\u{a}durariu, Cesar E. Silva, and Sahana Vasudevan
math.DS