Conjugacy of a group action to its inverse

Abstract

~(X, ~). The space ~(X, ~,Z ~) is a comwe introduce the weak topology induced from ~=: plete metrizable separable space in this topology. We will call the action T a lattice if there exist a measurable set C and positive numbers q:, "-', qk such that the sets {~..g~ C} for ij = 0 .... , qj 1 and j = 1 ..... k form a partition ~ of the space X (their union need not be all of X) and for each j we have T~ = Id. The set of lattices is everywhere dense in 9 (X, ~, Z~). The action T is conjugate to its inverse if the set