Endohomeomorphisms decomposing a space into disjoint copies of a subspace

Abstract:

The existence (conjectured by R. Levy in a private communication) of a space X and an endohomeomorphism, f, of $\beta X$, such that $f[X] = \beta X\backslash X$ is demonstrated. It is shown that if G is one of the topological groups ${{\mathbf {2}}^\alpha },{{\mathbf {Q}}^\alpha },{{\mathbf {R}}^\alpha }$ or ${{\mathbf {T}}^\alpha }$, where $\omega < \alpha$, then G has a dense C-embedded subgroup H and an autohomeomorphism, f, such that G is the union of disjoint sets, ${A_0}$ and ${A_1}$, where for $\{ i,j\} = \{ 0,1\} f[{A_i}] = {A_j}$, and ${A_i}$ is a union of cosets of H.