We show that all rank-one transformations are subsequence boundedly
rationally ergodic and that there exist rank-one transformations that are not
weakly rationally ergodic.Comment: Added references, minor update
We define the anisotropic Sobolev spaces as Hs1β,s2β(MΓN)={gβL2(MΓN):β₯gβ₯Hs1β,s2ββ=β₯gβ(ΞΎ,Ξ·)[(1+ΞΎ2)2s1ββ+(1+Ξ·2)2s2ββ]β₯L2(MβΓNβ)β2, in Hs,s(RΓT) if l=3 and s>2, in Hs,s(TΓT) if l=3 and s>819β, and in Hs,s(RΓT) if l=5 and s>25β. All four results require the initial data to be small