This chapter presents several further topics concerning recurrence and mixing. First is a direct construction (based on the theory of almost periodic functions) of eigenfunctions for m.p.t.s which are not weakly mixing. The second section presents the purely topological analogues of recurrence, ergodicity, and weak and strong mixing, and the third introduces Furstenberg's theory of multiple recurrence and his proof of the Szemeredi Theorem. In 4.4 we give the Jewett–Bellow–Furstenberg proof of the existence of uniquely ergodic topological representations of ergodic m.p.t.s. The final section examines two examples of weakly mixing m.p.t.s which are not strongly mixing: an example of Kakutani (which we show is not even topologically strongly mixing), and one of Chacon, which is also prime (i.e. without proper invariant sub-σ-algebras). as was discovered by del Junco.