DIAGONAL POINTS HAVING DENSE ORBIT

Let f : X -> X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m epsilon N, f x f(2) x ... x f(m) : X(m) -> X(m) is transitive, (ii) for each m epsilon N, there exists x epsilon X such that the diagonal m-tuple (x, x, ... , x) has a dense orbit in X(m) under the action of f x f(2) x ... x f(m). We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying (ii).