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Abstract

Let $f:X\to X$ 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 $m\in \mathbb{N}$, $f\times f^2\times \cdots \times f^m:X^m\to X^m$ is transitive, (ii) for each $m\in \mathbb{N}$, there exists $x\in X$ such that the diagonal $m$-tuple $(x,x,\ldots, x)$ has a dense orbit in $X^m$ under the action of $f\times f^2\times \cdots \times f^m$. 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).