TITLE: Equivalent definitions of topological weak mixing
QUESTION [8 upvotes]: A dynamical system $f:X\to X$ is said to be topologically transitive if for any two nonempty open sets $U,V$ there exists $n \in \mathbb{Z}$ such that $f^{\circ n}(U) \cap V \neq \emptyset$. The following two definitions of topological weak mixing can be found in the literature:

$f$ is topologically weak mixing if $f\times f$ is topologically transitive on $X\times X$ (e.g here)
$f$ is topologically weak mixing if it has no non-constant continuous (with respect to the topology) eigenfunctions of the shift operator. (e.g. here)

How are these definitions related? Can they be shown to be equivalent?

REPLY [4 votes]: Which are your regularity assumptions on $f$?
In general there are some subtle differences between the two definitions. If $X$ is a Baire space, then 1. implies (a slightly modified version of) 2. See on this Theorem 2.3 in H.B. Keynes and J.B. Robertson, Eigenvalue Theorems in Topological Transformation Groups.
In the same paper, Theorem 2.5 provides conditions making the two definitions basically equivalent. Theorem 2.8 shows instead a case of non-equivalence.
Notice that:

the mentioned paper concerns the more general setting of topological transformation groups;
the authors mean "almost everywhere" in a topological sense (= everywhere except on a meagre set);
they allow (unless differently specified) a meagre discontinuity set.