to some natural class of measures related to the structure of the underlying space $X$. Especially significant is the requirement that $\mu$ should have *full support* ( $\mu (V)>0$ for every open set $V\neq\emptyset$) since in this case any ergodicity property implies its topological counterpart. There is, of course, a huge literature on these matters since the classical work of Oxtoby and Ulam ([@OU]).

In recent years, the above problem has received a lot of attention in the specific setting of *linear* dynamics, when the transformation $T$ is a continuous linear operator acting on a topological vector space $X$ ([@F], [@BG2], [@BG3], [@BoGE], [@Sophie]). The main reason is that people working in linear dynamics are mostly interested in studying *hypercyclic* operators, operators having dense orbits. When the space $X$ is second-countable, it is very easy to see that if a continuous map $T:X\to X$ happens to be ergodic with respect to some Borel probability measure $\mu$ with full support, then almost every $x\in X$ (relative to $\mu$) has a dense $T$-orbit. (In fact, one can say more: it follows from Birkhoff’s ergodic theorem that almost all $T$-orbits visit every non-empty open set along a set of integers having positive lower 