Let $f \colon X \to X$ be a dynamical system. Last lecture we were interested in points $x \in X$ whose orbit was as small as possible: fixed points and periodic points. In this lecture we are interested in cases where the orbit is as large as possible.

Since an orbit $\mathcal{O}_f(x)$ is certainly at most countable (and most interesting metric spaces are not countable), it doesn't make sense to investigate points whose orbit is the entire space. Therefore we look at the next best thing: points $x \in X$ whose orbit $\mathcal{O}_f(x)$ is dense in $X$.

To wit, we introduce the notion of a transitive dynamical system. This is a dynamical system $f$ with the property that for any pair $U,V$ of non-empty open subsets, there exists $k \ge 0$ such that

$$f^k(U) \cap V \ne \emptyset.$$

The main result of today's lecture states that—under reasonable assumptions on the metric space $X$—a dynamical system is transitive if and only if it has a dense orbit.