Let $f:X\to X$ be a continuous map on compact metric space $(X, d)$. For $x\in X$ and open set $U$ of $x$, take $N(x, U)=\{n\in \mathbb{N}: f^{n}(x)\in U\}$. The point $x\in X$ is called positive upper density, if for every open set $U$ of $x$, $N(x, U)$ has positive upper density i.e. if $$\bar{d}(N(x, U)=\limsup_{n \to \infty} \frac{|N(x, U) \cap \{1,2,\dots,n\}|}{n},$$
then $\bar{d}(N(x, U))>0$
Let $X$ be a compact metric space without isolated point and dynamical system $(X, f)$ does not admit an invariant measure of full support.
If orbit of $x$ is dense in $X$, then it is not a positive upper density?