Linear operators with wild dynamics

If $X$ is a separable infinite-dimensional Banach space, we construct a bounded and linear operator $R$ on $X$ such that

$$A_R=\{x \in X, \Vert R^tx\Vert \rightarrow \infty \}$$

is not dense and has a non-empty interior with the additional property that $R$ can be written $I+K$, where $I$ is the identity and $K$ is a compact operator. This answers two recent questions of Hájek and Smith.