Is there a dynamical system such that every orbit is either periodic or dense?

In the following paper the authors give an almost 1-1 extension for a minimal system $(X,\mathbb{Z})$ which is transitive and the only non-transitive point is a fixed point.

For $\mathbb{N}$ action they can have a similar one with positive topological entropy.

T.Downarowicz, X. Ye: When every point is either transitive or periodic, Colloq. Math. 93 (2002) pp. 137-150.

I do not know whether hese examples can exists on manifolds.

I believe you will find such examples for $X=\mathbb{C}$ and $T$ a rational map in

Mary Rees, Ergodic rational maps with dense critical point forward orbit, Ergodic Theory and Dynamical Systems 4 (1984), 311-322. official version.

In my Ph.D. thesis, I showed that some of these even support a metric with respect to which these dynamical systems are ``hyperbolic''. This metric gives a notion of length of curves comparable to the usual metric on the Riemann sphere, but is defined by a function which is singular on a dense set of points on the sphere (the forward orbit of the critical point).

This reminds me of the theorem of Le Calvez and Yoccoz: There is no minimal homeomorphism on the multipunctured sphere, i.e. there is no homeomorphism on the 2-sphere such that every orbit is dense except a finite set. Clearly, the finite set consists of periodic points.

Now, to question. Google leads to https://arxiv.org/abs/1605.08873 This article is exactly the idea of Andrey Gogolev. This technique shows that there are diffeomorphisms on all orientable surfaces (of course, except the sphere) such that the non dense orbits form a finite set. I am sure that there are older proofs.