What is an example of a compact manifold which does not admit a diffeomorphism with at least one dense orbit?


Moreover, is it true to say that every isometry of $\mathbb{C}P^n$ with the Fubini-Study metric do not possess any dense orbit?



Compact manifolds which do not admit a diffeomorphism with a dense orbit
Dolgopyat and Pesin proved ([ETDS 2002](https://doi.org/10.1017/S0143385702000202)) that "every compact manifold of dimension $\geq 2$ admits a Bernoulli diffeomorphism with non-zero Lyapunov exponents". That is, on any such manifold $M$ there is a $C^\infty$ diffeomorphism $f$ that preserves volume $m$ and has the property that $(M,m,f)$ is a measure-preserving transformation that is measure-theoretically isomorphic to a Bernoulli shift. In particular, by the Birkhoff ergodic theorem, $m$-a.e. point $x\in M$ has the property that its orbit equidistributes with respect to $m$, meaning that for every continuous $\phi\colon M\to \mathbb{R}$, we have $\frac 1n \sum\_{k=0}^{n-1} \phi(f^kx) \to \int \phi\,dm$ as $n\to\infty$. In particular, any such $x$ has a dense orbit.


