There are many results (usually connected to specification-like properties) about density of periodic measures in the space of all invariant ones. However some questions that seem to be easy (at first glance at least) always puzzle me.


Consider dynamical system generated by a continuous self-map $f$ of a connected compact topological manifold $M$.


Denote by $\mathcal{M}\_i(M)$ the set of $f$-invariant probability measures. 


Denote by $\mathcal{M}\_p(M)$ the set of measures of the form $(1/k)\sum\_{i=0}^{k-1} \delta\_{f^i (x)}$ where $x$ is a $k$-periodic orbit of $f$.


Denote by $\mathcal{M}'\_p(M)$ the set of measures of the form $\sum\_{i=0}^{k-1} \alpha\_i \delta\_{f^i (x)}$ where $x$ is a $k$-periodic orbit of $f$ and $\alpha\_i > 0$ and $\sum\_{i=0}^{k-1} \alpha\_i = 1$. 


My question: is there an example of $f$ such that one of the following holds?


1. the set of periodic points is dense in $M$ but $\mathcal{M}\_p(M)$ is not dense in $\mathcal{M}\_i(M)$
2. $\mathcal{M}\_p(M)$ is dense in $\mathcal{M}\_i(M)$ but the set of periodic points is not dense in $M$
3. $\mathcal{M}'\_p(M)$ is dense in $\mathcal{M}\_i(M)$ but $\mathcal{M}\_p(M)$ is not dense in $\mathcal{M}\_i(M)$



Density of periodic points and density of periodic measures
For (1) and (3), as mentioned by Christian, consider the identity map $I:S^1\to S^1$.


For (2), consider the map $f:S^1\to S^1$ fixing $1\in S^1$, and $f^n x\to 1$ (as $n\to\pm\infty$) for all other points. Then $\mathcal{M}\_p=\mathcal{M}\_i=\{\delta\_1\}$, but there is no other periodic point beside $1\in S^1$. 


