Return to Answer

Just to clarify, the set of recurrent points may not be closed. There exists some transitive homeomorphism, whose uniquely ergodic measure is a Dirac measure. For example start with an irrational vector field $X$ on $\mathbb{T}^2$ and put a stop at $o\in\mathbb{T}^2$. That is, $Y=f\cdot X$ with $f(o)=0$. Then let $\phi_1$ be the time-1 map of the flow induced by $Y$. We can choose $f$ such that $\delta_o$ is the only invariant measure of $\phi_1$, and $\phi_1$ is transitive. In particular the set of recurrent points are dense on $\mathbb{T}^2$.

Edit: See the following paper for some flow-version examples. In particular see Proposition 1 and 2 there.

Just to clarify, the set of recurrent points may not be closed. There exists some transitive homeomorphism, whose uniquely ergodic measure is a Dirac measure. For example start with an irrational vector field $X$ on $\mathbb{T}^2$ and put a stop at $o\in\mathbb{T}^2$. That is, $Y=f\cdot X$ with $f(o)=0$. Then let $\phi_1$ be the time-1 map of the flow induced by $Y$. We can choose $f$ such that $\delta_o$ is the only invariant measure of $\phi_1$, and $\phi_1$ is transitive. In particular the set of recurrent points are dense on $\mathbb{T}^2$.

Edit: See the following [paper][1] for some flow-version examples. In particular see Proposition 1 and 2 there.

[1]: R. Saghin, W. Sun, E. Vargas. On Dirac Physical Measures for Transitive Flows. Commun. Math. Phys. 298, 741--756 (2010). https://doi.org/10.1007/s00220-010-1077-9

Just to clarify, the set of recurrent points may not be closed. There exists some transitive homeomorphism, whose uniquely ergodic measure is a Dirac measure. For example start with an irrational vector field $X$ on $\mathbb{T}^2$ and put a stop at $o\in\mathbb{T}^2$. That is, $Y=f\cdot X$ with $f(o)=0$. Then let $\phi_1$ be the time-1 map of the flow induced by $Y$. We can choose $f$ such that $\delta_o$ is the only invariant measure of $\phi_1$, and $\phi_1$ is transitive. In particular the set of recurrent points are dense on $\mathbb{T}^2$.

Edit: See the following paper for more detailed examples. In particular see Proposition 1 and 2 there.

Just to clarify, the set of recurrent points may not be closed. There exists some transitive homeomorphism, whose uniquely ergodic measure is a Dirac measure. For example start with an irrational vector field $X$ on $\mathbb{T}^2$ and put a stop at $o\in\mathbb{T}^2$. That is, $Y=f\cdot X$ with $f(o)=0$. Then let $\phi_1$ be the time-1 map of the flow induced by $Y$. We can choose $f$ such that $\delta_o$ is the only invariant measure of $\phi_1$, and $\phi_1$ is transitive. In particular the set of recurrent points are dense on $\mathbb{T}^2$.

Edit: See the following paper for some flow-version examples. In particular see Proposition 1 and 2 there.

Just to clarify, the set of recurrent points may not be closed. There exists some transitive homeomorphism, whose uniquely ergodic measure is a Dirac measure. For example start with an irrational vector field $X$ on $\mathbb{T}^2$ and put a stop at $o\in\mathbb{T}^2$. That is, $Y=f\cdot X$ with $f(o)=0$. Then let $\phi_1$ be the time-1 map of the flow induced by $Y$. We can choose $f$ such that $\delta_o$ is the only invariant measure of $\phi_1$, and $\phi_1$ is transitive. In particular the set of recurrent points are dense on $\mathbb{T}^2$.

Just to clarify, the set of recurrent points may not be closed. There exists some transitive homeomorphism, whose uniquely ergodic measure is a Dirac measure. For example start with an irrational vector field $X$ on $\mathbb{T}^2$ and put a stop at $o\in\mathbb{T}^2$. That is, $Y=f\cdot X$ with $f(o)=0$. Then let $\phi_1$ be the time-1 map of the flow induced by $Y$. We can choose $f$ such that $\delta_o$ is the only invariant measure of $\phi_1$, and $\phi_1$ is transitive. In particular the set of recurrent points are dense on $\mathbb{T}^2$.

Edit: See the following paper for more detailed examples. In particular see Proposition 1 and 2 there.