# Problem 21

## Primary tabs

Assume $\varphi _t$ $C$-dense. If $\nu$ is $\varphi _1$ invariant is $\nu$ $\varphi$-invariant?

## Tags

No, cf [1]

### References

1. [quas2011weak] Quas A, Soo T.  2011.  arXiv preprint arXiv:1110.1113.

### BHM

For an Axiom A basic set, C-dense means that the flow is mixing; I think that the terminology comes from the connection with the density of strong stable (contracting) leaves.

### Ponce and Varao PonceVarao

Ponce and Varao [1] recently proved a rigidity result that characterizes when an invariant measure for the time-1 map of a flow will be invariant measure for the flow.  Surprisingly, the result is very general and does not require hyperbolic properties for the flow.