Let $M$ be a closed manifold, $f:M\to M$ be a homeomorphism, and $\phi_t:M\to M$ be a flow. .

The map $f$ is said to be (point)-transitive if some orbit $\lbrace f^nx:n\in\mathbb{Z}\rbrace$ is dense in $M$.

The flow $\phi_t$ is said to be (flow)-transitive if some flow line $\lbrace\phi_tx:t\in\mathbb{R}\rbrace$ is dense in $M$.

My first question is:

- is there any sufficient condition under which the time-1 map of a transitive flow is still transitive?

In the following we assume $f$ is (point)-transitive and consider a special class of flows: suspensions of $f$.

Let $r\le R$ be positive numbers and $c:M\to[r,R]$ be a continuous suspension function. Let

$M_c=\lbrace(x,r):0\le r\le c(x),x\in M\rbrace/\sim$ where $(x,c(x))\sim(fx,0)$,

and $f_t:M_c\to M_c$ represented by the time-translation $(x,r)\mapsto(x,r+t)$.

According to our definition, the suspension flow $f_t:M_c\to M_c$ is (flow)-transitive.

If $c\equiv1$, it is easy to see that the time-1 map $f_1$ is not transitive (as a homeomorphism on $M_1$). So my second question is:

- is the time-1 map $f_1:M_c\to M_c$ (point)-transitive whenever $c$ is not constant?

Any proof or reference are good. Thank you!

As suggested by Zarathustra, it is worth to point out that the time-1 map of constant suspension flow with $c=\sqrt{2}$ is also (point)-transitive.