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?




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In the following we assume $f$ is (point)-transitive and consider a special class of flows: [suspensions](http://en.wikipedia.org/wiki/Suspension_%28dynamical_systems%29) 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! 




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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.



Transitivity of a flow and its time-1 map
I think the answer for the second question is no. We can construct a counterexample as follows: 


Let $f\colon M\to M$ be an arbitrary transitive homeomorphism and $u\colon M\to\mathbb (0,1/4)$ be an arbitrary non-constant continuous function. Then, let's define $$c(x):=u(x)-u(f(x))+1,\quad\forall x\in M,$$
and consider the flow $f\_t\colon M\_c\to M\_c$ as above.


We claim $f\_1$ is not transitive. In fact, given any $x\in M$ and any $t\in (0,1/4)$ it holds:
$$f\_1(x,u(x)+t)=(x,u(x)+t+1)=(x,c(x)+u(f(x))+t)=(f(x),u(f(x))+t).$$
That means that every compact set $U\_t:=\lbrace (x,u(x)+t) : x\in M\rbrace \subset M\_c$ is $f\_1$-invariant and hence it cannot be transitive. Notice the function $c$ is not constant because $f$ is transitive $u$ is not constant itself.


