I am seeing the definition of flow in a metric space:
$f:M\times \mathbb{R}\rightarrow M$ is one flow if $M$ is metric space, $f$ is continuous and $f(x,t+s)=f(f(x,t),s)$.
Note that the condition does not require $f(x,0)=x$. My question is this: If $f_t(x):=f(x,t)$ is a homeomorphism for each t, Then we can conclude that $f(x,0)=x$ for all x?
If one can suggest examples of flows under this definition I will be very grateful.
Let us put $f_t(x)=f(x,t)$. First note that since $f$ is assumed to be a flow
$$f_0=f_t\circ f_{-t}=f_{-t}\circ f_t,$$
so that each time-$t$ map $f_t$ of the flow $f$ is invertible. Then as mentioned in the comments
$$x= f_0^{-1}\circ f_0(x)= f_0^{-1}\circ f_{0+0}(x) = f_0^{-1}\circ f_0\circ f_0(x)=f_0(x),$$
that is to say,
$$f_0=\text{id}_M.$$
Note that in the case of a monoid action one needs to explicitly state this property; see Dynamical System Cocycles: (Semi)Group vs. Monoid Definitions - Examples When is Distinction Important? .
