There is practical sense in the further restriction that \$$\phi^t\$$ should be an action of T over the smooth evolution functions.

That means:

* \$$\phi^0\$$ is the identity function
* \$$\phi^{s+t} = \phi^s \circ \phi^t\$$

The latter statement can be interpreted as follows. \$$\phi^s: M \rightarrow M\$$ gives the transformative effect after \$$s\$$ units of time. So the transformative effect after \$$s + t\$$ units of time equals the composition of the transformative functions for \$$s\$$ units of time and \$$t\$$ units of time, separately.

Note this is a semi-group action. Stipulating a group action is too strong a prescription for the general case, as it would require that, for each \$$t\$$ the function \$$\phi^t\$$ is invertible. That would rule out, for example, the evolution which collapses everything to a point after a fixed amount of time. Furthermore a group action would require that \$$\phi^t\$$ be defined for negative \$$t\$$, which is off the table for semi-flows.

From [Dynamical System](https://mathworld.wolfram.com/DynamicalSystem.html) at Wolfram:

> Technically, a dynamical system is a smooth action of the reals or the integers on another object (usually a manifold).