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).

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).