Let's say I have a function that takes some real argument $t$ and some other argument ${\bf x}_0 \in D$, that returns ${\bf x}_1 \in D$ (for some domain D).
${\bf x}_1 = f({\bf x_0}, t)$
I'm interested in functions of this form that have the property where composing them with themselves is equivalent to evaluating them once with the sum of the $t$ arguments, so we get this identity:
$f\left( f( {\bf x_0}, t_1), t_2 \right) = f( {\bf x_0}, t_1 + t_2)$
But I don't know if this property has a standard name or way of describing it.
tl;dr is above; below, I show some examples and motivate why I'm looking for this property.
Here are two functions with this property:
$\require{cancel}\begin{align} g(a, t) &= a + ct\\ g(g(a, t_1), t_2) &= (a + ct_1) + ct_2\\ &=a + c(t_1 + t_2)\\ &= g(a, t_1 + t_2)\\ \\ h(a, t) &= a + (b - a)(1 - d^t)\\ h(h(a, t_1), t_2)&= \left[a + (b - a)(1 - d^{t_1})\right] + \left(b - \left[a + (b - a)(1 - d^{t_1})\right]\right)(1 - d^{t_2})\\ &=a + (b-a)(1 - d^{t_1}) + (b-a)(1-d^{t_2}) - (b-a)(1-d^{t_1})(1 - d^{t_2})\\ &= a + (b-a)(1 \cancel{- d^{t_1}} \cancel{+ 1} \cancel{- d^{t_2}} \cancel{- 1} \cancel{+ d^{t_1}} \cancel{+ d^{t_2}} + d^{t_1 + t_2})\\ &= a + (b - a)(1 - d^{t_1 + t_2})\\ &= h(a, t_1 + t_2) \end{align}$
I'm coming at this from game development, where we often want functions with this property to step some initial game state ${\bf x}_0$ forward in time by a duration of time $t \ge 0$. In that application:
The single evaluation form $f({\bf x_0}, t_1 + t_2)$ corresponds to the game running at a low framerate, so we try to simulate a big jump forward in time all at once.
The composed form $f\left( f( {\bf x_0}, t_1), t_2 \right)$ corresponds to the game running at a high framerate, so we step the simulation forward in two consecutive frames over the same real-time interval, taking smaller sub-steps.
We'd like to get the same answer, regardless of how many steps we took along the way. This property of equivalence between the two forms gives us "framerate independence" (or approximately so, since in practice we have rounding errors and other complications that aren't relevant here), helping the game achieve a consistent look and feel on different hardware or under different load conditions. That makes this property desirable in our game simulation math, but I don't know the proper terms to discuss it mathematically, or if there's a neater/more standardized way to formalize it than the way I've presented it above.
Is there a standard name for this property, or functions with this property? Or a neater way to show that a function of a particular form must have this property than the algebra I used above?