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Given the following standard definitions of the two concepts, I fail to see how the chronological future differs from the future domain of dependence?

Chronological future: The chronological future of $S$, denoted $I^+(S)$, is the set of points that can be reached from $S$ by following a future-directed timelike curve.

Future domain of dependence: Let $(\mathcal{M},g)$ be a spacetime and $S \subset \mathcal{M}$ such that no two points on $S$ can be connected by a timelike curve (we also say that $S$ is achronal). The future domain of dependence of $S$, denoted $D^+(S)$ is the set of all points $p \in \mathcal{M}$ with the property that every past-directed inextendible (i.e. with no endpoints) timelike curve starting at $p$ intersects $S$.

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    $\begingroup$ It is the word every that makes them different. In the first definition it says that a point can be reached by a timelike curve, but it doesn't mean that every timelike curve that goes through it must also go through S. $\endgroup$
    – MBN
    Jun 25, 2020 at 12:03

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The chronological future is the light cone. It's a cone that expands outwards as time goes on, because it consists of all the events that $S$ can influence. In other words, the events in $I^+(S)$ have at least one point of $S$ in their past:

(Taken from Hawking & Ellis, The Large Scale Structure of Space-Time)

On the other hand, the domain of dependence, also called the Cauchy development, consists of those events that are completely determined by what happens in $S$, because all their past intersects $S$. This means that it's a cone that closes: as you go far enough into the future you can't predict anymore, because information from events outside $S$ has had enough time to arrive:

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