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I have just begun reading Introduction to Dynamical Systems by Brin and Stuck, and I have some uneasiness about their definition of continuous-time dynamical system. Here's what they say:

My concerns are:

  1. There are two parts to the definition. For one, we need an indexed family $S = \{f^t: X \to X \}, t \in \mathbb{R}$ or $t \in \mathbb{R}^+_0$, that forms a group or a semigroup. Secondly, we need the indexing to be done in such a way $f^s \circ f^t = f^{s + t}$ and $f^0 =$ Id. This seems to imply that if we reindex $S$ differently so that $f^s \circ f^t$ may not always equal $f^{s+t}$, then we do not have a dynamical system anymore. Is my understanding correct? If so, this seems kind of strange, because by reindexing, we are not really changing $S$.

  2. I also gather that the set being a group or semigroup is entirely dependent on whether $t \in \mathbb{R}$ or $t \in \mathbb{R}^+$. That is, the set is strictly a group $\iff$ $t$ ranges over $\mathbb{R}$, and the set is a semigroup $\iff$ $t$ ranges over $\mathbb{R}^+_0$ Is this correct? Also, it seems like even when $t$ ranges over $\mathbb{R}^+_0$, there is still an identity. Why do they call it a semigroup and not a monoid?

Thank you very much.

Ovi
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  • @Dmitry I see. I guess the reason for caring about this map, and not just its image, will become apparent as I study the subject? – Ovi Aug 09 '20 at 00:29
  • I'm not sure I can confirm it... I guess this is the way to put it: $f^t$ shows how your system changes after $t$ time passes: it accepts an initial state and returns the new state after time $t$. So you do care about indexing: in the sense that it really matters if 1 second passed or 2 second passed. It also explains why composition is " additive". – Dmitry Aug 09 '20 at 00:32
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    Reindexing $S$ is a substantial change. The indexing reflects the time evolution. For instance, rescaling it would correspond to increasing or decreasing the rate of the process you're studying. Also, if the index doesn't matter, you might suppose $S$ doesn't have an index at all, but then you'd just have continuum-many functions from $X$ to $X$ floating around without much structure. As far as (2), you are correct. –  Aug 09 '20 at 00:32
  • Thank you both! I understand it better now. – Ovi Aug 09 '20 at 00:33