How does this work in Arrival

No convoluted time travel logic is required to explain the events in Arrival.

The two keys to understanding the movie are that the alien language rewires the brain so it perceives all of time happening at once and the movie makers are playing with the chronology and linearity of the script (either to mess with the audience or to enhance the oddness of the alien perception of time).

Once you accept that your mind can see all of time at once, you no longer need any time travel to explain knowledge of the future. Louise knows the wife's dying words because of this, not because anyone has travelled in time.

But the director plays with the audience's understanding of time by unveiling the story in a non-linear sequence (which makes a point using movie technique about just how odd it is to see time this way). The family scenes with Louise and her daughter, for example, are seen at the start of the movie. The audience assumes they are in the past: they are not; they are from the future. It isn't clear whether the scene with Louise and the Chinese general at the book launch (which the audience assumes happens long after the main action) is a future scene or Louise's memory in the present of a future event (there are odd features in it about what she remembers from the past). I suspect the ambiguity is deliberate (again to highlight just how odd the alien perception of time is in terms that an observant cinema audience can feel and perceive).

All of this makes sense given the nature of the alien perception of time and none of it requires time travel. The director has masterfully used to art of cinema to muddle our normal perception of linear narrative to create some of the same oddness in our heads as an audience.

Ref - Arrival Explained

I believe that incredibly complex time travel movie was Predestination. While films like those involve actual time travel... Arrival involves no time travel at all.

Just like we can see our past and present simultaneously... Louise can see her past, present and future simultaneously. Why? Because she has learned to understand the alien language.

Ian mentions a theory about how immersing oneself in a language can rewire the brain. We can see that Louise has been really connecting with the alien language by trying to interpret the nuances of each of the symbols. She's immersing herself into the alien language and as a result her brain is getting rewired to perceive reality like the aliens. Here's the catch about the aliens and their language. They don't see time as linear.

Looks like the Chinese leader has eventually learned to understand the language too. So over the 18 months after the aliens leave he sees visions of the future where he gives Louise his number and message.

Since time is non-linear for Louise, she gets information from her future and uses it in her present. On the other side, Shang is probably also learnt the alien language (over 18 months) and is able to see time non-linearly. 18 months later, he understands the importance of giving his private number and the Mandarin message to Louise so that she can use it in the past to change Shang’s mind.

Both of them have combined the usage of their gift to solve a crisis.

From my perspective, one paradigm behind the time travel topic in Arrival is the idea of a static, predefined time, where beings travel through like on railtracks.

Still, I believe this is not th main point of Arrival. From my perspective, the main point is the idea how the languages we use determine the way we think, even determine what we are able to think. One example for this coming into my mind is the possibility to express politeness in the Japanese language (https://en.m.wikipedia.org/wiki/Honorific_speech_in_Japanese).

The „Gift“ of the Aliens is the universal language allowing those who are able to use it in a proficient way to think independent of time. This is how Louise has her visions of the future: she starts to dream and think in the universal language and by doing so she can express thoughts in any direction in time.

However, time or the chain of events happening in time are predetermined to become logical.