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Update: the question has now been asked on the main site - Applications of number theory in dynamical systems

I am considering asking a reference-request type question on MO about significant and/or recent applications of number theory in dynamical systems/nonlinear dynamics. I see possible overlap with arithmetic dynamics (interesting in its own right - see https://arxiv.org/abs/1806.04980 for recent progress), but I am angling more at traditional dynamical systems theory.

Note that Lagarias writes in The Unreasonable Effectiveness of Number Theory, Proceedings of Symposia in Applied Mathematics, Volume 46, 1992, American Mathematical Society in the Chapter Number Theory and Dynamical Systems: "Number theoretic problems have occurred repeatedly in dynamical systems". Is this still true in the last twenty years or so? (In the question itself, I could summarize or at least list some topics, including small divisors, continued fractions, Farey sequences, Diophantine approximation, etc.)

Would the above be appropriate on MO? If not, how could the question be improved to make it suitable, if possible?

Proposed GRP 2019.08.24.

I am looking for references (or ways to find references) on significant and/or recent applications of techniques in number theory to problems in the areas of dynamical systems and nonlinear dynamics. While there may be overlap with arithmetic dynamics ([add your arxiv example here]), I want examples leaning more towards traditional dynamical systems.

[Add the paragraph on Lagarias's remarks here, followed by] A citation search using this reference was unrewarding. Other nonrecent papers which might yield a successful citation search would be welcome. Also useful search terms would be accepted.

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    $\begingroup$ Right now it looks too brief/too broad to be a good question. Also, what have you tried in answering it? What could the MathOverflow community do that a Web search could not? Are you looking for search terms? Gerhard "Reference Requests Are Often Specific" Paseman, 2019.08.20. $\endgroup$ – Gerhard Paseman Aug 20 at 14:03
  • $\begingroup$ @GerhardPaseman: I believe that the MathOverflow community might either be able to suggest how to target my search better or have one or more members who are up to date in the areas of dynamical systems and/or arithmetic dynamics and can point me in the direction of significant applications. I have found a chapter "Number Theory and Dynamical Systems" in The Unreasonable Effectiveness of Number Theory, but it dates back to 1991/1992. $\endgroup$ – J W Aug 20 at 14:18
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    $\begingroup$ I see my question has garnered a downvote. How should I interpret that? For instance, does it mean that my proposed question in its current form is regarded as inappropriate? I would prefer constructive critcism to help me improve my question, or failing that, clear advice that it is not a good question for MO and should be asked either elsewhere (MathematicsStackExchange?) or not at all. $\endgroup$ – J W Aug 20 at 16:30
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    $\begingroup$ Some users downvote meta questions as a way of answering "no". Personally I don't like that habit (and I think it's one of the little things that leads newcomers to find MO hostile). In my opinion, you should be upvoted for taking the care to ask on meta first, including by users who think your question on dynamical systems would be unsuitable for MO. If they think that, they can write it as a comment/answer. $\endgroup$ – Tom Leinster Aug 20 at 18:27
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    $\begingroup$ I trust @GerhardPaseman's judgement as a more experienced member of the community, but it sure seems to me like we've had plenty of successful questions of this sort. Nothing will stop people from downvoting a question if they don't like it or find it inappropriate, but, in general, I think that worrying about your question enough to seek guidance here on whether it's suitable and how to make it so is a good sign that you've thought about and have some feeling for community standards. My worth-probably-less-than-two-cents is that you should go for it. $\endgroup$ – LSpice Aug 21 at 16:42
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    $\begingroup$ The change improves it and makes it more specific. If you did a citation search on the Lagarias article and came up empty, then mention that. There are other edits, but I will let others suggest them. Give this a couple more days on meta before posting. Gerhard "Time Heals All Reference Requests" Paseman, 2019.08.21. $\endgroup$ – Gerhard Paseman Aug 21 at 17:32
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    $\begingroup$ @GerhardPaseman: Thank you for the feedback and advice. Indeed, my citation search does turn up relatively little, although I could have missed something. I am happy to wait a few more days before posting my question and would gladly receive further suggestions for edits. $\endgroup$ – J W Aug 22 at 9:16
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    $\begingroup$ I hope I have captured the intent of your question. I think the rephrasing above makes things more definite and more clear. I use [] for additional instructions. Gerhard "Thanks For Checking With Meta" Paseman, 2019.08.24. $\endgroup$ – Gerhard Paseman Aug 24 at 17:40
  • $\begingroup$ @GerhardPaseman: Thanks for the suggested rephrasing! Much appreciated. $\endgroup$ – J W Aug 26 at 5:47

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