Dynamical Systems & Symbolic Dynamics: Memory and Substitutions | Nathan Dalaklis

[@luciak9447]

Great video! Thanks for making it. :)

[@CHALKND]

Thanks Lucia! I’m happy you enjoyed it!!

[@ewfq2]

Hey I really like this video, but it actually is a bit too fast for me, which is a bit of a first for me. On the one hand it feels a nice intermediate level difficulty (rare, but appreciated), but the speed at which you speak makes it difficult to appreciate it. in other words, I like how this is not overly dumbed down and content-wise accessible for people like me with a decent conceptual grasp on maths, but it's difficult to let it seep in without due time to process. Just some feedback, because I think this channel deserves more viewers!

[@ewfq2]

Also I would personally love more videos on this specific intersection! (Dynamical Systems + Symbolic Dynamics + Topological entropy)

[@CHALKND]

Hi Jin Lin! I'm really glad you enjoyed this video, and I really appreciate the feedback. Looking back, I was definitely trying to cover way WAY too much material in a very short amount of time which makes this video more a survey of ideas or brief introduction than a teaching focused video. I have been thinking about doing more things related to Symbolic Dynamics in more detail lately, and I have done some other videos that explain some other places where these ideas come up as well! The most recent videos that I have done that use ideas from symbolic dynamics or relate to it are the ".999... = 1 and Fractal Geometry" video and the two part problem video about "Strange Circles" . So I would check those out if you enjoyed this one.

[@ewfq2]

Thank you for the pointers! I look forward to the new videos! I'll check out those you mention. If you have any recommended (accessible) reading on that triplet I mentioned, but isn't in the vid descriptions, I would love to hear!

[@CHALKND]

Sure thing! The go to text on Symbolic dynamics is called "An Introduction to Symbolic Dynamics and Coding Theory" by Douglas Lind and Brian Marcus. If you're looking for more information on Topological Entropy, the Symbolic Dynamics version is much more accessible, (and is in Marcus and Lind's text) but an overview of the general version can be found on the Wikipedia or Scholarpedia entries on Topological Entropy. I only really have had dynamical systems experience in Symbolic dynamics, but readings on Differential Equations tend to lead on to give more of a Dynamical systems 'flavor' to them the higher and higher you go with those.

[@ewfq2]

@@CHALKND Thanks so much for the recommendations, will check them out!

[@second_second_]

Thank you for this interesting video on this subject. I find it's hard to find videos or free books about this.. I would like to ask, is symbolic dynamics the same as finite automata? (Is there any difference?)

[@CHALKND]

That’s a great question! To my understanding they are not the same. However they are very closely related. In Symbolic dynamics there is no start or end state. Also, for a shift space (i.e. one kind of symbolic dynamic system) to be non-empty it’s strings must be infinite in length so while a finite automata can not accept an infinite string without also accepting infinitely many finite length ones (due to the requirement of finite states). Shift Spaces only “accept” infinite strings. You can however talk about a shift spaces underlying language. Which does include those finite strings, but then you’re talking about something that generates a symbolic system and not the system itself and languages of symbolic systems can get pretty hairy pretty quickly if you’re creative enough with how you decide to generate a shift space.

[@norahalhunayshil1762]

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