Open mrpenguln2 opened 2 years ago
3b1b
commented
2 years ago Hi Rob, this looks very interesting, I had not been familiar with symbolic dynamics before. And thanks for sharing all the links, those provide helpful context.
Do you have any favorite examples of instances where there's a difficult problem (but where it's accessible to phrase the problem) where it's hard to make progress, but once one introduces ideas of symbolic dynamics to the picture a clean solution emerges?
mrpenguln2
commented
2 years ago Hey Grant, thanks for the question! A few things come to mind...
Most people (who have some familiarity with the world of math) have probably heard of dynamical systems through its examples of chaotic systems: the logistics map, strange attractors, pretty fractals, etc. But when one is analyzing a particular dynamical system, it can actually be pretty tricky or tedious to check if the system is actually, mathematically 'chaotic' (whatever that means formally).
When one is able to find a correspondence from the given system to a symbolic system, it becomes much easier to analyze and ultimately answer all those questions. Which points are periodic with period $n$, how many such points are there, is the system topologically mixing, what is its entropy, etc.
In the general setting, one likely has to talk about piecewise formulas, slopes, intersections of graphs, etc; while in the symbolic setting, one only needs to talk about sequences of symbols and which transitions/local patterns of symbols are allowed or disallowed by the dynamics.
The most basic introductory example is the map $T : [0,1) \to [0,1)$ given by $T(x) = 2x \bmod 1$ or equivalently $T(x) = 2x - \lfloor 2x \rfloor$. If one considers how $T$ acts on the base 2 expansion of $x$, then suddenly all its dynamics are illuminated. The base 2 expansion of $x$ is an infinite (one-sided) sequence of the symbols $0$ and $1$, and the base 2 expansion of $T(x)$ is the very same sequence, just shifted one place to the left (and with the leftmost bit deleted). Which points are periodic under $T$? Those which have periodic base 2 expansions.
This correspondence from a general system to a symbolic system can be understood as an example of the more general idea of a Markov partition. Given a dynamical system $T : X \to X$, we cut up the space $X$ into finitely many discrete pieces. Then, for each point $x \in X$, consider its orbit: the sequence of points $x, T(x), T^2(x), \ldots$, and produce a sequence of symbols by tagging the $i$-th coordinate with which element of the partition that $T^i(x)$ belongs to.
By 'discretizing' the base space into finitely many (carefully selected) components, from orbits we derive a system of sequences of symbols, which is a homomorphic (sometimes isomorphic) image of the original system, and therefore illuminates many of its most fundamental dynamical properties.
See this paper (especially from page 20 on) for similar techniques applied to studying the bifurcations of the logistics map.
Another example similar to the one above (but which uses two-sided or bi-infinite sequences) is the Baker's map. The above mapping is non-invertible, which is why we only look at the forward orbit, but the Baker's map 'invertiblizes' it by considering a map on the unit square, stretching, cutting, and squishing it back into itself in an invertible way.
One of the most successful classical campaigns of Markov partitions for studying dynamics is the class of 'hyperbolic toral automorphisms', of which the cat map is an example.
fbunc
commented
2 years ago Wow, this is really interesting! I must check all the links you shared, I never heard of the term symbolic dynamics. I've been playing with emerging patterns (as a hobby) for the last years, and I'd really love to learn more with the correct guide.
Is this kind of structure of any use in the field?
About the author
My name is Rob, I am a doctoral candidate and instructor of record at the University of North Carolina at Charlotte. I have worked in higher math education for my entire adult life. My strengths include mathematical exposition, writing, and breaking down complex topics to make them more approachable and intuitive. My doctoral minor is in computer science and I have some basic programming skills, but I lack time, high quality equipment, and technical know-how to animate and edit videos.
Quick Summary
I want to bring some attention to the area of my dissertation research: symbolic dynamics (especially multidimensional symbolic systems). This particular niche is often hidden (unnecessarily, I think) beyond many graduate level prerequisites, including topology and measure theory. But, most of the basic foundational elements can be understood by any undergrad! In the symbolic framework, one only needs to talk about sequences of symbols (in the case of dimension 1), labelled/colored lattices (in the case of multidimensional systems), and the combinatorics of which patterns of symbols are permissible and which are not in a given system.
The role of symbolic systems in the broader study of topological dynamical systems is analogous to the role of lab mice in the broader study of mammalian biology: symbolic systems are "clean", abstracted frameworks which are tractable and easier to study than general dynamical systems. Symbolic dynamics also has connections to fields outside of math, especially statistical mechanics in physics and theoretical computer science.
For example, Conway's Game of Life can be seen as a shift of finite type (SFT) over $\mathbb{Z}^2 \times \mathbb{Z}$ (two spatial dimensions and one time dimension), which means that the allowed configurations are determined by a finite number of 'local rules'. SFTs are the basic example of tractable, well-behaved symbolic systems, and they alone span a wide class of interesting examples (and are the focus of my own research).
Target medium
I want to make a video, more conversational in tone than strictly educational, which takes the viewer through some high-level descriptions of ideas, examples, and problems from symbolic dynamics. I am open to discussion about the particular route and content of the video, in fact I specifically request it! I think that the natural intrigue and curiosity of a novice will make a great vehicle for this discussion.
There is a lot one can explore in this area, as it is very old and mature, stretching back to Claude Shannon's seminal work developing information theory in the 1950s. At the same time, the area of multidimensional symbolic dynamics is fresh (since the 1990s) and wild, much different and far less understood than symbolic dynamics in dimension 1.
Here are some examples of topics which are included in/adjacent to symbolic dynamics in dimensions $d \geq 1$:
I am looking for someone who is interested to learn more about this topic and can bring it to life with good questions, animations and video edits. I am looking for someone who will collaborate creatively and analytically with regards to the presentation and style of the video.
More details
See this Scholarpedia page for a discussion of basic ideas and examples of symbolic systems in dimension 1.
See this paper for an admittedly drier discussion of ideas and examples of multidimensional symbolic systems.
See these slides which I made to introduce this area and present my own preliminary dissertation results at a dynamics conference earlier this year (this goes even beyond symbolic systems over $\mathbb{Z}^d$ all the way to symbolic systems over arbitrary countable amenable groups).
I wish there were more novice friendly resources I could post here, but one of my motivations to make this video is the lack of such resources in the first place.
Contact details
Feel free to friend/message me on Discord @ rob#1522. I am also in the SoME2 server, so it should be easy to find me.