What is a topological dynamical system? The doubling map and other basics.

What is a topological dynamical system? Here we go over the basics of discrete dynamics of metrizable spaces, and we will give a handful of examples including the doubling map. Other examples come from two families of well studied cases, those of circle maps and directed graphs. After giving the definition of a topological dynamical system we go through some basic computations for these systems and then introduce the idea of topological conjugacy — the notion of equivalence we use to distinguish these systems along with an infinite family of conjugacy invariants. At the end of the video we will show that there are infinitely many non-conjugate circle maps.

The video can be broken up into the following chapters:
00:00 Intro
00:25 What is a topological dynamical system?
02:10 Some examples, The doubling map and directed graphs
05:15 Basic computations for topological dynamical systems
09:58 Why is the doubling map the "doubling" map
13:40 Where do we start in mathematics? Topological Conjugacy and Invariants
17:00 Count of periodic points of a certain period is a conjugacy invariant
18:40 There are infinitely many non-conjugate circle maps.

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[ruinenlust_]

This is SO interesting. Love the video!

[jpmourax]

You explained in a 21-minute video everything I studied as an undergraduate with my advisor lol. You are incredible! Unfortunately I gave up and ended up going into statistics and data analysis. But I still love topics related to topology/geometry.

[millamulisha]

I call ‘graphs of functions’ plots now, after I learned ‘graph theory’. 😅

Loved your video. 🥂

[NikolajKuntner]

I don't know if your research is more like analysis, measure theory, algebra or combinatorics.
Made me think, though: I don't know how common those notions are outside of computer science and philosophy debts, but, e.g. for describing machine system states, there are modal logics syntactically abstracting various notions into modalities.
E.g. if T:X->X and if P(x) is a predicate, you may by 〇P(x) denote the predicate P(T(x)), saying "P holds at the next state".
Similarly, you may let ☐P(x) denote the statement that "P holds for all future states".
So e.g. ¬☐¬P(x) claims that "it's not the case that for all future states, P is false". Or in other words (making use of some Markov's principle) it claims that there will be some future state where P holds. You write ◊P for ¬☐¬P, as is common in modal logic. On the "Temporal logic" Wikipedia page you see ♢ defined from a higher order operation (namely QuP or "Q holds until P holds")

More broadly, if t:(X x A)->(X x O) maps a given state and an action to a new state and am observation, you're already in some reinforcement learning type scenario (maybe think Markov decision process with observation queuing you in on some correlated consequence of your action) and can prove theorems with just logic and some extra symbols.
I remember when I learned this I unpromtedly came up with this basic propositional logic derivation
(¬B(x) ∧ ¬P(x)) → ☐¬B(x)
¬☐¬B(x) → ¬(¬B(x) ∧ ¬P(x))
◊B(x) → (¬¬B(x) ∨ ¬¬P(x))
◊B(x) → (¬B(x) → P(x))
(◊B(x) ∧ ¬B(x)) → P(x)
giving a theorem with the intended meaning of taking you from
"If you have no dark-squared bishop (B) and no pawn (P) on the board, then, till the end of the game (☐), you'll have no dark-squared bishop."
to
"If it's still possible that you can have a dark-squared bishop on the board
despite not having one now, then you have a pawn on the board."

Maybe this is somehow of use.

[LucasDimoveo]

Within 15 seconds the simple explanation was far beyond my comprehension 😅