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Hello frNdZz!! ...... I recently came across a important branch of studying dynamic systems .... ERGODIC THEORY .... I had started with some pdf from the internet but they're too complex so I need ya guys to help me out.....

HOW DO I START. ..... Can someone guide me through the pre requisite material required ?....

kindly reply 😊

Note by Abhinav Raichur
2 years, 10 months ago

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How about this one? I think it's a nice introduction, explaining ergodic theory from a mathematical standpoint, and then later progresses to applications in dynamical systems.

Ergodic Theory

I agree with you, almost everywhere you look, ergodic theory gets real dense right off at the start, requiring much more than basic skills in calculus. Michael Mendrin · 2 years, 10 months ago

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@Michael Mendrin I went through a few pages.... I think I can grasp it 😃... Thanks.... But it is still not clear as to what ergodic th contains... What are your views on it 😐? Abhinav Raichur · 2 years, 10 months ago

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@Abhinav Raichur Can you restate your question more clearly? Michael Mendrin · 2 years, 10 months ago

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@Michael Mendrin i mean ..... what is ergodic theory all about ? (im asking for your view particularly)........ and also, if you were my teacher how would you tell me to approach it. Abhinav Raichur · 2 years, 10 months ago

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@Abhinav Raichur Ergodic theory is one of the hardest things to try to explain in elementary terms, but I can try. First, let's start with the origin of the term, specifically the "ergodic hypothesis". If you recall, Boltzman, in developing his statistical interpretation of entropy and thermodynamics, envisioned a dynamical system as something constantly moving from one state to another, so he looked at "all the states the system could at any time be in", and considered its abstract hyperdimensional "volume", which in fact grew over time. Thus, the genesis of his famous entropy law \(S=kLog(W)\), where \(S\) is entropy, \(k\) is Boltzman's constant, and \(W\) is the number of "available or accessible microstates", that's related to this hyperdimensional volume. Thus was born the beginning of the "ergodic hypothesis", which supposes that the time a dynamical system spends in any particular volume of microstates is proportional to that volume. As a real simple example, if a state of a system can be represented by any one of \(52\) cards in a deck, then the percentage of the time that the cards are spades is \(1/4\). Okay, that was only just the beginning . Since then, the idea of looking at the behavior of an [extremely] complex system in terms of the "different states" that it can be found in, and how those states can be represented by a kind of an abstract "mathematical cloud" that evolves over time, has become very popular and fruitful over the past century since Boltzman's seminal work in statistical mechanics, and it's not limited to just physical systems. It's found applications in pure mathematics.

As an example, we could imagine, say, a complicated planetary system, which, in spite of Isaac Newton's idealization of it as some kind of beautiful and precise "clockwork", is in fact slightly chaotic. Suppose we plot all of its possible states in some configuration space. We would have a kind of a volume or a cloud in that configuration space, which, in fact, over a very long period of time, can change and evolve over time. Ergodic theory attempts to analyze the behavior of this cloud over time. Many other examples exists in chaotic theory, where something seemingly random isn't totally random---patterns show up when states are plotted, and we see "strange attractors" , i.e., unexpected structures when we'd normally be expecting featureless scatterplots. It's a difficult field of study, but after a while you get the idea that nothing is truly as random as you think. It's really hard to achieve perfect randomness---some order is happening, if you know where to find it, and that order evolves over time. Michael Mendrin · 2 years, 10 months ago

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@Michael Mendrin Fantastic! I never believed I could get it (at least the elementary stuff) .... Thank you very much for taking all the time in writing this ..... * you gave me the best gift on my birthday* !!!! Abhinav Raichur · 2 years, 10 months ago

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