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I have recently been interested in going deeper into ergodic theory, beyond an introductory level of knowledge. Background wise, my training has mostly been in stochastic analysis, and I have a cursory knowledge of general dynamical systems and ergodic theory, say at the level of the book by Brin and Stuck for general dynamics, and Ward for ergodic theory.

I am particularly interested in those aspects of ergodic theory that intersect stochastic analysis, especially stochastic control and filtering. Some examples of papers that deal with this intersection that I have found interesting are Disjointness in Ergodic Theory by Furstenberg and Ergodicity, Decisions, and Partial Information by van Handel.

However, I would also be interested in more general aspects of ergodic theory, though I do not know enough about the field right now to list particular subareas of interest.

Thus my question is twofold :

  1. What landmark papers would you recommend to read for someone to get their feet wet in the field? I am especially interested in those with links to stochastic analysis, but if there are more general papers that you think would be important to read, please don’t hesitate to list them!

  2. What are the current main research directions in ergodic theory? I would really appreciate a short list, together with a brief description of the subfield.

Thanks in advance!

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    $\begingroup$ I have found the book of Eisner, Farkas, Haase and Nagel (Operator theoretic aspects of ergodic theory) to be very enlightening. It treats ergodic theory from the abstract point of view of functional analysis, which make a lot of arguments very easy to understand. The ergodic theory can then be applied to dynamical systems (measure preserving, topologic or stochastic) by means of composition operators called Koopman operators. Furthermore, it contains lots of interesting and open questions in the field. $\endgroup$
    – Julian Hölz
    Commented Jul 28, 2023 at 9:45

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  1. Concerning "landmark papers", Cosma Shalizi has documented their own pathway into ergodic theory, with an eye towards applications in statistical learning theory (but with many side branches). You might find their annotated bibliography helpful.

  2. Concerning research directions, I understand the question as being motivated by a plan to embark on original research at the interface of ergodic theory and stochastic analysis. For such a purpose it seems helpful if the topic is less mature, less well established, and perhaps less well covered by "landmark papers".
    One example: Ergodicity economics is a branch of mathematical finance at this interface. A foundational paper is by Ole Peters (2019). There are intriguing connections with machine learning.

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    $\begingroup$ Thanks for the informative answer! I indeed hope to do some work in the area sometime. $\endgroup$
    – Nate River
    Commented Nov 16, 2023 at 18:40

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