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I have no background in any of these areas and was wondering whether these topics have any significant applications in applied mathematics and/or probability/statistics? From what I’ve read it seems ergodic theory has some application to statistical mechanics and thermodynamics? Are there other specific concrete examples and areas in applications where these areas in maths are used?

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One key application of topological/symbolic dynamics is to formally analyze the presence of chaos in a dynamical system. The key idea is this:

  1. Describe trajectories in the dynamical system using sequences of symbols.
  2. Determine all the symbol sequences associated with admissible trajectories.
  3. Most of the time, if any symbol sequence is an admissible trajectory, then chaos is present in the system.

This process, and its associated ideas like the Smale horseshoe, is basically the only way we can formally diagnose chaos in a system. That being said, topological/symbolic dynamics is an extremely powerful and general tool for making statements about qualitative properties of dynamical systems.

Ergodic theory is very closely related, and has key applications in understanding the behavior of many-body systems (i.e. what collective features should we expect given the physics associated with each individual constituent?)

At the risk of self-promotion, I’ll mention that these topics don’t just have to be applied to abstract mathematical problems—I recently published a paper using symbolic dynamics to discern the behavior of particles in peculiar microfluidic channels.

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