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In a recent lecture on ergodicity and many-body localization, the presenter, Dmitry Abanin, mentioned that it is possible for a classical dynamical system to be chaotic but still fail to obey the ergodic hypothesis, which is frankly a pretty remarkable combination of properties. Unfortunately, the lecture had a lot of ground to cover and Abanin did not elaborate.

So:

  • Are any explicit examples known that have been shown to be both chaotic and non-ergodic?
  • Is there some clear explanation for what properties of those systems allow them to show this behaviour?
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  • $\begingroup$ Maybe the two first chapters of this: arxiv.org/abs/1509.06411 can give some hints for research and also this : arxiv.org/abs/nlin/0411062 for more info. $\endgroup$ – Constantine Black May 12 at 16:42
  • $\begingroup$ As an anecdote, I used to work with a mechanical test rig whose dynamics were supposed to be non-chaotic, but which in reality also had two different chaotic modes of operation, and it would flip randomly between the three states with very little provocation - e.g somebody walking around the room was often enough. We had no interest in whether it was or wasn't an ergodic system, but if the question had been asked, it would have been hard to believe that it was ergodic! $\endgroup$ – alephzero May 12 at 21:09
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A trivial example of a non-ergodic, chaotic system is a 2D conservative system that is not fully chaotic, i.e., with a mix of regular and chaotic regions in its phase space: each individual chaotic region is ergodic in itself, but since trajectories cannot cross the regular, invariant barriers between those regions, the systems as a whole is not ergodic.

An example of such a system is Chirikov's Standard Map:

enter image description here

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