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Article is seriously deficient
editThis encyclopedia article is seriously deficient.
Instead of immediately launching into a technical discussion of flows that are ergodic, the article really ought to begin by defining what an ergodic flow is.
It seems that this article as it is currently constituted has forgotten the role of an encyclopedia, instead reading as though this is a survey article in a mathematics journal.
It is not.47.44.96.195 (talk) 17:39, 27 October 2020 (UTC)
- I agree. The opening paragraph should be retitled "overview", and preceded by an informal description of a flow (as a low-brow, non-mathematical description of the action of a continuous monoid on a smooth manifold. Avoiding the use of the word "monoid".) The definition of "ergodic" could be a one-sentence "space-filling curve", and then a link to ergodicity to explain what that means. The ergodicity article now provides a pretty good overview or ergodicity in fairly plain-simple terms, I think. (I re-wrote it last week to offer this hopefully "simple-enough" review.) The conclusion to this first paragraph should say "to discuss ergodic flow in a meaningful manner, it must be done using the formal definitions of geodesic, geodesic flow, hyperbolic manifold and the modular group." That's it, just one paragraph. I think that's enough - leave the reader with some warm fuzzies about things flowing. (Ideally, it would be possible to find some every-day example of ergodic flow. It's almost like, ummm, turbulent flow of water mixing into a tub. Yes, that's a flawed example in two or three different ways... (its high-dimensional, ... its dissipative...)... flow on a torus is easy to understand, but not very compelling... flow in a double pendulum might be a good example, its simple, intuitive, low-dimensional, and meets the strict mathematical definition ... and that article already has at least one nice graphic. It's just not obviously a "flow", but maybe with hand-waving, it is... Hmmm.... 67.198.37.16 (talk) 21:43, 8 November 2020 (UTC)
Translation surfaces are flat, but have ergodic flows. This follows from the fact that they can be mapped to Riemann surfaces. It would be interesting to somehow "explain" that it is not the metric, not negative curvature in-and-of-itself that is making things hyperbolic, but rather that ergodicity follows from there being diverging orbits. Hmmm. I guess one could also say that for a translation surface, you've taken all the curvature and shoved it around and mashed it down into a finite number of very singular points. ... that two geodesics passing on either side of such a point diverge. Not sure how to say this. Jacobi fields are not insightful, here... whatever ... (this article is not actually claiming that curvature has something to do with ergodicity, but it almost is... which is an almost-correct claim, except that the example of translation surface needs to be handled.) I'm not sure what point I am trying to make here, other than, .. it's an interesting special case... 67.198.37.16 (talk) 21:57, 8 November 2020 (UTC)
typo
editI'm pretty sure that pasing in "and, pasing to translates by Z, which by assumption are null sets, to Z-invariant null sets" is a typo. I'd guess for either passing or phasing, but it would be a complete guess. Anyone know? ϢereSpielChequers 17:53, 25 September 2022 (UTC)