Talk:Poincaré recurrence theorem

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Contents 1 is this topological or measurable dynamics? 2 Expanding the article 3 This doesn't seem right 4 questionable paragraph

[edit] is this topological or measurable dynamics?

I vaguely recall that the poincare recurrence theorem is a statement in topological dynamics, not measurable dynamics. But I might be wrong. Haven't thought about this stuff for a while. Dmharvey Talk 01:42, 27 August 2005 (UTC)

I vaguely recall a defintion based on measure. I don't think they had topological dynamics in Poincare's time. But I dunno. I'll keep my eye out for a reference. linas 00:56, 30 August 2005 (UTC)

And BTW, have you seen the baker's map images? They're stunning, I remember the first time I saw them. First umpteen-ten-or-hundred-thousand iterations, you have pure, unadulterated white noise. And then, all of a sudden, wham, the original image re-emerges (although its tainted by speckles.) Very dramatic. The speckles are clearly show the recurrance is "measure based". linas 01:01, 30 August 2005 (UTC)

[edit] Expanding the article

Hi, all

I took the liberty to start expanding this article. It definitely needs much more. On the other hand, the last paragraph contains material which is a result of my own thinking about the subject, so it may not be appropriate for inclusion in WP. I hope there are people here who can judge it better.

Dmanin 07:53, 26 November 2005 (UTC)

I think it is appropriate to have a formal statement of the theorem. I don't know the formal statement -- I did once upon a time -- and I'd like to see it there. Dmharvey 14:21, 27 November 2005 (UTC)

I agree completely. Dmanin 23:40, 27 November 2005 (UTC)

[edit] This doesn't seem right

Hi.

I noticed this:

"In mathematics, the Poincaré recurrence theorem states that a system having a finite amount of energy and confined to a finite spatial volume will, after a sufficiently long time, return to an arbitrarily small neighborhood of its initial state."

But this does not seem to work. For example, consider the simple system given by iterating the function f(z) = z/2, in, say, the unit disk of the complex plane. (This can be expressed as a function of time as f(z, t) = z/(2^t), which generalizes it to continuous time as well.) This is confined to a finite amount of (complex) space, and yet all points inevitably tend toward zero, and except for zero itself, never return to an "arbitrarily small neighborhood" of the initial state. The unit disk shrinks inexorably to zero under the iteration. But Poincaré's theorem is true (otherwise it would not be a theorem), so I must be missing something here. Does the "energy" of the system need to be conserved in order for this to work, and so does my system above not have a constant "energy" (what is the general, mathematical definition of "energy" for a dynamical system, anyway?) with respect to time t? If so, then should not the statement of the theorem in the opener be modified? mike4ty4 20:37, 11 September 2007 (UTC)

The first line is a rather imprecise formulation of the theorem, meant to be accessible to as many people as possible. The precise statement, later on, explains that the map has to be volume-preserving. In your example, the map shrinks sets and thus Poincaré's theorem does not apply.

I understand that the first sentence of the article, that you quoted above, is misleading, so I rewrote it. I hope that helps. -- Jitse Niesen (talk) 07:51, 15 September 2007 (UTC)

[edit] questionable paragraph

This paragraph:

One possible way of reconciling entropy and recurrence is the following. Poincaré's theorem hinges on the fact that phase trajectories don't intersect. But this premise breaks if there is environment-induced noise in the system. Roughly speaking, environment influence introduces a timescale for the duration of the period for which the system can be considered isolated. If the system is chaotic, this isolation timescale grows only logarithmically with decreasing noise level. Hence, Poincaré recurrence timescale has to be compared with this isolation timescale (and not with such an extrinsic timescale as human lifespan). The limit of large system (recurrence timescale ) which is perfectly isolated (isolation timescale ) is therefore ill-defined, and since isolation timescale grows "very slowly", while recurrence timescale grows "very quickly", physically speaking, we can't have large isolated systems. That is, "large isolated system" is a result of two idealizations, which depends on the order in which they are applied, and realistically, if the system is large, it can't be isolated for the purposes of proving the recurrence theorem.

Seems to me to be questionable, and in any case is not sourced to any citation, and so unless a citation is given we'd have to treat it as a piece of OR. So I will remove it (someone can add it back in if they have a proper cite). --SJK (talk) 06:38, 20 April 2008 (UTC)