Talk:Arnold's cat map

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Are we sure that after a certain number of iterations the original image is restored (see Poincaré recurrence theorem)? The Poincaré recurrence theorem doesn't say that a set of states (i.e. points) terurn close to it's original form, it says that every point return close to itself but the returning time for any single point of a set is completely different.--Pokipsy76 21:26, 28 June 2006 (UTC)

If the individual points return in time t1, t2, t3, ... then the whole configuration returns in time t = t1 * t2 * t3 ...

It can be proven that the integer cat map returns in at most 3N discrete timesteps. JocK 12:46, 30 June 2006 (UTC)

But that product is infinity if you consider a set that contains an infinite number of points (as the image of the cat is).--Pokipsy76 12:55, 30 June 2006 (UTC)

In reality it isn't. However, based on your comments I have decided to limit to discussion to integer cat maps. The restoring of the image is trivial for such maps. JocK 13:37, 8 July 2006 (UTC)

Why the standard cat map on the torus shouldn't be covered by this article? If you want to limit the discussion to the integer cat map maybe you should rename the article.--Pokipsy76 14:38, 8 July 2006 (UTC)

You're right. I think the latest edits address this issue. JocK 18:22, 9 July 2006 (UTC)

But the "original" Arnold's cat map is that on the torus. The discrete map is a subsequent development of the origional idea. The original cat map on the torus should come first, and only after that we can introduce the further developments.--Pokipsy76 09:55, 10 July 2006 (UTC)

I think that is beside the point. The current formulation applies to continuous and discrete domains. If discrete, the domain is a toroidal grid. JocK 19:03, 9 November 2006 (UTC)

Discrete and continuous cat map need different formulations and we should first give the formulation of the continuous one because it is the mist important and most known in the dynamical systems theory. Moreover properties of the classical (continuous) cat map like Anosov diffeomorphism and mixing are not shared by the discrete one and are erroneously attributed to it in the article.--Pokipsy76 14:36, 15 November 2006 (UTC)

Can you substantiate your claims? Are you suggesting that the integer cat map has no clearly marked local directions of 'expansion' and 'contraction'? Also, virtually each article on the cat map illustrates the mapping using a discrete image of a cat (or otherwise). I can't see why the continuous map should 'come first'. Anyway, the current article covers both. having said that: feel free to incorporate your ideas. JocK 19:20, 15 November 2006 (UTC)

Anosov diffeomorphism is a diffeomorphism i.e. a particular map un a manifold with some regolarity properties like differentiability. The discrete cat map is not defined on a manifold and is not differentiable in any sense. Having been studying dynamical system since some years as a graduate student I have seen the continuos cat map a lot of times (every book introducing ergodicity and mixing cite it) but I never heard or read of the discrete cat map before. This is why I think the continuos cat map is the origina one and the most known.--Pokipsy76 07:29, 24 November 2006 (UTC)

Nice contribution, Claudio! JocK 18:57, 9 November 2006 (UTC)