Talk:Arnold's cat map

message: In the "properties"-section: the term "preperiodic" point is used. Please give an explanation, link or reference clarifying its meaning (or correct it if it was a typing error).  — Preceding unsigned comment added by 193.190.253.144 (talk) 08:28, 7 May 2013 (UTC)
thread_title: What is a preperiodic point?

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

message: What is \mathbb{T}? —Centrx→talk • 16:47, 7 April 2007 (UTC)
replies: text: \mathbb{T}^n is the standard notation for the n-torus.--Pokipsy76 08:45, 18 April 2007 (UTC)
replies: text: Okay. —Centrx→talk • 19:56, 24 April 2007 (UTC)

text: The cross reference to quotient space near the start of this article refers to Quotient Space (linear algebra). Is that what was intended? By comparison the article regarding n-torus refers to the topological Quotient space. Daleswift (talk) 23:10, 4 October 2012 (UTC)
thread_title: T-space

message: Concerning the picture that illustrates Poincare recurrence, the short time recurrences observed are not typical of chaotic systems. Actually, they are observed only because the trajectories used are all periodic (strictly speaking there is no chaos in the simulations). If chaotic trajectories are used the period increases dramatically. To clarify this point and estimate the recurrence times we made the following web page in our group. I think a link to this web-page and/or the use of some of its material would be useful in the main article.
Edugalt 13:03, 19 July 2007 (UTC)
replies: text: Thanks, have added a link. Would be good to have some of the content in the article itself. However, don't think we can really classify the trajectories of the integer cat-map as non-chaotic because these have a recurrence time scaling with the linear dimension of the area being mapped. On the contrary, I would argue that chaos in integer 2D maps of size N x N typically manifests itself as a number of orbits and cycle times per orbit both scaling with N. JocK 08:54, 20 July 2007 (UTC)
thread_title:  Recurrence time and initial conditions 

message: I think we need to split the article in two section, one for the continuous map and one for the discrete one because the actual form seems a little bit confusing. For example it seems to say that the continuous map becomes the identity map after a finite number of steps, while this is true just for the discrete one.--Pokipsy76 07:45, 30 July 2007 (UTC)
replies: text: Both the real and the integer cat-map exhibit Poincaré recurrence. The consequence of this is that integer cat maps yield the identity map after a finite number of steps, whilst real-valued cat-map after a finite lapse of time return to a state arbitrarily close to the initial state. I.e. a "noisy" identity map is obtained, the noise level of which can be made arbitrarily small by cycling through more and more "close recurrences". having said that, I agree that the discussions on the continuous and discrete cat map need a bit of a re-write to make them more consistent. If I find the time, I will try and make some edits to accomplish that. Regards, JocK 19:21, 31 July 2007 (UTC)
replies: text: Mmmmmh.... I don't think that a "Noisy identity map" is obtained and I will explain why. Poincaré recurrence theorem says that for the (continuous) cat map we have this: if we fix ε then for every point x there exists a number N(x,ε) such that iterating the map on x for N(x,ε) times will give a new point that is close to x less than ε. But the number N(x,ε) depends on the point x! There is not a number N(ε) depending only on ε such that the iteration of the map for N(ε) times on *every* point x is close to x less than ε!!--Pokipsy76 14:43, 1 August 2007 (UTC)

text: I agree that a separation is needed. I think the more relevant case is the continuous version, what is not clear from the present version. I would say that one paragraph at the end, with reference to the picture, should be enough for the discrete case. Moreover, I don't know what the terms "mixing" and "chaotic" mean for the discrete case. Concerning the Poincare recurrences, I would say they are trivial in the discrete case since all initial conditions correspond to periodic orbits and the recurrence time is just the period. In the continuous time case the periodic orbits have zero measure and in this sense all initial conditions are chaotic. I would be extremely careful with the terms "noisy" and "random" since these terms seem to contradict the obvious deterministic nature of the map (Edugalt 13:03, 9 September 2007 (UTC)
thread_title:  Stronger separation betwen continuous and discrete map 

message: The image shows a behaviour of the discrete map which is not shared by the continuous map, therefore I think it should be moved on the appropriate section. Objections?--Pokipsy76 (talk) 14:46, 2 June 2008 (UTC)
replies: text: --Pokipsy76 (talk) 22:25, 7 June 2008 (UTC)
thread_title:  Image should be moved 

message: The color of the piece that is not moved in the first image should be changed, I think. It looks like the tan and blue overlayed, and it confused me for a while. After all, it's only the colors that show where the pieces are moved to, in this image, so they should be distinct. Salvar (talk) 15:48, 22 June 2008 (UTC)
thread_title:  First Image Colors 

message: Hi all,
In Arnold's original paper from 1967, the map is called \phi, not \Gamma, and its matrix is \begin{pmatrix} 1 & 1 \\ 1 & 2\end{pmatrix}, not \begin{pmatrix} 2 & 1 \\ 1 & 1\end{pmatrix}. This also makes the 2D diagram wrong. Of course this does not change anything to the principle, but the article does not describe the original Arnold's cat map.
In the hope that helps, --MathsPoetry (talk) 08:47, 30 July 2012 (UTC)
thread_title:  Historically incorrect 

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thread_title:  External links modified (January 2018)