Let B be the matrix associated to the Arnold map, so B =[1 1;1 2] . Then with some effort you can show that B^48 mod 576 is equal to the identity matrix. This explains the result.
So you can write B as B=invXDX where X is the eigen matrix with corresponding eigenvalues on the diagonal of the diagonal matrix D. It is then easy compute B^m you then need to find an m such that B^m mod size of your matrix is equal to the identity matrix. So this gives you equations to find the periodicity
When you iterate you only have to compute D^m D is a diagonal matrix so that is pretty easy to compute. X and invX stay the same. More specifically B^m=invX D^m X
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Do you have by any chance examples of practical applications of this kind of maps? Thanks!
PS: I love the cat output :D
I just saw that this transform is applied to encrypt images. Attaching the researchgate link to a related publication: https://www.researchgate.net/publication/309463059_Arnold's_Cat_Map_Algorithm_in_Digital_Image_Encryption
I am yet to see other applications in real world.
Thanks! Don't hesitate to share anything you could you hear about future applications of this.
Okay 😃
Thank you😻.
Reminds me of my old TV. ;-)
We used to call those noise patterns as grains 😂
What is the size of the matrix s for your cat jpg? I am guessing that it has just the right size for periodicity to occur.
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the original size was 576x640. But for simplicity of coding, I resized it to 576x576.
that is a right guess. there are some bounds for this periodicity. From https://arxiv.org/pdf/1111.2984.pdf
![](https://steemitimages.com/640x0/https://cdn.steemitimages.com/DQme1mQX6Z3iVJ4LXw1WpacSNNKfmkJd54oLMGeEvngNNmA/image.png)
And some samples:
Let B be the matrix associated to the Arnold map, so B =[1 1;1 2] . Then with some effort you can show that B^48 mod 576 is equal to the identity matrix. This explains the result.
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Oh ok. That is a nice find. But is there a general rule to find periodicity of any number sized image?
So you can write B as B=invXDX where X is the eigen matrix with corresponding eigenvalues on the diagonal of the diagonal matrix D. It is then easy compute B^m you then need to find an m such that B^m mod size of your matrix is equal to the identity matrix. So this gives you equations to find the periodicity
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Again to find m I will have to iterate till I see identity matrix right? So that means it is sometimes computationally intensive right?
When you iterate you only have to compute D^m D is a diagonal matrix so that is pretty easy to compute. X and invX stay the same. More specifically B^m=invX D^m X
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Ah so I guess it is 576. 576=48x12. Probably there is some kind of reason for this.
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