Timeline for cat map re-transformation
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 11, 2012 at 16:47 | comment | added | Shanti | Here is the simple explanation. Is the map defined above have associative property? Is it possible to do Map(Map($A_{p,q}$)$_{p1,q2}$)=Map($B_{p2,q2}$? If it is possible, how $p_1$ and $q_1$ can be calculated based on only $p,q,p_2$, and $q_2$ ? | |
| Oct 11, 2012 at 14:45 | comment | added | Shanti | @Quas: I mean instead of permuting $B_p$ back to $B$ with its parameters and then permuting the resulted $B$ with $A$'s parameter to get the same permuted matrix as $A_p$, directly go from $B_p$ to $A_p$ without doing back and forth. | |
| Oct 11, 2012 at 10:57 | comment | added | Anthony Quas | Still don't understand the question: what do you mean by "moving from one transformation to another"? | |
| Oct 11, 2012 at 3:05 | comment | added | Shanti | Then, we want to use $B_p$ and $p_2$, $q_2$ ,$p_1$, and $q_1$ to permute $B_p$ further such that the result is like permuting $B$ with $p_1$ and $q_1$ using the above mapping. | |
| Oct 11, 2012 at 3:02 | comment | added | Shanti | @Vaughn: Thanks for your comments and sorry for the confusion. Yes I meant "toral automorphism" with the application of something like Arnold Cat Map where we have a map $ \begin{bmatrix} x_n\\y_n \end{bmatrix} =\begin{bmatrix} 1&p\\ q&1+pq \end{bmatrix} \begin{bmatrix} x_{n-1}\\y_{n-1} \end{bmatrix} mod~ n$. Since $det\begin{pmatrix} \begin{bmatrix} 1 & p\\ q & pq+1 \end{bmatrix} \end{pmatrix}=1 $ then the map is area preserving. So we permute $A$ and $B$ using the map with two different parameter $p_1$,$q_1$ and $p_2$ and "q_2" which gives us $A_p$ and $ B_p$. | |
| Oct 10, 2012 at 21:12 | comment | added | Vaughn Climenhaga | Can you clarify what your definitions are? By "cat map" I assume you mean "toral automorphism", that is a map $\mathbb{R}^d/\mathbb{Z}^d$ given by a matrix $A\in SL(d,\mathbb{Z})$, and probably you want $A$ to have no eigenvalues on the unit circle. But I don't know what you mean by "cat map parameters" or "permuted matrix", so right now I don't understand your question. | |
| Oct 10, 2012 at 19:21 | history | asked | Shanti | CC BY-SA 3.0 |