Timeline for Beginners level question : symbolic dynamics and notations
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Jun 21, 2017 at 20:15 | vote | accept | SKM | ||
| Apr 5, 2017 at 5:16 | comment | added | Gerhard Paseman | It may be doing the same thing, but in a different way, like multiplication with Roman numerals instead of Arabic. I think they are different but similar because they take different inputs but do a similar job. Until I know what "same" or "alike enough" means, I am going to be cautious and not say they are the same. Gerhard "We Are Not All One" Paseman, 2017.04.04. | |
| Apr 5, 2017 at 3:48 | comment | added | SKM | In my question $f^{-1}$ is doing the same thing that is given a symbolic sequence, it is mapping it to a same point. I am confused if $f^{-1}$ which is called as the inverse map or inverse image of the map $f()$ is the same thing as $S^{-1}$ or not because $f^{-1}$ seems to be doing the same thing as $S^{-1}$. Thank you for your feedback. | |
| Apr 5, 2017 at 3:42 | comment | added | SKM | $f^{-1}$ is used to encode a symbol string into a real number using the reverse interval mapping, $x \mapsto f^{-1}(0,x)$. If you may kindly see the pdf document in the link web.math.rochester.edu/people/faculty/edummit/docs/… , on Pg13, using an example of quadratic map there is an equivalence between shift map and the quadratic map. Given a symbol string $S(x) = (d_0,d_1,...)$ we can construct $x$ and on pg 14 last line the document says that using $S^{-1}$ is the map that maps the sequence to a single point. | |
| Apr 4, 2017 at 18:53 | comment | added | Gerhard Paseman | Every time I see f inverse in your post, it seems to take a symbol or symbol string and a real number or set of real numbers as input. Does S inverse take a real number as input? Gerhard "Let's Get Real Basic Here" Paseman, 2017.04.04. | |
| Apr 4, 2017 at 18:45 | comment | added | SKM | In my question i have mentioned that $S^{-1}$ is function from: $\Sigma_2 \rightarrow R$. So, $f^{-1}$ is not the same as $S^{-1}$? By substituting $f(.)$ by $\sigma(.)$ Since the dynamics of the chaotic orbit is studied by the shift map, so by substitute I meant can we replace and use $f(.)$ by $\sigma(.)$ as the symbolic dynamics are obtained directly from $f(.)$ and $\sigma(.)$ is the shift map. | |
| Apr 4, 2017 at 18:27 | comment | added | Gerhard Paseman | The intuition is that (from a certain perspective) the map f behaves a lot like the shift map, especially if you look at (technically a modification of) f as just mapping symbols. Conjugate points should behave the same way, or very similarly, under this blurry view of f. I think an expert should weigh in on your questions, as I have not worked through the subject. Gerhard "Definitely Not A Dynamics Expert" Paseman, 2017.04.04. | |
| Apr 4, 2017 at 18:21 | comment | added | Gerhard Paseman | (1) from context, I infer the difference since the inverses start from different domains. Actually, you do not give me S inverse, so I am guessing based on your report. (2) I don't know if it safe since I am unsure what substitute means. For me, a gun is safe only if it is kept away from all ammunition. (3) I don't recall conjugacy well enough to trust my explanation. The intuition though should be that it is a way to determine how similar things are. In group Theory, conjugate elements have the same order, for example. Gerhard "Analogies Are Much Like Guns" Paseman, 2017.04.04. | |
| Apr 4, 2017 at 18:09 | comment | added | Gerhard Paseman | I will look at your other question, but I no longer use an account there, so I will not respond there. Gerhard "It's A Long Boring Story" Paseman, 2017.04.04. | |
| Apr 4, 2017 at 17:11 | comment | added | SKM | (2) Based on your explanation on the purpose of symbolic dynamics, is it safe to say that the chaotic map, $f(.)$ can be substituted by $\sigma(.)$ and the output of $\sigma(.)$ map are the symbols. (3) what does conjugacy tell us and how is it useful? Can you please explain these and also take a glance at my new Question math.stackexchange.com/questions/2216318/… that is in continuation to this one? | |
| Apr 4, 2017 at 17:09 | comment | added | SKM | thank you for your reply. However some points are still unclear because of some technical terms. Could you please clarify the following?(1) A chaotic sequence can be obtained from a symbolic sequence through inverse operation of $f^{-1}$. Since, it is an operation from symbolic space to real numbers, so I thought it is the same thing as $S^{-1}$ which is also an operation from symbols to real numbers. But you say that these two are different. Why? | |
| Apr 4, 2017 at 15:47 | history | edited | Gerhard Paseman | CC BY-SA 3.0 |
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| Apr 4, 2017 at 15:41 | history | edited | Gerhard Paseman | CC BY-SA 3.0 |
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| Apr 4, 2017 at 15:35 | history | answered | Gerhard Paseman | CC BY-SA 3.0 |