Timeline for Entropy of $f^{m(x)+n}$ of full shift
Current License: CC BY-SA 4.0
19 events
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| Oct 19, 2022 at 7:09 | vote | accept | Adam | ||
| Oct 18, 2022 at 20:16 | answer | added | Ronnie Pavlov | timeline score: 3 | |
| Oct 18, 2022 at 16:27 | history | edited | Adam | CC BY-SA 4.0 |
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| Oct 18, 2022 at 16:10 | comment | added | Ilkka Törmä | @Adam In my example we have $S=\{1,2\}$. Anyway, if you know $T$ to be measure-preserving, then it would be a good idea to add that as an assumption to the question. | |
| Oct 18, 2022 at 15:59 | comment | added | Adam | @IlkkaTörmä The reason that I asked questions is: I have a preserving function like $T$, where it is some iteration of $f$, and I want to see whether the above entropy relation holds or not. Unfortunately, I can not exactly write the function as it is so complicated, but I knew $f$, which is some iteration of $T$ is preserving. | |
| Oct 18, 2022 at 15:56 | comment | added | Adam | @IlkkaTörmä. Thanks for your example. One more question: assume there is a finite set of natural numbers; let's call $S$. We define $T(x)=f^{m(x)+n}(x)$, where $m(x) \in S.$ Again, isn't it preserving? | |
| Oct 18, 2022 at 15:02 | comment | added | Ilkka Törmä | @Adam $T$ need not preserve even the uniform product measure on the binary full shift $\{0,1\}^\mathbb{Z}$. Namely, for $n=1$ and $m(x)=1+x_3$ we have $T(x)_0=0$ iff $x_2=x_3=0$, which has probability $1/4$ instead of $1/2$. | |
| Oct 18, 2022 at 13:46 | comment | added | Adam | @VilleSalo No, it is not! Feel free to ask any questions | |
| Oct 18, 2022 at 13:23 | comment | added | Ville Salo | Is $0$ a natural number? (Sorry, I tried to be concise.) | |
| Oct 18, 2022 at 13:18 | comment | added | Adam | @VilleSalo Natural numbers | |
| Oct 18, 2022 at 12:58 | comment | added | Ville Salo | What is $\mathbb{N}$? | |
| Oct 18, 2022 at 11:45 | history | edited | Adam | CC BY-SA 4.0 |
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| Oct 18, 2022 at 11:38 | history | edited | Adam | CC BY-SA 4.0 |
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| Oct 18, 2022 at 11:37 | comment | added | Adam | @RW Thanks for your comment. Even, isn't it a measure preserving when $(X, \mu, f)$ is a full shift? | |
| Oct 18, 2022 at 11:33 | history | edited | Adam | CC BY-SA 4.0 |
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| Oct 18, 2022 at 11:24 | comment | added | R W | To begin with, your new transformation need not be measure preserving. | |
| Oct 18, 2022 at 11:07 | history | edited | Adam | CC BY-SA 4.0 |
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| Oct 18, 2022 at 9:51 | history | edited | Adam | CC BY-SA 4.0 |
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| Oct 18, 2022 at 9:43 | history | asked | Adam | CC BY-SA 4.0 |