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Is the $T$-function $f(x) = (x \text{AND} c) + (x^2 \text{OR} c)$, where $c$ is positive integer ergodic in the space $Z_2$ (p-adic numbers)?

What is the measure of this function?

I am trying to use this theorem

My progress:

  1. x = x^2 in $Z_2$

Found out that f1(x) = (x AND c), f2(x) = (x OR c) are uniformly differentiable in $Z_2$

  1. It remains to prove transitivity of f(x) ( Based on this theorem)

transitivity means that the function is a full-cycle permutation.

  1. I am trying to prove transitivity based on this example
mascai
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    You seem to be applying Boolean operations "and" and "or" to numbers like $c$; what does that mean? – Andreas Blass Apr 25 '20 at 13:58
  • @AndreasBlass, thank you,

    added that we work in space Z_2 (binary numbers {0, 1})

     – mascai Apr 25 '20 at 14:02
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    But you also still say that $c$ is a positive integer, rather than being in $\mathbb Z/2$. Also, it seems silly to write $x^2$ when dealing with binary numbers since $x^2=x$ for both values of $x$ in ${0,1}$. – Andreas Blass Apr 25 '20 at 14:12
  • @AndreasBlass added link to what i mean by Z_2 space. I agree that x= x*x in the task – mascai Apr 25 '20 at 14:31
  • You had already written that you mean ${0,1}$ by $Z_2$, but that still makes no sense since $c$ is allegedly just some positive integer. – Andreas Blass Apr 25 '20 at 14:32
  • @AndreasBlass what should I clarify? Maybe some assumption about constant c will help? – mascai Apr 25 '20 at 14:44
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    You already have an assumption about $c$, namely that it's a positive integer. That conflicts with your other assumption that you're working in $\mathbb Z_2$. What needs to be clarified is what you're actually asking about, i.e., what the symbols and words in the question are supposed to mean – Andreas Blass Apr 25 '20 at 17:01
  • Judging from the link, $\mathbb Z_2$ likely means the $2$-adic integers, not the quotient $\mathbb Z/2$. But even with that I do not understand what "and" and "or" in the function are supposed to mean (I have seen them being used for maximum resp. minimum of two numbers, but that makes no sense since there is no natural order on $p$-adics). – Torsten Schoeneberg Apr 25 '20 at 17:30
  • @AndreasBlass I mean 2-adic integers – mascai Apr 26 '20 at 11:09
  • @TorstenSchoeneberg "and", "or" -- bitwise operations – mascai Apr 26 '20 at 11:10
  • @TorstenSchoeneberg Also the word "positive" in the question makes no sense in the $p$-adic context. Furthermore, the question still says "binary numbers ${0,1}$". And I don't know of any meaning for "measure of this function". As far as I can see, the question, combined with the OP's comments, is simply meaningless. – Andreas Blass Apr 26 '20 at 12:47
  • @mascai: I don't understand your comment. Also, now that you have changed the meaning of $\mathbb Z_2$ to the $2$-adic integers, of course in general $x\neq x^2$ for $x\in \mathbb Z_2$. – Torsten Schoeneberg Apr 26 '20 at 15:26
  • @mascai: Maybe you can help my understanding like this: Let's just take $c=5$, i.e. $101$ in $2$-adic expansion, and $x=$ the $2$-adic square root of $17$ which is congruent $1$ mod $4$, i.e. $...11101001$ in $2$-adic expansion (cf. https://math.stackexchange.com/a/2299828/96384). Then what are the values of $c ; AND ; x$ resp. $c ; OR ; x$? I'm sure this is defined somewhere in those slides you link to, because it is used there, I really just don't know it. – Torsten Schoeneberg Apr 26 '20 at 16:13
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    @TorstenSchoeneberg thank you for your time c=101, x = . ...11101001 --> c AND x = 00...01 (bitwise and, similar to c++ commands https://www.geeksforgeeks.org/bitwise-operators-in-c-cpp/) – mascai Apr 26 '20 at 19:29

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