I am wondering if the following is true:

Suppose $T:X\to X$ is a transformation with probability measure $\rho$. Suppose tht $T$ is mixing. i.e., $|\rho(A\cap T^{-n}B)-\rho(A)\rho(B)|\to 0$

If $Y\subset X$ with positive measure and $R:Y\to \mathbb{N}$ is the first return time to $Y$ such that $T^{R(y)}(y)\in Y$ for all $y\in Y$, we can define the induced transformation by $F=T^R:Y\to Y$.

Can we say that $F$ is mixing (with respect to $\rho|_Y$)?

  • $\begingroup$ is it obvious that $F$ preserves $\rho\mid_Y$? $\endgroup$ – mathworker21 Oct 11 at 1:10
  • $\begingroup$ For example, take $X = \{1,2\}$ with uniform measure, $T(1)=2,T(2)=1$, $Y=X$, $R(1) = 1, R(2)=2$. Then $F(1)=2,F(2)=2$, so $F$ is not measure-preserving. $\endgroup$ – mathworker21 Oct 11 at 4:40
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    $\begingroup$ Have a look at mathoverflow.net/questions/160955/…. I edited the question to refer to the first return time. $\endgroup$ – John B Oct 11 at 15:37
  • $\begingroup$ @mathworker21 there is a theorem in "Laws of Chaos" by Boyarsky and Gora which confirms $F$ preserves the measure. I can not remember the number off the top of my head. $\endgroup$ – Mr Martingale Oct 11 at 17:25
  • $\begingroup$ @JohnB this is perfect, thanks! $\endgroup$ – Mr Martingale Oct 11 at 17:25

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