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 $\rho|_Y$)?
If $Y=X$, then the first return map is $T$, so $F = T$. The answer is that the induced transformation $F$ is not necessarily mixing. The easiest example I can think of is $T$ equal to the staircase transformation: https://community.ams.org/journals/proc/1998-126-03/S0002-9939-98-04082-9/S0002-9939-98-04082-9.pdf. This transformation is mixing due to staircases whose heights increase by one being added infinitely many times. To get the induced transformation $F$, fix one of the columns and let $Y$ be just that column without including any other staircases. The transformation $F$ will be rigid and therefore not mixing.