Suppose $(X,\mathcal{F},\mu,T)$ is an ergodic measure preserving dynamical system.
Let $Y\subset X$ be such that $\mu(Y)>0$ and suppose there is an integrable function $R:Y\to \mathbb{N}$ such that $T^{R(y)}(y)\in Y$.


Then we can define a function $F:Y\to Y$ by $F=T^R$ and consider the induced system $(Y,\mathcal{F}\cap Y, \mu|\_Y,F)$.


Can we say that $F$ is ergodic?


When $R(x)=\inf\{n\ge 1 : T^n(x)\in Y\}$, this induced system is very well studied and the answer to my question is yes (see for example <http://www.weizmann.ac.il/math/sarigo/sites/math.sarigo/files/uploads/ergodicnotes.pdf>, Theorem 1.7). However, the given proof does not extend to the general return times I am considering above.


Thanks :)



Ergodicity of induced system
In general $F$ is not ergodic. A very simple example can be constructed as follows: let $X=\mathbb{Z}\_3=\{0,1,2\}$ and $\mu =1/3(\delta\_0+\delta\_1+\delta\_2)$ and $T(x):=x+1$. This is an ergodic system. Let us define $Y:=\{0,1\}$ and $R\equiv 3$. Since $F=T^3=id$, it is not ergodic. 


