I am interested in a "dynamical" modification of the cardinals $\mathfrak r$ and $\mathfrak r_\sigma$, well-known in the theory of cardinal characteristics of the continuum.

For a compact metrizable space $X$, let $\mathfrak r_X$ be the smallest cardinality of a family $\mathcal R$ of infinite subsets of $\omega$ such that for every sequence $(x_n)_{n\in\omega}\in X^\omega$ there exists $R\in\mathcal R$ such that the subsequence $(x_n)_{n\in R}$ converges in $X$.

It is easy to see that $\mathfrak r_2\le \mathfrak r_X\le\mathfrak r_{2^\omega}$ for every compact metric space $X$ containing more than one point. Moreover, it can be shown that
$$\mathfrak r_X=\begin{cases}\mathfrak r_2&\mbox{if $|X|\le\omega$};\\
\mathfrak r_{2^\omega}&\mbox{if $|X|>\omega$}.
\end{cases}
$$
Here $2$ is the doubleton $\{0,1\}$, endowed with the discrete topology.

In the theory of cardinal characteristics of the continuum, the cardinals $\mathfrak r_2$ and $\mathfrak r_{2^\omega}$ are denoted by $\mathfrak r$ and $\mathfrak r_\sigma$, respectively. 
As written in [the survey of Blass][1], it is not known wheter $\mathfrak r=\mathfrak r_\sigma$ in ZFC.

Now I introduce a dynamical modification of the cardinal $\mathfrak r_X$.  

Let us recall that a *dynamical system* is a pair $(X,f)$ consisting of a compact metric space $X$ and a continuous function $f:X\to X$. Define the iterations $f^n$ of $f$ by the recursive formula: $f^0$ is the identity map of $X$ and $f^{n+1}=f\circ f^n$ for $n\in\omega$. 

**Definition.** For a dynamical system $(X,f)$, let $\mathfrak r_{(X,f)}$ be the smallest cardinality of a family $\mathcal R$ of infinite subsets of $X$ such that for every $x\in X$ there exists $R\in\mathcal R$ such that the sequence $(f^n(x))_{n\in R}$ coverges in $X$. 

It is clear that $\mathfrak r_{(X,f)}\le\mathfrak r_X$. If $f$ is the identity map of $X$, then $\mathfrak r_{(X,f)}=1$, so the inequality can be strict.

>**Problem.** Is it consistent that $\mathfrak r_{(2^\omega,f)}<\mathfrak r_{2^\omega}$ for the  shift map $f:2^\omega\to 2^\omega$, $f:(x_n)_{n\in\omega}\mapsto (x_{n+1})_{n\in\omega}$?


  [1]: http://www.math.lsa.umich.edu/~ablass/hbk.pdf