# The cardinal characteristic $\mathfrak r_{(X,f)}$ of a dynamical system

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, it is not known whether $$\mathfrak r=\mathfrak r_\sigma$$ 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 $$\omega$$ 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 $$f:2^\omega\to 2^\omega$$, $$f:(x_n)_{n\in\omega}\mapsto (x_{n+1})_{n\in\omega}$$?

Unfortunately, $$\mathfrak r_{(2^\omega,f)}\ge\mathfrak r$$. Indeed, let $$\mathcal R$$ be a family of infinite subsets of $$\omega$$ such that $$|\mathcal R|=\mathfrak r_{(2^\omega,f)}$$ and for any $$x=(x_n)_{n\in\omega}\in 2^\omega$$ there exists $$R\in\mathcal R$$ such that the sequence $$(f^n(x))_{n\in R}$$ converges in $$2^\omega$$. Then the sequence $$(x_n)_{n\in R}$$ of the first coordinates of the sequence $$(f^n(x))_{n\in R}$$ stabilizes, so the family $$\mathcal R$$ witness that $$\mathfrak r\le \mathfrak r_{(2^\omega,f)}$$.