Find an $(n,\varepsilon)$-spanning set $A\subset [0,1)^{\mathbb{N}}$.

Question

Let $X:=[0,1)^{\mathbb{N}}=[0,1)\times[0,1)\times\ldots$ be equipped with the metric $$d(x,y):=\sum_{j\geq1}10^{-j}|x_{j}-y_{j}|.$$ Consider the map $f\colon X\to X$ defined by $$f(x_{1},x_{2},x_{3},\ldots):=(10x_{1} \ \text{mod} \ 1,x_{1},x_{2},\ldots).$$ Let $n\in\mathbb{N}$ and $\varepsilon>0$ be given. A finite subset $A\subset X$ is called $(n,\varepsilon)$-spanning for $f$ iff for all $x\in X$ there exists an $a\in A$ such that $d(f^{i}(x),f^{i}(a))<\varepsilon$ for all $0\leq i\leq n$. The problem is to find such a set $A$.

I encountered this problem in an old exam of a course on dynamical systems I'm currently following. I think this question is inspired by the proof of proposition 2.6.5 in the book Brin-Stuck, Introduction to Dynamical Systems. (But I think the proof they give is wrong, since they use the wrong definition of the map $\alpha$.)

My attempt: Choose $k\in\mathbb{N}$ such that $10^{k}<\varepsilon$. Then $$d(f^{i}(x),f^{i}(y))=\sum_{j=1}^{k}10^{-j}|f^{i}(x)_{j}-f^{i}(a)_{j}|+\sum_{j\geq k+1}10^{-j}|f^{i}(x)_{j}-f^{i}(a)_{j}|.$$ Then one can use the geometric series to estimate the infinite sum $<\varepsilon/9$ (if I made no mistakes). To estimate the finite sum, I think I need to find the subset $A\subset X$. This is where I got stuck.

Any suggestions are greatly appreciated.

Answer 1

To begin with, notice the way the distance changes for close points as we apply $f$. Let $d(f^{i}(x),f^{i}(a))= d_i(x, a)$. If the first $k$ coordinates of $x_1$ and $a_1$ are the same, then $$d_k(x, a) = d(x, a) 10^{-k} + |x_1 - a_1| \sum_{i=1}^{k} 10^{k-i} = d(x, a) 10^{-k} + |x_1 - a_1| \frac{(10^k -1)}{9}$$ or something very like that. The first term more or less controls itself. The second means you need to have a bunch of points for each initial part of the decimal expansion.

So, start with a $(0, 5 \varepsilon)$ cover, say $C$, and then get together $10^n \lfloor 2\varepsilon^{-1} \rfloor$ in $[0, 1)$ to deal with the first entry in the product, call them $P$. And then form $P \times C$ to construct the $(n, \varepsilon)$ cover.

Does that help?