# Example(s) of a symbolic dynamical system with proximal but not asymptotic points

Can anybody give me (an) example(s) of a symbolic dynamical system (preferably arising from a substitution) which has a pair of points which are proximal but not asymptotic? I would prefer to work with the space of two-sided infinite words $\mathcal{A}^\mathbb{Z}$, since the shift operator $\sigma$ there is invertible.

By a proximal pair $x,y$, I mean that for every $\epsilon>0$ there is an $n\in\mathbb{Z}$ such that $d(\sigma^n(x),\sigma^n(y))<\epsilon$, where $d$ is the usual metric on $\mathcal{A}^\mathbb{Z}$.

By an asymptotic pair $x,y$, I mean that for every $\epsilon>0$ there is an $N\in\mathbb{N}$ such that for all $n\in\mathbb{Z}$, $|n|>N$ implies that $d(\sigma^n(x),\sigma^n(y))<\epsilon$.

An example with (the negation of) a 'weaker' version of asymptoticity, meaning that there is an $N\in\mathbb{N}$ such that either for all $n\in\mathbb{Z}$, $n>N$ implies $d(\sigma^n(x),\sigma^n(y))<\epsilon$, or for all $n\in\mathbb{Z}$, $n<-N$ implies $d(\sigma^n(x),\sigma^n(y))<\epsilon$, is also welcome.

Any help or reference will also be appreciated.

• Doesn't the following example obviously work: $A=\{0,1\}$, $x_n=0$ for all $n$, $y_{\pm n^2}=1$, $y_n=0$ for other $n$, so $x,y$ are proximal but not asymptotic.
– YCor
Commented Dec 7, 2015 at 8:45
• @YCor Yes, it does - if we take the system to be the shift-orbit closure of $y$ in $A^\mathbb{Z}$, which is what I assume you meant. Thank you. Commented Dec 7, 2015 at 15:51
• @YCor, how about you post your comment as an answer, so Simon can accept it? This will prevent the software automatically returning this question to the front page. Commented Dec 9, 2015 at 0:23
• @YCor Can you post an answer now so I can accept it? I'm new here, so have no idea how to migrate things... Also, can your example be modified to one of a minimal system which has proximal not asymptotic points? Commented Dec 15, 2015 at 8:43

A (non-minimal) example is given the full shift $A^\mathbf{Z}$ with $A=\{0,1\}$, $x_n=0$ for all $n$, $y_{\pm n^2}=1$, $y_n=0$ for other $n$, so $x$ and $y$ are proximal but not asymptotic. (Alternatively, we can restrict to the orbit closure of $y$.)

Here's now a minimal example. It's a bit more complicated but is very classical in topological dynamics.

If $Y$ is a subset of the circle $C=\mathbf{R}/\mathbf{Z}$, let $X=C_Y$ denote the set which informally is obtained from the circle by doubling all points in $Y$; formally, say $C_Y=C\times\{0\}\cup Y\times\{1\}$, and denote the elements $(c,0)$ as $c=c^+=c^-$ if $c\notin Y$, as $c^-$ if $c\in Y$, and denote $(c,1)$ as $c^+$. Endow $C_Y$ with the obvious cyclic ordering (so $a^\pm<b^\pm<c^\pm$ whenever $a<b<c$, and $a^-<a^+<b$ whenever $a\neq b$), and the corresponding topology. This is a compact topology.

Now let $t$ be an irrational, and denote $(t)=\mathbf{Z}t\subset C$. So the self-homeomorphism $\sigma:x\mapsto x+t$ on $C$ lifts to a self-homeomorphism of $C_{(t)}$. Then the pair $(0^-,0^+)$ is proximal but not asymptotic. (Indeed, if $|n_i|\to\infty$ and $n_it$ converges to an element $y$ of $C$, then if $y\notin (t)$, $\sigma^{n_i}(0^\pm)\to y$, while if $y\in (t)$ then $\sigma^{n_i}(0^\pm)\to y^\pm$ so $\sigma^{n_i}(0^+)$ and $\sigma^{n_i}(0^-)$ have distinct limits.)

Added: this is symbolic. Indeed you can embed it as a subshift on two letters $\{0,1\}$ by mapping a point $x$ to the sequence $(x_n)$, where $x_n=1$ if and only if $\sigma^n(x)\in [0^+,t^-]$ (where $t$ is chosen with $0<t<1$).

• I accepted the answer for the non-minimal example; as for the minimal one, can you please give a symbolic dynamical system (as that was the original question)? And thank you for the answer. Commented Dec 16, 2015 at 23:06
• It's symbolic by standard coding. I added the argument.
– YCor
Commented Dec 16, 2015 at 23:21