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.

  • 2
    $\begingroup$ 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. $\endgroup$
    – YCor
    Commented Dec 7, 2015 at 8:45
  • $\begingroup$ @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. $\endgroup$ Commented Dec 7, 2015 at 15:51
  • $\begingroup$ @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. $\endgroup$ Commented Dec 9, 2015 at 0:23
  • $\begingroup$ @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? $\endgroup$ Commented Dec 15, 2015 at 8:43

1 Answer 1


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 $0<t<1$).

  • $\begingroup$ 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. $\endgroup$ Commented Dec 16, 2015 at 23:06
  • $\begingroup$ It's symbolic by standard coding. I added the argument. $\endgroup$
    – YCor
    Commented Dec 16, 2015 at 23:21

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