Existence of a homeomorphism that does not return much

Question

Let $f:X\rightarrow X$ a homeomorphism where $X$ is a compact metric space.

Fix $x\in X$, denote $O(f,x)=\{ f^n(x):n\in \mathbb{Z}\}$ the orbit of $f$ by $x$.

For $m\in \mathbb{N}$ denote $O(f,x,m)=\{ f^j(x): \vert j\vert \leq m\}$ and $\#(A)$ is the cardinal of $A$.

I am interested in the existence of an example of $f$ such that: there is $\delta>0$ with the following property

$$\displaystyle{\lim_{m\to\infty}}\frac{\#(B[z,\delta]\cap O(f,x,m))}{\#(O(f,x,m))}=0$$ where $z\in \overline{O(f,x)}\setminus O(f,x)$ and $B[z,\delta]$ is the ball closed.

I appreciate if you could give me some suggestion to know if such homeomorphism exists.

Answer 1

It's easy to find such points for the full shift: $X=\{0,1\}^{\mathbb Z}$, $f(x)_n=x_{n+1}$. Take an $x$ that is mainly zero, but also has arbitrarily long intervals of $1$'s, for example $$ x_n = \begin{cases} 1 & 2^k\le n \le 2^k + k \\ 0 & \textrm{otherwise} \end{cases} . $$ Then $z_n=1$ is in the orbit closure, but $f^n x$ can only be close to $z$ if $n\approx 2^k$, so the limit is indeed zero for all small $\delta>0$.