This almost periodic condition implies equicontinuity?

Question

Let $X$ be a metrizable compact space and $T\colon X\to X$ a minimal homeomorphism, i.e. $$ \mathrm{orb}(x) := \{T^kx:k\in\mathbb{Z}\}$$ is dense in $X$ for every $x \in X$. Assume that the following condition is met:

• There exist $\varepsilon_n \to 0$ and $s_n \in \mathbb{N}$ such that $d(T^{s_n}x,x) < \varepsilon_n$, for every $n \in \mathbb{N}$ and $x \in X$.

This implies that $X$ is equicontinuous? When $X$ is a subshift it is easy to see that this is true (beacuse, then, every point is periodic), but for a general system I couldn't prove it.

Answer 1

In general, such a homeomorphism is not necessary equicontinuous.

The existence of such examples on $X=\mathbb{T}^2$, i.e. the $2$-torus, can be shown as follows: let $\mathcal{O}$ be the $C^\infty$ closure of the set $\{h\circ R_\alpha\circ h^{-1} : h\in\mathrm{Diff}^\infty(\mathbb{T^2}),\ \alpha\in\mathbb{T}^2\}$, where $\mathbb{T}^2$ denotes the $2$-torus and $R_\alpha : \mathbb{T}^2\ni x\mapsto x+\alpha$.

Fathi and Herman showed in

Fathi, Albert; Herman, Michael R., Existence de difféomorphismes minimaux, Astérisque 49(1977), 37-59 (1978). ZBL0374.58010.

that there is residual set $C_0\subset \mathcal{O}$ such that every diffeomorphism of $C_0$ is minimal.

On the other hand, in

Kocsard, Alejandro; Koropecki, Andrés, A mixing-like property and inexistence of invariant foliations for minimal diffeomorphisms of the 2-torus, Proc. Am. Math. Soc. 137, No. 10, 3379-3386 (2009). ZBL1179.37063.

we proved the existence of a residual set $C_1\subset\mathcal{O}$ such that any $f\in C_1$ is weak-spreading, i.e. if $\tilde f\colon\mathbb{R}^2\to\mathbb{R}^2$ is a lift of $f$, then for every non-empty open set $U\subset\mathbb{R}^2$, every $\epsilon>0$ and any $R>0$, there exist $n>0$ and a ball $B_R\subset\mathbb{R}^2$ of radius $R$ such that $\tilde f^n(U)$ is $\epsilon$-dense in $B_R$. It is clear that every weak spreading homeomorphism is not equicontinuous.

Finally, one can show that rigid diffeomorphisms are generic in $\mathcal{O}$, i.e. there is residual set $C_2\subset\mathcal{O}$ such that for every $f\in C_2$ there is a sequence of natural numbers $n_j\to \infty$ so that $f^{n_j}\to id$ in the $C^0$ uniformly, when $j\to \infty$.

So, any diffeomorphism in $C_0\cap C_1\cap C_2$ is minimal and rigid, but not equicontinuous.