# This almost periodic condition implies equicontinuity?

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.

• Hi, Ia writing just to signal a little typo: perhaps it is minimal, not minimial. Feb 24, 2020 at 18:20
• Here is a relevant statement: If $T$ is an isometry and $X$ is proper, then $X$ is compact --- it is simple, but not trivial; see “On conditions under which...” by Aleksander Całka. Feb 24, 2020 at 18:29
• @AntonPetrunin you are right! In fact in every equicontinuous system this can happen. What about if this condition implies equicontinuity? I will edit the question. Feb 24, 2020 at 18:30
• This is called uniform topological rigidity if I recall correctly, and arxiv.org/abs/1508.03366 should contain some info. Feb 24, 2020 at 19:06
• I think your specific question is answered by their statement that rigidity of minimal systems does not imply uniform rigidity, but I can't check now. Feb 24, 2020 at 19:16

## 1 Answer

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.

• Thank you very much! Feb 24, 2020 at 23:11