# Are these topological sequence entropy definition equivalent?

I am working on Möbius disjointness for models of topological dynamic systems. In that purpose, I try to understand the notion of topological entropy. We know, for a t.d.s $$(X,T)$$ that it is defined by

Definition 0

$$h(f)=\lim_{\varepsilon\to 0}\left(\limsup_{n\to+\infty}\frac{1}{n}\log\left(N(n,\varepsilon)\right)\right)$$

with $$N(n,\varepsilon) = \max(\mathrm{Card}(E(n,\varepsilon))$$ where

$$\forall(x,y)\in E(n,\varepsilon)^2,\ d_n(x,y)>\varepsilon$$

(Source: https://arxiv.org/abs/1803.06201 - First definition)

I think I am clear with this definition. Now we dive into topological $$\textbf{sequence}$$ entropy. The common definition is the following

Definition 1

Let $$A=\{a_0 be an increasing subsequence of $$\mathbb{N}$$, and $$\mathscr{U}$$ an open cover of $$X$$, define

$$h^A(T,\mathscr{U})=\limsup_{n\to\infty}\frac{1}{n}\log\mathscr{N}\left(\bigvee_{i=0}^{n-1} T^{-a_i}(\mathscr{U})\right)$$

where $$\mathscr{N}(V)$$ is the minimal possible cardinality of a subcover chosen from a cover $$V$$ Then topological sequence entropy is the following quantity

$$h^A(T)=\sup\{h^A(T,\mathscr{U}):\mathscr{U}\text{ is an open cover of }X\}$$

(Source: https://arxiv.org/abs/1810.00497 - Definition 2.1)

Up to this point, I am still ok with these definition, but now comes the definition from Terence Tao blog

Definition 2

Let $$(X,T)$$ be a t.d.s and $$f:\mathbb{N}\to\mathbb{C}$$ an application verifying $$\forall x\in X,\ f(n)=f(T^nx)$$. We denote the topological sequence entropy $$h_{\mathrm{seq}}(f)$$ of $$f$$ by the smallest $$\sigma$$ such as for all $$\varepsilon>0$$, when $$m\to+\infty$$, the set

$$\{f(n+1),\dots,f(n+m):n\in\mathbb{N}\}\subset \mathbb{C}^m$$

can be covered by $$O(\exp(\sigma m+o(m)))$$ balls of radius $$\varepsilon$$.

(Source: https://terrytao.wordpress.com/tag/topological-entropy/ - Just after Remark 2)

Now, here are my questions:

Question 1: Are definition 1 and definition 2 equivalent? I mean, do we know if they are or not?

Question 2: Terence Tao claims in the same post : "For minimal t.d.s definition 0 and definition 2" are equivalent. My goal is to work on that claim for which I found no proof so far. If I have the equivalence between definition 1 and definition 2, I think it would help!

• I don’t see the sequence $(a_n)$ in definition 2. Also do you mean$f(n)=f(T^nx)$ for all $n\in\mathbb N$? May 4 at 17:02
The topological entropy I am assigning to a sequence $$f$$ (Definition 2) is not directly related to the similar-sounding concept of topological sequence entropy (Definition 1), but is instead related to Definition 0.
If $$f(n) = F(T^n x)$$ for some continuous $$F: X \to {\bf C}$$ on a compact metric space $$X$$, we see from the uniform continuity of $$F$$ (and taking contrapositives) that for every $$\varepsilon > 0$$ there exists $$\delta>0$$ such that whenever $$f(n), f(n')$$ are $$\varepsilon$$-separated, then $$T^n x, T^{n'} x$$ are $$\delta$$-separated in the metric $$d$$ on $$X$$. This implies that for any $$m$$, whenever two tuples $$(f(n+1),\dots,f(n+m)), (f(n'+1),\dots,f(n'+m)) \in {\bf C}^m$$ are $$\varepsilon$$-separated (in the $$\ell^\infty$$ metric), the corresponding points $$T^n x, T^{n'} x \in X$$ are $$\delta$$-separated in the metric $$d_m$$ defined by $$d_m(y,z) := \sup_{1 \leq i \leq m} d(T^i y, T^i z)$$. From this it is a routine matter to verify that the entropy of $$f$$ in Definition 2 is less than or equal to the the entropy of the system in Definition 0. (Informally: small entropy in the Definition 0 sense means that there are not that many $$\delta$$-separated points in $$X$$ with the $$d_m$$ metric, hence there are not that many $$\varepsilon$$-separated orbits $$(f(n+1),\dots,f(n+m))$$ in the $$\ell^\infty$$ metric.)
Conversely, assume that the system $$(X,T)$$ is minimal and also that the function $$F$$ and its shifts generate the topology of $$X$$ (this latter hypothesis was omitted by mistake in my remark, otherwise there is no equivalence as can be seen for instance by considering the case of constant $$F$$). The latter hypothesis implies that for any two distinct points $$y,z \in X$$, the tuples $$(F(y), F(Ty), \dots, F(T^{n_0} y))$$, $$(F(z), F(Tz), \dots, F(T^{n_0} z))$$ must be distinct for some $$n_0$$. By compactness this implies that for every $$\varepsilon>0$$ there exists $$n_0$$ and $$\delta>0$$ such that whenever two points $$y,z \in X$$ are $$\varepsilon$$-separated in the original metric $$x$$, then $$(F(y), F(Ty), \dots, F(T^{n_0} y))$$, $$(F(z), F(Tz), \dots, F(T^{n_0} z))$$ are $$\delta$$-separated in $$\ell^\infty$$. If $$y_1,\dots,y_N \in X$$ are $$2\varepsilon$$-separated in the $$d_m$$ metric for some large $$m$$, then by minimality (which makes the orbit of $$x$$ dense) we can find $$n_1,\dots,n_N$$ such that $$T^{n_1} x, \dots, T^{n_N} x$$ is $$\varepsilon$$-separated in the $$d_m$$ metric, which combined with the preceding remark means that the tuples $$(f(n_1+1),\dots,f(n_1+m+n_0)), \dots, (f(n_N+1),\dots,f(n_N+m+n_0))$$ are $$\delta$$-separated in the $$\ell^\infty$$ metric (this is perhaps easiest to see by taking contrapositives). From this it is a routine matter to verify that the entropy of the system in Definition 0 is less than or equal to the entropy of $$f$$ in Definition 2.