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 $A=\{a_0<a_1<\dots\}$ 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!

Thanks in advance for reading!