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!

  • $\begingroup$ I don’t see the sequence $(a_n)$ in definition 2. Also do you mean$f(n)=f(T^nx)$ for all $n\in\mathbb N$? $\endgroup$ 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.

  • $\begingroup$ Hello Terry, thank you very much for your detailed answer. Helped me a lot to understand these notions. Much more clear now! $\endgroup$
    – Jean DELI
    May 5 at 15:04
  • $\begingroup$ I would like to thank again Prof. Terrance Tao. These notions are much clear to me now. Let me add that I learnt later from my advisor el Houcein el Abdalaoui that the definitions 0 and 1 are not equivalent by the work of Kushnirenko (iopscience.iop.org/article/10.1070/RM1967v022n05ABEH001225) and Canovas (researchgate.net/publication/…”). In fact, the variational principale does not hold for the topological entropy along a sequence (definition 1). I plan to add this part to my master thesis. $\endgroup$
    – Jean DELI
    May 14 at 11:27

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