Groups of dynamical origin

    Let $A$ be a finite alphabet, $A^*$ the set of finite words in $A$ and $\mathcal{F} \subseteq A^*$ a set of forbidden patterns that contains the set $\left\{aa: a \in A\right\}$. Let $$S_\mathcal{F} = \left\{x \in A^\mathbb{Z}: \text{ $x$ contains no subword in $\mathcal{F}$}\right\}.$$

Define $\varphi_a \in Homeo(S_\mathcal{F})$ as follows:

$$\varphi_a: \left \{ \begin{matrix} \text{ shift $x$ to the left} & \text{ if $x(1) = a$} \\ \text{ shift $x$ to the right} & \text{ if $x(0) = a$} \\ x & \text{ otherwise} \end{matrix}\right.$$

and $G_\mathcal{F} = \left\langle\varphi_a: a \in A \right\rangle$. \\

Recall that the topological entropy is defined as $$\mathcal{H}_\mathcal{F} := \lim_{n \rightarrow +\infty} \frac{\log \left( \text{\# of words of length $n$ that appear in some $x \in S_\mathcal{F}$ } \right)}{n}.$$ and $F(G_\mathcal{F})$ is the full topological group. A group $G$ is called residually finite if $$\bigcap_{H \leq G: [G:H] < \infty} H = 1$$

    Let $A$ be a finite alphabet and $\Omega$ be a 2-sided subshift, i.e., a closed subset of $A^\mathbb{Z}$ that is invariant under the shift map $\sigma$. An element $g \in Homeo(\Omega)$ belong to the full group of the shift ($F(\Omega)$) if each point in $\Omega$ has an open neighborhood $U$ such that $g|_U = (\sigma^n)|_U$ for some $n \in \mathbb{Z}$ depending on $U$ is equal to resembles a power of the shift map.

    Let $G$ be a finitely generated infinite ameanable group, $A$ be a finite alphabet and define $$\mathcal{F} = \left\{f:B \rightarrow A: B \subseteq G \text{ finite}\right\}$$ a finite set of forbidden patterns and $$S_{G,\mathcal{F}, A} = \left\{x \in A^G: x \text{ contains no forbidden subpatterns}\right\}.$$

$G$ acts on $S_{G,\mathcal{F}, A}$ on both sides as the regular representation, namely, permuting the coordinates of the sequences by right or left translation. Choose $\left\{F_n\right\}_{n \in \mathbb{N}}$ a Følner sequence for $G$ and define $$\mathcal{H}_{G, \mathcal{F}, A} = \lim_{n \rightarrow +\infty} \frac{\log \left( \# \left\{f: F_n \rightarrow A: \text{ $f$ appears in some $x \in S_{G,\mathcal{F}, A}$}\right\} \right)}{n}.$$ \\

It is a fact that $\mathcal{H}_{G, \mathcal{F}, A}$ does not depend on the selected Følner sequence. Define also $$E_G = \left\{\mathcal{H}_{G, \mathcal{F}, A}: \mathcal{F} \text{ is a finite set of forbidden subwords}\right\}.$$

It is known (see [MR2680402]) that $$E_{\mathbb{Z}} = \left\{q\log(\lambda): q \in \mathbb{Q}^+, \text{ and $\lambda$ is a Perron eigenvalue}\right\}$$ and $$E_{\mathbb{Z}^2} = \left\{r \in \mathbb{R}^+: \text{ $r$ is a right-recursively enumerable number}\right\},$$ where a Perron eigenvalue is the eigenvalue given by the Perron-Frobenius theorem of a square matrix with non-positive entries such that for some power of the matrix, the entries are all positive, and a right recursively enumerable number is the infimum of the image of a computable function $f: \mathbb{N} \rightarrow \mathbb{Q}$.