Entropy of axial product of multiplicative subshifts

Abstract

We obtain the entropy and the surface entropy of the axial products on $\mathbb{N}^d$ and the $d$-tree $T^d$ of two types of systems: the subshift and the multiplicative subshift.

Figures (8)

• Figure 1: The types of $\Delta_3$ of $T^2$.
• Figure 2: The $\Delta_2^{(2,1)}$ with respect to the multiplicative constraints $p_1=2$ and $p_2=3$.
• Figure 3: The map $\phi$ on $\Delta_3^{(1,1)}$ with respect to the multiplicative constraints $p_1=2$ and $p_2=3$.
• Figure 4: The decomposition of $\Delta_3$ of the 2-tree with $p_1=2$ and $p_2=3$.
• Figure 5: The graph of $\Delta_k^{(i,1)}$ (resp. $\Delta_k^{(1,j)}$), where $\ell =\left\lfloor \log_{p_1}\frac{i+k}{i}\right \rfloor$ (resp. $\ell =\left\lfloor \log_{p_2}\frac{j+k}{j}\right \rfloor$).

Theorems & Definitions (27)

• Theorem 2.1: Ban et al., ban2021entropy
• Corollary 2.2
• proof
• Theorem 2.3
• Remark 2.4
• proof
• Corollary 2.5
• proof
• Definition 3.1
• Lemma 3.2