Entropy on regular trees (Q2176549)

Let \(k\geq 2\) and let \(\Sigma_k =\{0,1, \dots, k-1\}\). The set \(\Sigma_k^{*}\) of all finite words on the alphabet \(\Sigma_k\) is the \(k\)-tree, which is naturally visualized as the Cayley graph of the free semigroup on \(k\) generators. The function \(\sigma: \Sigma_k^{*} \rightarrow A\) is called labeled tree, where \(A =\{0,1, \dots, d-1\}\) is an alphabet on \(d\) symbols, \(d\geq 2\). In the present paper the authors prove that the limit in the definition of tree shift topological entropy is actually the infimum: \[ h(\tau )=\lim_{n\rightarrow \infty } \frac{\log p_{\tau}(n)}{1+k+\cdots +k^n}=\inf \left\{\frac{\log p_{\tau}(n)}{1+k+\cdots +k^n}:n\in \mathbb{N}\right\},\] where \(h(\tau)\) is the topological entropy of the labeled tree \(\tau\). Let \(X\subset A^{\mathbb{Z}}\) be a subshift on a finite alphabet, let \(k = 2\), and let \(Z(X)\) be the tree shift on the binary tree associated with \(X\). The authors prove that \(h_{\text{top}}(X)\leq h(Z(X))\). By using the strip method for the golden mean shift of finite type on the \(k\)-tree, \(k\geq 2\), the authors show that the entropy \(h^{(k)}\) is strictly increasing in \(k\). Then, for each fixed \(k = 2, 3,\dots,\) the authors prove that the site specific strip approximation entropies \(h^{(k)}_n\) converge to the entropy \(h^{(k)}\) of the golden mean SFT on the \(k\)-tree. For all \(k\geq 2\), the entropy of the golden mean SFT on the \(k\)-tree is given by a formula. The authors study the monotonicity of the strip approximation entropies for \(k = 2,\dots, 8\). By means of some numerical calculations, they show that these approximations increase strictly with \(n\). They also show that the intermediate entropy \(h^{i} = \lim_{n\rightarrow \infty}\frac{\log (q(n))}{n}\) exists. An infinite series formula of the entropy of the \(k\)-tree shift corresponding to an irreducible \(d\times d\) (0,1)-matrix \(M\) is given. It is shown that \(h^{i}(Z) = h(Z)\). In the last section, by using the strip method, the authors study the topological entropy \(h(Z_M)\) of the system \(Z_M\).