$k$-type entropy of $\mathbb{Z}^d$ actions
Anshid Aboobacker, Sharan Gopal
TL;DR
This work generalizes entropy to higher-rank actions by introducing $k$-type entropy for $\\mathbb{Z}^d$-actions on compact metric spaces. It defines the $k$-type metric $\rho_{n,k}$ and proves that $h_k(T)=\lim_{\epsilon\to0^+}\limsup_{n\to\infty}\frac{1}{n}\log cov(n,k,\epsilon,T)$ is well-defined, metric-independent, and invariant under topological conjugacy, with foundational properties such as product additivity $h_k(T_1\times T_2)=h_k(T_1)+h_k(T_2)$ and a max-over-invariant-subspaces relation. The paper demonstrates that $h_k$ respects iterations and factors, and provides a concrete toral computation: for a $\\mathbb{Z}^2$-action on $\mathbb{T}^2$ given by $T((m_1,m_2),x)=A^{m_1}B^{m_2}x$ with commuting hyperbolic $A,B$, one has $h_k(T)=\log|\lambda_A|+\log|\lambda_B|$, where $\lambda_A,\lambda_B$ are the expanding eigenvalues. This result shows that $k$-type entropy recovers the expected sum of entropies in a toral setting and extends entropy concepts to multi-parameter actions.
Abstract
We introduce the concept of \(k\)-type entropy for dynamical systems generated by \(\mathbb{Z}^d\)-actions on compact metric spaces. We investigate its fundamental properties and establish connections with classical entropy and other \(k\)-type dynamical notions. The $k$-type entropy of some $\mathbb{Z}^2$-actions on a two dimensional torus is also calculated.