Effect of quantified irreducibility on the computability of subshift entropy

Computability at zero temperature

In this paper, we investigate the computability of thermodynamic invariants at zero temperature for one-dimensional subshifts of finite type. In particular, we prove that the residual entropy (i.e., the joint ground state entropy) is an upper semi-computable function on the space of continuous potentials, but it is not computable. Next, we consider locally constant potentials for which the zero-temperature measure is known to exist. We characterize the computability of the zero-temperature measure and its entropy for potentials that are constant on cylinders of a given length k. In particular,…

On the computability of rotation sets and their entropies

Given a continuous dynamical system $f:X\to X$ on a compact metric space $X$ and an $m$-dimensional continuous potential $Φ:X\to \mathbb R^m$, the (generalized) rotation set ${\rm Rot}(Φ)$ is defined as the set of all $μ$-integrals of $Φ$, where $μ$ runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy ${\mathcal H}(w)$ to each $w\in {\rm Rot}(Φ)$. In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. We then apply our resul…

Computability of topological pressure on compact shift spaces beyond finite type

We investigate the computability (in the sense of computable analysis) of the topological pressure $P_{\rm top}(ϕ)$ on compact shift spaces $X$ for continuous potentials $ϕ:X\to {\mathbb R}$. This question has recently been studied for subshifts of finite type (SFTs) and their factors (Sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalized gap shifts, and particu…

$\mathbb{Z}$的类平移作用、子群成员问题与有效子转移的Medvedev度

Translation-like actions by $\mathbb{Z}$, the subgroup membership problem, and Medvedev degrees of effective subshifts

研究证明无限局部有限连通图存在Z的类平移作用（单/两端图可传递且d(v,v*1)≤3，强化Seward定理），还证明具可判定字问题的有限生成无限群有可计算类平移作用且满足可判定轨道成员问题，应用得此类群的有效子转移可达所有Π₁⁰Medvedev度，扩展Miller对Zᵈ的分类。