论文标题
拓扑压力在紧凑型偏移空间上超出有限类型的可计算性
Computability of topological pressure on compact shift spaces beyond finite type
论文作者
论文摘要
我们研究了拓扑压力$ p _ {\ rm top}（ϕ）$的可计算性（在可计算分析的意义上）在紧凑型偏移空间上$ x $，用于连续电位$ ϕ：x \ to {\ mathbb r} $。最近对这个问题进行了研究，以进行有限类型（SFTS）及其因素（SOFIC Shifts）的子缩短。我们开发一个框架来解决一般偏移空间上拓扑压力的可计算性，并将此框架应用于编码偏移。特别是，我们证明了在S-GAP偏移，广义间隙偏移和特定β档的所有连续电势方面的拓扑压力的可计算性。我们还构建了偏移空间，取决于电势，表现出拓扑压力的可计算性和不可兼容性。我们进一步证明了广义压力函数$（x，ϕ）\ mapsto p _ {\ rm top}（x，ϕ，ϕ \ vert_ {x}）$对于大型偏移空间$ x $和电位$ ϕ $无法计算。特别是，熵映射$ x \ mapsto h _ {\ rm top}（x）$在shift space $ x $时可计算，并且仅当$ x $具有零拓扑熵。在开发这些可计算性结果的过程中，我们得出了编码移位的几种千古理论特性，这些偏移范围超出了可计算性范围。
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 particular Beta-shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalized pressure function $(X,ϕ)\mapsto P_{\rm top}(X,ϕ\vert_{X})$ is not computable for a large set of shift spaces $X$ and potentials $ϕ$. In particular, the entropy map $X\mapsto h_{\rm top}(X)$ is computable at a shift space $X$ if and only if $X$ has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability.
