We study the parametrized complexity of fundamental relations between multidimensional subshifts, such as equality, conjugacy, inclusion, and embedding, for subshifts of finite type (SFTs) and effective subshifts. We build on previous work of E. Jeandel and P. Vanier on the complexity of these relations as two-input problems, by fixing one subshift as parameter and taking the other subshift as input. We study the impact of various dynamical properties related to periodicity, minimality, finite type, etc. on the computational properties of the parameter subshift, which reveals interesting differences and asymmetries. Among other notable results, we find choices of parameter that reach the maximum difficulty for each problem; we find nontrivial decidable problems for multidimensional SFT, where most properties are undecidable; and we find connections with recent work relating having computable language and being minimal for some property, showing in particular that this property may not always be chosen conjugacy-invariant.

翻译：本文研究多维子移位之间基本关系（如相等性、共轭性、包含性与嵌入性）对于有限型子移位和有效子移位的参数化复杂性。我们在E. Jeandel和P. Vanier先前关于这些关系作为双输入问题的复杂性研究基础上，通过固定一个子移位作为参数并以另一个子移位作为输入展开分析。我们探讨了与周期性、极小性、有限型等相关的各类动力学性质对参数子移位的计算特性的影响，揭示了显著的差异性与非对称性。在多项重要结果中，我们发现了使每个问题达到最大计算难度的参数选择；为多维有限型子移位找到了非平凡的可判定问题（而大多数性质在该背景下是不可判定的）；并与近期关于"具有可计算语言"和"对某些性质具有极小性"之间关联的研究建立了联系，特别证明了该性质未必总能选择为共轭不变的。