Computability of extender sets in multidimensional subshifts

Abstract: Subshifts are colorings of $\mathbb{Z}^d$ defined by families of forbidden patterns. Given a subshift and a finite pattern, its extender set is the set of admissible completions of this pattern. It has been conjectured that the behavior of extender sets, and in particular their growth called extender entropy (arXiv:1711.07515), could provide a way to separate the classes of sofic and effective subshifts. We prove here that both classes have the same possible extender entropies: exactly the $\Pi_3$ real numbers of $[0,+\infty)$. We also consider computational properties of extender entropies for subshifts with some language or dynamical properties: computable language, minimal and some mixing properties.