arXiv subjects

Antonin Callard

Publications and source records attributed to Antonin Callard.

3 recordsLinked to original sources

arXiv2024

Computability of extender sets in multidimensional subshifts: asymptotic growths, dynamical constraints

Subshifts are sets of colorings of $\mathbb{Z}^d$ defined by families of forbidden patterns. In a given subshift, the extender set of a finite pattern is the set of all its admissible completions. Since soficity of $\mathbb{Z}$ subshifts is equivalent to having a finite number of extender sets, it had been conjectured that the number of extender sets could provide a way to separate the classes of sofic and effective subshifts in higher dimensions. We investigate some computational characterizations of extender sets in multidimensional subshifts, and in particular their growth, in terms of extender entropies (arXiv:1711.07515) and extender entropy dimensions. We prove here that sofic and effective subshifts have the same possible extender entropies (exactly the $\Pi_3$-computable real numbers of $[0,+\infty)$) and extender entropy dimensions, and investigate the computational complexity of these growth-type quantities under various dynamical and combinatorial constraints.

cs.DM↗

arXiv2022

Distortion element in the automorphism group of a full shift

We show that there is a distortion element in a finitely-generated subgroup $G$ of the automorphism group of the full shift, namely an element of infinite order whose word norm grows polylogarithmically. As a corollary, we obtain a lower bound on the entropy dimension of any subshift containing a copy of $G$, and that a sofic shift's automorphism group contains a distortion element if and only if the sofic shift is uncountable. We obtain also that groups of Turing machines and the higher-dimensional Brin-Thompson groups $mV$ admit distortion elements; in particular, $2V$ (unlike $V$) does not admit a proper action on a CAT$(0)$ cube complex. The distortion element is essentially the SMART machine.

math.GR↗

arXiv2022

The aperiodic Domino problem in higher dimension

The classical Domino problem asks whether there exists a tiling in which none of the forbidden patterns given as input appear. In this paper, we consider the aperiodic version of the Domino problem: given as input a family of forbidden patterns, does it allow an aperiodic tiling? The input may correspond to a subshift of finite type, a sofic subshift or an effective subshift. arXiv:1805.08829 proved that this problem is co-recursively enumerable ($Π_0^1$-complete) in dimension 2 for geometrical reasons. We show that it is much harder, namely analytic ($Σ_1^1$-complete), in higher dimension: $d \geq 4$ in the finite type case, $d \geq 3$ for sofic and effective subshifts. The reduction uses a subshift embedding universal computation and two additional dimensions to control periodicity. This complexity jump is surprising for two reasons: first, it separates 2- and 3-dimensional subshifts, whereas most subshift properties are the same in dimension 2 and higher; second, it is unexpectedly large.

cs.DM↗