Part 2: Regular PapersInternational audienceWe consider Turing machines as actions over configurations in ${\mathrm{\Sigma <!--\; \Sigma \; -->}}^{{\mathbb{Z}}^d}$which only change them locally around a marked position that can move and carry a particular state. In this setting we study the monoid of Turing machines and the group of reversible Turing machines. We also study two natural subgroups, namely the group of finite-state automata, which generalizes the topological full groups studied in the theory of orbit-equivalence, and the group of oblivious Turing machines whose movement is independent of tape contents, which generalizes lamplighter groups and has connections to the study of universal reversible logical gates. Our main results are that the group of Turing machines in one dimension is neither amenable nor residually finite, but is locally embeddable in finite groups, and that the torsion problem is decidable for finite-state automata in dimension one, but not in dimension two

A theorem of Gao, Jackson and Seward, originally conjectured to be false by Glasner and Uspenskij, asserts that every countable group admits a 2-coloring. A direct consequence of this result is that every countable group has a strongly aperiodic subshift on the alphabet {0, 1}. In this article, we use Lovász local lemma to first give a new simple proof of said theorem, and second to prove the existence of a G-effectively closed strongly aperiodic subshift for any finitely generated group G. We also study the problem of constructing subshifts which generalize a property of Sturmian sequences to finitely generated groups. More precisely, a subshift over the alphabet {0, 1} has uniform density α ∈ [0, 1] if for every configuration the density of 1's in any increasing sequence of balls converges to α. We show a slightly more general result which implies that these subshifts always exist in the case of groups of subexponential growth.

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We generalize the classical definition of effectively closed subshift to finitely generated groups. We study classical stability properties of this class and then extend this notion by allowing the usage of an oracle to the word problem of a group. This new class of subshifts forms a conjugacy class that contains all sofic subshifts. Motivated by the question of whether there exists a group where the class of sofic subshifts coincides with that of effective subshifts, we show that the inclusion is strict for several groups, including recursively presented groups with undecidable word problem, amenable groups and groups with more than two ends. We also provide an extended model of Turing machine which uses the group itself as a tape and characterizes our extended notion of effectiveness. As applications of these machines we prove that the origin constrained domino problem is undecidable for any group of the form G × Z subject to a technical condition on G and we present a simulation theorem which is valid in any finitely generated group.

We provide a unifying approach which links results on algebraic actions by Lind and Schmidt, Chung and Li, and a topological result by Meyerovitch that relates entropy to the set of asymptotic pairs. In order to do this we introduce a series of Markovian properties and, under the assumption that they are satisfied, we prove several results that relate topological entropy and asymptotic pairs (the homoclinic group in the algebraic case). As new applications of our method, we give a characterization of the homoclinic group of any finitely presented expansive algebraic action of (1) any elementary amenable, torsion-free group or (2) any left orderable amenable group, using the language of independence entropy pairs.

We generalize a result of Hochman in two simultaneous directions: Instead of realizing an effectively closed Z d action as a factor of a subaction of a Z d+2 -SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with Z 2 . Let H be a finitely generated group and G = Z 2 ⋊ H a semidirect product. We show that for any effectively closed H-dynamical system (Y, f ) where Y is a Cantor set, there exists a G-subshift of finite type (X, σ) such that the H-subaction of (X, σ) is an extension of (Y, f ). In the case where f is an expansive action of a recursively presented group H, a subshift conjugated to (Y, f ) can be obtained as the H-projective subdynamics of a G-sofic subshift. As a corollary, we obtain that G admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of H is decidable.

We formulate and prove a very general relative version of the Dobrushin-Lanford-Ruelle theorem which gives conditions on constraints of configuration spaces over a finite alphabet such that for every absolutely summable relative interaction, every translation-invariant relative Gibbs measure is a relative equilibrium measure and vice versa. Neither implication is true without some assumption on the space of configurations. We note that the usual finite type condition can be relaxed to a much more general class of constraints. By "relative" we mean that both the interaction and the set of allowed configurations are determined by a random environment. The result includes many special cases that are well known. We give several applications including 1) Gibbsian properties of measures that maximize pressure among all those that project to a given measure via a topological factor map from one symbolic system to another; 2) Gibbsian properties of equilibrium measures for group shifts defined on arbitrary countable amenable groups; 3) A Gibbsian characterization of equilibrium measures in terms of equilibrium condition on lattice slices rather than on finite sets; 4) A relative extension of a theorem of Meyerovitch, who proved a version of the Lanford-Ruelle theorem which shows that every equilibrium measure on an arbitrary subshift satisfies a Gibbsian property on interchangeable patterns.