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We revisit the problem of algorithmically deciding whether a given infinite connected graph has an Eulerian path, namely, a path that uses every edge exactly once. It has been recently observed that this problem is $D_3^0$-complete for graphs that have a computable description, whereas it is $\Pi_2^0$-complete for graphs that have a highly computable description, and that this same bound holds for the class of automatic graphs. A closely related problem consists of determining the number of ends of a graph, namely, the maximum number of distinct infinite connected components the graph can be separated into after removing a finite set of edges. The complexity of this problem for highly computable graphs is known to be $\Pi_2^0$-complete as well. The connection between these two problems lies in that only graphs with one or two ends can have Eulerian paths. In this paper we are interested in understanding the complexity of the infinite Eulerian path problem in the setting where the input graphs are known to have the right number of ends. We find that in this setting the problem becomes strictly easier, and that its exact difficulty varies according to whether the graphs have one or two ends, and to whether the Eulerian path we are looking for is one-way or bi-infinite. For example, we find that deciding existence of a bi-infinite Eulerian path for one-ended graphs is only $\Pi_1^0$-complete if the graphs are highly computable, and that the same problem becomes decidable for automatic graphs. Our results are based on a detailed computability analysis of what we call the Separation Problem, which we believe to be of independent interest. For instance, as a side application, we observe that K\"onig's infinity lemma, well known to be non-effective in general, becomes effective if we restrict to graphs with finitely many ends., Comment: 33 pages, 27 figures

The Medvedev degree of a subshift is a dynamical invariant of computable origin that can be used to compare the complexity of subshifts that contain only uncomputable configurations. We develop theory to describe how these degrees can be transferred from one group to another through algebraic and geometric relations, such as quotients, subgroups, translation-like actions and quasi-isometries. We use the aforementioned tools to study the possible values taken by this invariant on subshifts of finite type on some finitely generated groups. We obtain a full classification for some classes, such as virtually polycyclic groups and branch groups with decidable word problem. We also show that all groups which are quasi-isometric to the hyperbolic plane admit SFTs with nonzero Medvedev degree. Furthermore, we provide a classification of the degrees of sofic subshifts for several classes of groups., Comment: 25 pages

Let $G$ be a group with undecidable domino problem (such as $\mathbb{Z}^2$). We prove the undecidability of all nontrivial dynamical properties for sofic $G$-subshifts, that such a result fails for SFTs, and an undecidability result for dynamical properties of $G$-SFTs similar to the Adian-Rabin theorem. For $G$ amenable we prove that topological entropy is not computable from presentations of SFTs, and a more general result for dynamical invariants taking values in partially ordered sets., Comment: Only changes in exposition. Comments welcome!

Using tools from computable analysis we develop a notion of effectiveness for general dynamical systems as those group actions on arbitrary spaces that contain a computable representative in their topological conjugacy class. Most natural systems one can think of are effective in this sense, including some group rotations, affine actions on the torus and finitely presented algebraic actions. We show that for finitely generated and recursively presented groups, every effective dynamical system is the topological factor of a computable action on an effectively closed subset of the Cantor space. We then apply this result to extend the simulation results available in the literature beyond zero-dimensional spaces. In particular, we show that for a large class of groups, many of these natural actions are topological factors of subshifts of finite type., Comment: 40 pages, 1 gorgeous figure. Comments are welcome!

The Erd\H{o}s, Gr\"unwald and Weiszfeld theorem provides a characterization of infinite graphs which are Eulerian. That is, infinite graphs which admit infinite Eulerian trails. In this article we complement this theorem with a characterization of those finite trails that can be extended to infinite Eulerian trails. This allows us to prove an effective version of the Erd\H{o}s, Gr\"unwald and Weiszfeld theorem for a class of graphs that includes non locally finite ones, generalizing a theorem of D.Bean., Comment: 13 pages

We show that every infinite graph which is locally finite and connected admits a translation-like action by $\mathbb{Z}$ such that the distance between a vertex $v$ and $v\ast1$ is uniformly bounded by 3. This action can be taken to be transitive if and only if the graph has one or two ends. This strenghens a theorem by Brandon Seward. Our proof is constructive, and thus it can be made computable. More precisely, we show that a finitely generated group with decidable word problem admits a translation-like action by $\mathbb{Z}$ which is computable, and satisfies an extra condition which we call decidable orbit membership problem. As an application we show that on any finitely generated infinite group with decidable word problem, effective subshifts attain all effectively closed Medvedev degrees. This extends a classification proved by Joseph Miller for $\mathbb{Z}^{d},$ $d\geq1$.