arXiv subjects

Nicanor Carrasco-Vargas

Publications and source records attributed to Nicanor Carrasco-Vargas.

13 recordsLinked to original sources

arXiv2026

Isomorphisms and slow entropy of deterministic $[T,T^{-1}]$ systems

We study skew product measure-preserving systems driven by an irrational rotation and the step function with values $1$ and $-1$ on each half of the circle. These systems can be seen as a deterministic version of Kalikow's $[T,T^{-1}]$ system, and are particular instances of Rokhlin cocycle extensions. Under the assumption of ergodicity, we show that these systems obey an interesting rigidity property for isomorphisms. That is, we show that two skew products are isomorphic if and only if the rotations in the base are isomorphic, and the fiber transformations are flip isomorphic. Under mild extra assumptions, we are able to characterize all isomorphisms. We prove that by choosing the rotation angle suitably, these systems can realize arbitrarily low measure-theoretic complexity in the sense of lower slow entropy. We apply this result to obtain new examples of systems that fail the variational principle for slow entropy. These examples can have a set of invariant measures as rich as desired (any metrizable Choquet simplex, up to affine homeomorphism).

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arXiv2026

Medvedev degrees of SFTs on cocompact Fuchsian groups

We prove that for any cocompact Fuchsian group the class of Medvedev degrees attained by subshifts of finite type is the class of $Π_1^0$ degrees. This result relies on the construction of a rigid hierarchical structure in a graph model of the hyperbolic plane, and on the fact that every such group admits a representation in $\operatorname{PSL}_2(\mathbb{R})$ such that the matrices in the image have coefficients that are computable real numbers.

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arXiv2026

Medvedev degrees of subshifts on groups

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.

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arXiv2026

Parametrized complexity of relations between multidimensional subshifts

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.

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arXiv2026

Paths, Ends and The Separation Problem for Infinite Graphs

We introduce and study the Separation Problem for infinite graphs, which involves determining whether a connected graph splits into at least two infinite connected components after the removal of a given finite set of edges. We prove that this problem is decidable for every highly computable graph with finitely many ends. Using this result, we demonstrate that König's Infinity Lemma is effective for such graphs. We also apply it to analyze the complexity of the Eulerian Path Problem for infinite graphs, showing that much of its complexity arises from counting ends. Indeed, the Eulerian Path Problem becomes strictly easier when restricted to graphs with a fixed number of ends. Under this restriction, we provide a complete characterization of the problem. Finally, we study the Separation Problem in a uniform setting (i.e., where the graph is also part of the input) and offer a nearly complete characterization of its complexity and its relationship to counting the number of ends.

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arXiv2026

The strong topological Rokhlin property and Medvedev degrees of SFTs

We prove that if a recursively presented group admits a (nonempty) subshift of finite type with nonzero Medvedev degree then it fails to have the strong topological Rokhlin property. This result simplifies a known criterion and provides new examples of recursively presented groups without this property.

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arXiv2025

Topological slow entropy, sequence entropy, and generalized $[T,T^{-1}]$ systems

We consider topological dynamical systems given by skew products $S\rtimes_τ T$, where $S\colon Y\to Y$ is a subshift, $τ\colon Y\to\mathbb{Z}$ is a continuous cocycle, and $T$ is an arbitrary invertible topological system. For fixed $(Y,S,τ)$ it may happen that all systems of the form $S\rtimes_τT$ have the same topological entropy, and thus it arises the problem of distinguishing two such systems. We show that if $T_1$ and $T_2$ are invertible topological dynamical systems with different topological entropy then $S\rtimes_τ T_1$ and $S\rtimes_τ T_1$ can be distinguished using slow entropy as introduced by Katok and Thouvenot. We prove a similar result under the assumption that the fiber systems have different slow entropy at some scale (this can be applied if $T_1$ and $T_2$ have both zero entropy, or have the same entropy). These results require rather mild assumptions on $(Y,S,τ)$, and can be applied to some entropy-zero systems in the base. We generalize classical results in the theory of sequence entropy, which were proved by Goodman with the additional assumption of finite topological dimension. We show that the measure-theoretic sequence entropy of any system is bounded by its topological sequence entropy. Under an extra assumption on the sequence we establish a variational principle, and prove that the topological sequence entropy of a system equals its topological entropy multiplied by a positive constant that depends only on the sequence.

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arXiv2025

Subshifts on groups and computable analysis

The study of subshifts on groups different from $\mathbb{Z}$, such as $\mathbb{Z}^d$, $d\geq 2$, has been a subject of intense research in recent years. These investigations have unveiled aremarkable connection between dynamics and recursion theory. Different questions about the dynamics of these systems have been answered in recursion-theoretical terms. In this work we further explore this connection. We use the framework of computable analysis to explore the class of effective dynamical systems on metric spaces, and relate these systems to subshifts of finite type (SFTs) on groups. We prove that every effective dynamical system on a general metric space is the topological factor of an effective dynamical system with topological dimension zero. We combine this result with existing simulation results to obtain new examples of systems that are factors of SFTsWe also study a conjugacy invariant for subshifts on groups called Medvedev degree. This invariant is a complexity measure of algorithmic nature. We develop the basic theory of these degrees for subshifts on arbitrary finitely generated groups. Using these tools we are able to classify the values that this invariant attains for SFTs and other classes of subshifts on several groups. Furthermore, we establish a connection between these degrees and the distribution of isolated points in the space of all subshifts. Motivated by the study of Medvedev degrees of subshifts, we also consider translation-like actions of groups on graphs. We prove that every connected, locally finite, and infinite graph admits a translation by $\mathbb{Z}$, and that this action can be chosen transitive exactly when the graph has one or two ends. This generalizes a result of Seward about translation-like actions of $\mathbb{Z}$ on finitely generated groups.

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arXiv2025

Translation-like actions by $\mathbb{Z}$, the subgroup membership problem, and Medvedev degrees of effective subshifts

We show that every infinite, locally finite, and connected graph admitsa translation-like action by $\mathbb{Z}$, and that this action can be takento be transitive exactly when the graph has either one or two ends.The actions constructed satisfy $d(v,v\ast 1)\leq3$ for every vertex$v$. This strengthens a theorem by Brandon Seward. We also study the effective computability of translation-like actionson groups and graphs. We prove that every finitely generated infinitegroup with decidable word problem admits a translation-like actionby $\mathbb{Z}$ which is computable, and satisfies an extra condition whichwe call decidable orbit membership problem. As a nontrivial application of our results, we prove that for everyfinitely generated infinite group with decidable word problem, effectivesubshifts attain all $Π_{1}^{0}$ Medvedev degrees. This extends a classification proved by Joseph Miller for $\mathbb{Z}^{d},$ $d\geq1$.

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arXiv2025

Infinite Eulerian paths are computable on graphs with vertices of infinite degree

The Erdős, Grünwald, and Weiszfeld theorem is a characterization of those infinite graphs which are Eulerian. That is, infinite graphs that admit infinite Eulerian paths. In this article we prove an effective version of the Erdős, Grünwald, and Weiszfeld theorem for a class of graphs where vertices of infinite degree are allowed, generalizing a theorem of D.Bean. Our results are obtained from a characterization of those finite paths in a graph that can be extended to infinite Eulerian paths.

math.CO↗

arXiv2025

On a Rice theorem for dynamical properties of SFTs on groups

Let $G$ be a group with undecidable domino problem, such as $\mathbb{Z}^2$. We prove that all nontrivial dynamical properties for sofic $G$-subshifts are undecidable, that this is not true for $G$-SFTs, and an undecidability result for dynamical properties of $G$-SFTs similar to the Adian-Rabin theorem. Furthermore we prove a Rice-like result for dynamical invariants asserting that every computable real-valued invariant for $G$-SFTs that is monotone by disjoint unions and products is constant.

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arXiv2025

The automorphism group of a strongly irreducible subshift on a group

We study the automorphism group $\operatorname{Aut}(X)$ of a non-trivial strongly irreducible subshift $X$ on an arbitrary infinite group $G$ and generalize classical results of Ryan, Kim and Roush. We generalize Ryan's theorem by showing that the center of $\operatorname{Aut}(X)$ is generated by shifts by elements of the center of $G$ modded out by the kernel of the shift action. We generalize Kim and Roush's theorem by showing that if the free group $F_k$ of rank $k\geq 1$ embeds into $G$, then the automorphism group of any full $F_k$-shift embeds into $\operatorname{Aut}(X)$. If $X$ is an SFT, or more generally, if $X$ satisfies the strong topological Markov property, then we can weaken the conditions on $G$. In this case we show that the automorphism group of any full $\mathbb{Z}$-shift embeds into $\operatorname{Aut}(X)$ provided $G$ is not locally finite, and that the automorphism group of any full $F_k$-shift embeds into $\operatorname{Aut}(X)$ whenever $G$ is nonamenable. Our results rely on a new marker lemma which is valid for any nonempty strongly irreducible subshift on an infinite group. We remark that our results are new even for $G=\mathbb{Z}$ as they do not require the subshift to be an SFT.

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arXiv2024

Effective dynamical systems beyond dimension zero and factors of SFTs

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.

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