arXiv subjects

Ville Salo

Publications and source records attributed to Ville Salo.

At least 37 records · Page 2Linked to original sources

arXiv2025

Symbol Frequencies in Surjective Cellular Automata

We study the behavior of probability measures under iteration of a surjective cellular automaton. We solve the following question in the negative: if the initial measure is ergodic and has full support, do all weak-* limit points of the sequence of measures have full support as well? The initial measure of our solution is not a product measure, and in this case the question remains open. To this end, we present a tool for studying the frequencies of symbols in preimages of surjective cellular automata, and prove some basic results about it. However, we show that by itself it is not enough to solve the stricter question in the positive.

math.DS↗

arXiv2025

Finitary Ryan's and local $\mathcal{Q}$ entropy for $\mathbb{Z}^{d}$ subshifts

For the action of a group $G$ by homeomorphisms on a space $X$, the automorphism group $\mathrm{Aut}(X,G)$ consists of all self-homeomorphisms of $X$ which commute with $x \mapsto g \cdot x$ for every $g \in G$. A theorem of Ryan shows that for an irreducible $\mathbb{Z}$-shift of finite type $(X,σ_{X})$, the center of $\mathrm{Aut}(X,σ_{X})$ is generated by the shift $σ_{X}$. A finitary version of this for $\mathbb{Z}$-shifts of finite type was proved by the second author for certain full shifts, and later generalized by Kopra to irreducible $\mathbb{Z}$-shifts of finite type. We generalize these finitary Ryan's theorems to shifts of finite type over more general groups. We prove that for contractible $\mathbb{Z}^{d}$-shifts of finite type with a fixed point, there is a finitely generated subgroup of the automorphism group whose centralizer in the group of homeomorphisms is the subgroup of shifts. We also prove versions of this for full shifts over any infinite, finitely generated group on sufficiently nice alphabet sizes. The stabilized automorphism group $\mathrm{Aut}^{(\infty)}(X,G)$ is the union of $\mathrm{Aut}(X,H)$ over all finite index subgroups $H \subset G$. Aimed at studying stabilized automorphism groups for shifts of finite type, we introduce an entropy-like quantity for pointed groups which we call local $\mathcal{Q}$ entropy, a generalization of a notion called local $\mathcal{P}$ entropy previously introduced by the first author. Using the finitary Ryan's theorems, we prove that the local $\mathcal{Q}$ entropy of the stabilized automorphism group of a contractible $\mathbb{Z}^{d}$-shift of finite type recovers the topological entropy of the underlying shift system up to a rational multiple. We then use this to give a complete classification up to isomorphism of the stabilized automorphism groups of full shifts over $\mathbb{Z}^{d}$.

math.DS↗

arXiv2025

Self-simulable groups

We say that a finitely generated group $Γ$ is self-simulable if every effectively closed action of $Γ$ on a closed subset of $\{\texttt{0},\texttt{1}\}^{\mathbb{N}}$ is the topological factor of a $Γ$-subshift of finite type. We show that self-simulable groups exist, that any direct product of non-amenable finitely generated groups is self-simulable, that under technical conditions self-simulability is inherited from subgroups, and that the subclass of self-simulable groups is stable under commensurability and quasi-isometries of finitely presented groups. Some notable examples of self-simulable groups obtained are the direct product $F_k \times F_k$ of two free groups of rank $k \geq 2$, non-amenable finitely generated branch groups, the simple groups of Burger and Mozes, Thompson's $V$, the groups $\operatorname{GL}_n(\mathbb{Z})$, $\operatorname{SL}_n(\mathbb{Z})$, $\operatorname{Aut}(F_n)$ and $\operatorname{Out}(F_n)$ for $n \geq 5$; The braid groups $B_m$ for $m \geq 7$, and certain classes of RAAGs. We also show that Thompson's $F$ is self-simulable if and only if $F$ is non-amenable, thus giving a computability characterization of this well-known open problem. We also exhibit a few applications of self-simulability on the dynamics of these groups, notably, that every self-simulable group with decidable word problem admits a nonempty strongly aperiodic subshift of finite type.

math.GR↗

arXiv2024

Solitaire of Independence

In this paper, we study a reversible process (more precisely, a groupoid/group action) resembling the classical 15-puzzle, where the legal moves are to ``move the unique hole inside a translate of a shape $S$''. Such a process can be defined for any finite subset $S$ of a group, and we refer to such a process as simply ``solitaire''. We develop a general theory of solitaire, and then concentrate on the simplest possible example, solitaire for the plane $\mathbb{Z}^2$, and $S$ the triangle shape (equivalently, any three-element set in general position). In this case, we give a polynomial time algorithm that puts any finite subset of the plane in normal form using solitaire moves, and show that the solitaire orbit of a line of consecutive ones -- the line orbit -- is completely characterised by the notion of a so-called fill matrix. We show that the diameter of the line orbit, as a graph with edges the solitaire moves, is cubic. We show that analogous results hold for the square shape, but indicate some shapes (still on the group $\mathbb{Z}^2$) where this is less immediate. We then explain in detail the connection of the solitaire to TEP and more generally permutive subshifts. Namely, the solitaire is a closure property of various sets of subsets of the group that can be associated to such a subshift, such as the independence, spanning and filling sets.

math.CO↗

arXiv2024

Snakes can be fooled into thinking they live in a tree

We construct a finitely generated group which is not virtually free, yet has decidable snake tiling problem. This shows that either a long-standing conjecture by Ballier and Stein (the characterization of groups with decidable domino problem as those virtually free ones) is false, or a question by Aubrun and Bitar has a positive answer (there exists a group for which the domino and snake problems are of different difficulty).

math.GR↗

arXiv2024

Soficity of free extensions of effective subshifts

Let $G$ be a group and $H\leqslant G$ a subgroup. The free extension of an $H$-subshift $X$ to $G$ is the $G$-subshift $\widetilde{X}$ whose configurations are those for which the restriction to every coset of $H$ is a configuration from $X$. We study the case of $G = H \times K$ for infinite and finitely generated groups $H$ and $K$: on the one hand we show that if $K$ is nonamenable and $H$ has decidable word problem, then the free extension to $G$ of any $H$-subshift which is effectively closed is a sofic $G$-subshift. On the other hand we prove that if both $H$ and $K$ are amenable, there are always $H$-subshifts which are effectively closed by patterns whose free extension to $G$ is non-sofic. We also present a few applications in the form of a new simulation theorem and a new class of groups which admit strongly aperiodic SFTs.

math.DS↗

arXiv2024

Avoshifts

An avoshift is a subshift where for each set $C$ from a suitable family of subsets of the shift group, the set of all possible valid extensions of a globally valid pattern on $C$ to the identity element is determined by a bounded subpattern. This property is shared (for various families of sets $C$) by for example cellwise quasigroup shifts, TEP subshifts, and subshifts of finite type with a safe symbol. In this paper we concentrate on avoshifts on polycyclic groups, when the sets $C$ are what we call ``inductive intervals''. We show that then avoshifts are a recursively enumerable subset of subshifts of finite type. Furthermore, we can effectively compute lower-dimensional projective subdynamics and certain factors (avofactors), and we can decide equality and inclusion for subshifts in this class. These results were previously known for group shifts, but our class also covers many non-algebraic examples as well as many SFTs without dense periodic points. The theory also yields new proofs of decidability of inclusion for SFTs on free groups, and SFTness of subshifts with the topological strong spatial mixing property.

math.DS↗

arXiv2024

Nonexpansive chaotic almost minimal systems on residually finite groups

The question of existence of nonexpansive chaotic almost minimal (CAM) systems, and the existence of CAM systems on every residually finite group, were raised in a recent paper of Van Cyr, Bryna Kra and Scott Schmieding. We construct nonexpansive CAM systems on all finitely-generated residually finite groups.

math.DS↗

arXiv2024

Finding codes on infinite grids automatically

We apply automata theory and Karp's minimum mean weight cycle algorithm to minimum density problems in coding theory. Using this method, we find the new upper bound $53/126 \approx 0.4206$ for the minimum density of an identifying code on the infinite hexagonal grid, down from the previous record of $3/7 \approx 0.4286$.

math.CO↗

arXiv2024

SFT covers for actions of the first Grigorchuk group

We study symbolic dynamical representations of actions of the first Grigorchuk group $G$, namely its action on the boundary of the infinite rooted binary tree, its representation in the topological full group of a minimal substitutive $\mathbb{Z}$-shift, and its representation as a minimal system of Schreier graphs. We show that the first system admits an SFT cover, and the latter two systems are conjugate to sofic subshifts on $G$, but are not of finite type.

math.DS↗

arXiv2024

Shifts on the lamplighter group

We prove that the lamplighter group admits strongly aperiodic SFTs, has undecidable tiling problem, and the entropies of its SFTs are exactly the upper semicomputable nonnegative real numbers, and some other results. These results follow from two relatively general simulation theorems, which show that for a large class of effective subshifts on the sea-level subgroup, their induction to the lamplighter group is sofic; and the pullback of every effective Cantor system on the integers admits an SFT cover. We exhibit a concrete strongly aperiodic set with $1488$ tetrahedra. We show that metabelian Baumslag-Solitar groups are intersimulable with lamplighter groups, and thus we obtain the same characterization for their entropies.

math.DS↗

arXiv2023

Gate lattices and the stabilized automorphism group

We study the stabilized automorphism group of a subshift of finite type with a certain gluing property called the eventual filling property, on a residually finite group $G$. We show that the stabilized automorphism group is simply monolithic, i.e. it has a unique minimal non-trivial normal subgroup -- the monolith -- which is additionally simple. To describe the monolith, we introduce gate lattices, which apply (reversible logical) gates on finite-index subgroups of $G$. The monolith is then precisely the commutator subgroup of the group generated by gate lattices. If the subshift and the group $G$ have some additional properties, then the gate lattices generate a perfect group, thus they generate the monolith. In particular, this is always the case when the acting group is the integers. In this case we can also show that gate lattices generate the inert part of the stabilized automorphism group. Thus we obtain that the stabilized inert automorphism group of a one-dimensional mixing subshift of finite type is simple.

math.GR↗

arXiv2023

Conjugacy of transitive SFTs minus periodic points

It is a question of Hochman whether any two one-dimensional topologically mixing subshifts of finite type (SFTs) with the same entropy are topologically conjugate when their periodic points are removed. We give a negative answer, in fact we prove the stronger result that there is a canonical correspondence between topological conjugacies of transitive SFTs and topological conjugacies between the systems obtained by removing the periodic points.

math.DS↗

arXiv2023

Automorphism groups of random substitution subshifts

We prove that for a suitably nice class of random substitutions, their corresponding subshifts have automorphism groups that contain an infinite simple subgroup and a copy of the automorphism group of a full shift. Hence, they are countable, non-amenable and non-residually finite. To show this, we introduce the concept of shuffles and generalised shuffles for random substitutions, as well as a local version of recognisability for random substitutions that will be of independent interest. Without recognisability, we need a more refined notion of recognisable words in order to understand their automorphisms. We show that the existence of a single recognisable word is often enough to embed the automorphism group of a full shift in the automorphism group of the random substitution subshift.

math.DS↗

arXiv2023

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↗

arXiv2023

Universal groups of cellular automata

We prove that the group of reversible cellular automata (RCA), on any alphabet $A$, contains a subgroup generated by three involutions which contains an isomorphic copy of every finitely generated group of RCA on any alphabet $B$. This result follows from a case study of groups of RCA generated by symbol permutations and partial shifts (equivalently, partitioned cellular automata) with respect to a fixed Cartesian product decomposition of the alphabet. For prime alphabets, we show that this group is virtually cyclic, and that for composite alphabets it is non-amenable. For alphabet size four, it is a linear group. For non-prime non-four alphabets, it contains copies of all finitely generated groups of RCA. We also prove this property for the group generated by RCA of biradius one on any full shift with large enough alphabet, and also for some perfect finitely generated groups of RCA.

math.GR↗

arXiv2023

Diddy: a Python toolbox for infinite discrete dynamical systems

We introduce Diddy, a collection of Python scripts for analyzing infinite discrete dynamical systems. The main focus is on generalized multidimensional shifts of finite type (SFTs). We show how Diddy can be used to easily define SFTs and cellular automata, and analyze their basic properties. We also showcase how to verify or rediscover some results from coding theory and cellular automata theory.

cs.MS↗

arXiv2023

The group of reversible Turing machines: subgroups, generators and computability

We study an abstract group of reversible Turing machines. In our model, each machine is interpreted as a homeomorphism over a space which represents a tape filled with symbols and a head carrying a state. These homeomorphisms can only modify the tape at a bounded distance around the head, change the state and move the head in a bounded way. We study three natural subgroups arising in this model: the group of finite-state automata, which generalizes the topological full groups studied in topological dynamics and the theory of orbit-equivalence; 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; and the group of elementary Turing machines, which are the machines which are obtained by composing finite-state automata and oblivious Turing machines. We show that both the group of oblivious Turing machines and that of elementary Turing machines are finitely generated, while the group of finite-state automata and the group of reversible Turing machines are not. We show that the group of elementary Turing machines has undecidable torsion problem. From this, we also obtain that the group of cellular automata (more generally, the automorphism group of any uncountable one-dimensional sofic subshift) contains a finitely-generated subgroup with undecidable torsion problem. We also show that the torsion problem is undecidable for the topological full group of a full $\mathbb{Z}^d$-shift on a non-trivial alphabet if and only if $d \geq 2$.

math.GR↗