A generalization of the simulation theorem for semidirect products
We generalize a result of Hochman in two simultaneous directions: Instead of realizing an effectively closed action as a factor of a subaction of a -SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with . Let be a finitely generated group and a semidirect product. We show that for any effectively closed -dynamical system where is a Cantor set, there exists a -subshift of finite type such that the -subaction of is an extension of . In the case where is an expansive action of a recursively presented group , a subshift conjugated to can be obtained as the -projective subdynamics of a -sofic subshift. As a corollary, we obtain that admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of is decidable.
A dynamical system is a tuple where is a set and is a map which describes the evolution of points of in time. In the case where is bijective one can describe as a -action by associating . This can be generalized to a set of bijective maps which satisfy some set of relations –for instance, the relation which indicates and commute–. These actions and their relations can be expressed by the group action where and .
More than often dynamical systems arising from group actions are difficult to study, and a fruitful technique is to look at their subactions, that is, the restriction of the group action to a particular subgroup. For instance, see the study of expansive subdynamics of actions [6, 10]. It is thus appealing to ask the following question: What systems can be obtained as subactions of a class of dynamical systems? An interesting class is the one of subshifts of finite type (SFT), that is, the sets of colorings of a group along with the shift action which are defined by a finite number of forbidden patterns.
For the class of -SFTs there is still no characterization of which dynamical systems can arise as their subactions, nevertheless, it has been proven by Hochman  that every -action over a cantor set which is effectively closed – meaning that it can be described with a Turing machine– admits an almost trivial isometric extension which can be realized as the subaction of a -SFT. This result has subsequently been improved for the expansive case independently in  and  showing that every effectively closed subshift can be obtained as the projective subdynamics of a sofic -subshift. These kind of results yield powerful techniques to prove properties about the original systems. An example is the characterization of the set of entropies of -SFTs  as the set of right recursively enumerable numbers.
In this article we extend Hochman’s result to the case of group actions for groups which are of the form for some finitely generated group and an homomorphism . More specifically we prove the following result.
For every -effectively closed dynamical system there exists a -SFT whose -subaction is an extension of .
We remark the strong gap which occurs when passing from -SFTs to the multidimensional case. For instance, -SFTs contain periodic points, have regular languages and the possible set of entropies they can have is reduced to logarithms of Perron numbers . In the other hand multidimensional SFTs can be strongly aperiodic [5, 20, 16, 15], can be composed uniquely of non-computable points [11, 19] and their entropies are not even computable . Most of these differences can be put into evidence with simulation theorems by the fact that multidimensional SFTs can be projected onto effectively closed subshifts in one dimension. Our Theorem 3.1 allows analogously to extend properties of effectively closed subshifts in general groups and show that they also appear in SFTs when the group is replaced by . This is a powerful tool to construct examples of groups with admit subshifts of finite type with some desired property which is easier to realize in an effective subshift.
Readers who are not familiar with computability or the embedding of Turing machine computations in subshifts of finite type will be reassured by the fact that in the proof all of those aspects are hidden in black boxes. Namely, we use the result of [3, 9] that every effectively closed -subshift is the projective subdynamics of a sofic -subshift whose vertical shift action is trivial. We also make use of a theorem of Mozes  which states that subshifts arising from two-dimensional substitutions are sofic.
In the case when is a recursively presented group, Theorem 3.1 can be presented in a purely symbolic dynamics fashion for expansive actions, namely we show:
Let be an effectively closed -subshift. Then there exists a sofic -subshift such that its -projective subdynamics is .
It is known that every -SFT contains a periodic configuration . However, it was shown by Berger  that there are -SFTs which are strongly aperiodic, that is, such that the shift acts freely on the set of configurations. This result has been proven several times with different techniques [20, 16, 15] giving a variety of constructions. However, it remains an open question which is the class of groups which admit strongly aperiodic SFTs. Amongst the class of groups that do admit strongly aperiodic SFTs are: for , hyperbolic surface groups , Osin and Ivanov monster groups , and the direct product for a particular class of groups which includes Thompson’s group and . It is also known that no group with two or more ends can contain strongly aperiodic SFTs  and that recursively presented groups which admit strongly aperiodic SFTs must have decidable word problem .
As an application of Theorem 3.1 we present a new class of groups which admit strongly aperiodic SFTs, that is:
Every semidirect product where is finitely generated and has decidable word problem admits a non-empty strongly aperiodic SFT.
Amongst this new class of groups which admit strongly aperiodic SFTs, we remark the Heisenberg group which admits a presentation .
Consider a group and a compact topological space . The tuple where is a left action by homeomorphisms is called a -flow (or -dynamical system). Let , be two -flows. We say is a morphism if it is continuous and for all . A surjective morphism is a factor and we say that is a factor of and that is an extension of . When is a bijection and its inverse is continuous we say it is a conjugacy and that is conjugated to .
In what follows, we consider only cantor sets with the product topology and finitely generated groups. Without loss of generality, we consider actions over closed subsets of . Let be a group generated by a finite set . A -effectively closed flow is a -flow where:
is a closed effective subset: where is a recursively enumerable language. That means that is the complement of a union of cylinders which can be enumerated by a Turing machine.
is an effectively closed action: there exists a Turing machine which on entry and enumerates a sequence of words such that .
The idea behind the definition is the following: There is a Turing machine which given a word representing an element of and coordinates of returns an approximation of the preimage of by .
Let be a finite alphabet and a finitely generated group. The set equipped with the left group action given by: is the -full shift. The elements and are called symbols and configurations respectively. We endow with the product topology, therefore obtaining a compact metric space. The topology is generated by the metric where is the length of the smallest expression of as the product of some fixed set of generators. This topology is also generated by a clopen basis given by the cylinders . A support is a finite subset . Given a support , a pattern with support is an element of , i.e. a finite configuration and we write . We also denote the cylinder generated by centered in as . If for some we write .
A subset of is a -subshift if it is -invariant – – and closed for the cylinder topology. Equivalently, is a -subshift if and only if there exists a set of forbidden patterns that defines it.
That is, a -subshift is a subset of which can be written as the complement of a union of cylinders.
If the context is clear enough, we will drop the and simply refer to a subshift. A subshift is of finite type – SFT for short – if there exists a finite set of forbidden patterns such that . A subshift is sofic if there exists a subshift of finite type and a factor . A subshift is effectively closed if there exists a recursively enumerable coding of a set of forbidden patterns such that . More details can be found in  or in Section 4.
Any -flow over a cantor set can be seen as a subshift over an infinite alphabet: Indeed, can be seen as equipped with the shift action such that if and only if . In this setting, effectively closed -flows correspond to effectively closed subshifts in this infinite alphabet.
Let be a subgroup and a -flow. The -subaction of is where is the restriction of to , that is . In the case of a subshift there is also the different notion of projective subdynamics. The -projective subdynamics of is the set .
3 Simulation Theorem
The purpose of this section is to prove our main result.
Let be finitely generated group and . For every -effectively closed flow there exists a -SFT whose -subaction is an extension of .
We begin by introducing some general constructions. The general schema of the proof is the following: First we construct for each non-zero vector a substitution which encodes countable copies of as lattices with the property that any automorphism sends each of the lattices of to those of where is the automorphism obtained by projecting each component to . This structure allows us to pair lattices of when moving in by elements of .
Then we encode the elements of and the -flow in an effective Toeplitz -subshift. We do so in a way that the projections of the -th order lattice to the line in the previous construction always matches with the symbol . For technical reasons of matching all possible lattices, we do this coding in two different ways.
Afterwards, we extend the Toeplitz subshift to a -subshift by repeating rows (or columns). Using a known simulation theorem we obtain that this object is a sofic -subshift from which we extract an SFT extension.
Finally, we extend this construction to by adding local rules that ensure that if a -coset codes the point then the coset of given by the action of codes . This set of rules is coded as a finite amount of forbidden patterns.
Finally, we define the factor code, and show that it satisfies the required properties.
3.1 A substitution which encodes an action of .
Let . We define a substitution over a two symbol alphabet which generates a sofic -subshift encoding translations of for . In the proof of the simulation theorem we will only use the case where , but we prefer to proceed here with more generality.
Let and . The -substitution is defined by:
As an example, if and we get the following:
In this example we obtain that the patterns and are: