Factoring strongly irreducible group shift actions onto full shifts of lower entropy

A well-known result in the study of symbolic dynamical systems states that any subshift of finite type (SFT) with the action of $\mathbb{Z}$ and entropy greater than or equal to log $𝑁$ factors onto the full shift over $𝑁$ symbols—this was proven in [9] and [1] for the cases of equal and unequal entropy, respectively. Extending these results for actions of other groups has been difficult, and it is known that a factor map onto a full shift of equal entropy may not exist, even in the case of ${\mathbb{Z}}^{𝑑}\text{,}$ where 𝑑 > 1 (see [3]). Johnson and Madden [7] showed that any SFT with the action of ${\mathbb{Z}}^{𝑑}\text{,}$ which has entropy greater than log $𝑁$ and satisfies an additional mixing condition (known as corner gluing), has an extension which is finite-to-one (hence of equal entropy) and maps onto the full shift over $𝑁$ symbols. This result was later improved by Desai in [4] to show that such a system factors directly onto the full shift, without the intermediate extension, and then by Boyle, Pavlov, and Schraudner [2] to replace the corner gluing by a weaker mixing condition (block gluing).

In this paper, we use similar methods to adapt these constructions to symbolic dynamical systems with actions of amenable groups. Our approach requires three assumptions: that the group $𝐺$ has the comparison property (satisfied, for instance, for all countable amenable groups with subexponential growth), that the system $𝑋$ is strongly irreducible (which replaces the corner gluing condition and allows the construction to be valid without assuming that the underlying system be an SFT), and that it has nonperiodic blocks for all possible sets of periods. With these assumptions, we prove that $𝑋$ can be factored onto any full shift of smaller entropy (i.e., a full shift over $𝑁$ symbols, where log N < h(X)). We note that our method does not apply for the case of equal entropy (log N = h(X)); as we mentioned earlier, Boyle and Schraudner [3] have shown that there exist ${\mathbb{Z}}^{𝑑}$-shifts of finite type which do not factor onto full shifts of equal entropy.

In this section, we establish the definitions, notation, and standard facts we will use in this paper. Since this is mainly standard material, we omit most proofs and references.

The result below seems classical, but for the convenience of the reader, we include the elementary proof.

Let $𝐷$ denote the irreducibility distance of $𝑋\text{.}$ By Theorem 2.14, there exists a tiling 𝒯 of $𝐺$ such that h(𝒯)= 0 and the shapes of 𝒯 can have arbitrarily good invariance properties, which we will specify within the next few sentences. Enumerate the shapes of 𝒯 by ${𝑆}_1,{𝑆}_2,\dots ,{𝑆}_{𝐽}\text{.}$ There exist some ${𝛿}^{\ast },{𝛿}^{\ast \ast }\text{,}$ and $𝑛$ such that if the shapes of 𝒯 are $(𝐷,{𝛿}^{\ast })$ and $({𝐹}_{𝑛},{𝛿}^{\ast \ast })$-invariant, then for every $𝑗$ we can choose a family of blocks ${ℬ}_{𝑗}\subset {ℬ}_{({𝑆}_{𝑗}{)}_{𝐷}}(𝑋)$ such that if we denote ${𝑁}_{𝑗}=|{ℬ}_{𝑗}|\text{,}$ then we have $𝑎+𝜂<\frac{1}{|({𝑆}_{𝑗}{)}_{𝐷}|}\mathrm{𝑙𝑜𝑔}{𝑁}_{𝑗}<𝑏-𝜂\text{.}$ In addition, since for small enough ${𝛿}^{\ast }$ the relative difference between $|({𝑆}_{𝑗}{)}_{𝐷}|$ and $|{𝑆}_{𝑗}|$ can be arbitrarily small, we can even write

$𝑎+𝜂<\frac{1}{|{𝑆}_{𝑗}|}\mathrm{𝑙𝑜𝑔}{𝑁}_{𝑗}<𝑏-𝜂.$

 

Now, let $𝑌$ be the orbit closure of the set of all points $𝑥\in 𝑋$ such that for every $𝑇\in 𝒯$ $𝑥{|}_{{𝑇}_{𝐷}}\in {ℬ}_{𝑗}\text{,}$ where $𝑗$ is such that ${𝑆}_{𝑗}$ is the shape of $𝑇\text{.}$ Observe that strong irreducibility, combined with the fact that ${𝑇}_{𝐷}$ is disjoint from all other tiles of 𝒯, means (via a standard compactness argument) that we can choose the blocks $𝑥{|}_{{𝑇}_{𝐷}}$ independently of each other, and any such choice will yield a valid element of $𝑌\text{.}$

We will estimate the entropy of $𝑌$ by considering the number of blocks with domain ${𝐹}_{𝑛}$ as $𝑛$ increases to infinity, beginning with estimating this number of blocks described above. Fix some large $𝑛$ and note that every block $B$ with domain ${𝐹}_{𝑛}$ that occurs in $𝑌$ has the property that there exists some right-translate 𝒯′ of 𝒯 such that for every tile $𝑇$ of 𝒯′ such that $𝑇\subset {𝐹}_{𝑛}$ the block $𝐵{|}_{{𝑇}_{𝐷}}$ belongs to ${ℬ}_{𝑗}\text{,}$ where $𝑗$ is such that ${𝑆}_{𝑗}$ is the shape of $𝑇\text{.}$ Let ${𝐻}_{𝑛}$ be the number of ways in which the right-translates of 𝒯 can intersect ${𝐹}_{𝑛}\text{,}$ or equivalently, the number of ways in which the right-translates of ${𝐹}_{𝑛}$ can intersect 𝒯. Since 𝒯 has entropy 0, for large enough $𝑛$ we have $\frac{1}{|{𝐹}_{𝑛}|}\mathrm{𝑙𝑜𝑔}{𝐻}_{𝑛}<\frac{𝜂}{2}\text{.}$

For any such right-translate 𝒯′ let ${𝑙}_{𝑗}$ be the number of tiles of ${𝒯}^{\prime }$ with shape ${𝑆}_{𝑗}$ that are subsets of ${𝐹}_{𝑛}\text{.}$ Note that if $𝑛$ is large enough, then ${\sum }_{𝑗=1}^{𝐽}{𝑙}_{𝑗}|{𝑆}_{𝑗}|$ is almost equal to $|{𝐹}_{𝑛}|\text{.}$ Since for every tile of ${𝒯}^{\prime }$ the block $𝐵{|}_{{𝑇}_{𝐷}}$ is a block from ${ℬ}_{𝑗}\text{,}$ and thus one of the ${𝑁}_{𝑗}$ possible blocks, the $𝐷$-interiors of the tiles of ${𝒯}^{\prime }$ can be filled in at most $({\prod }_{𝑗=1}^{𝐽}{𝑁}_{𝑗}^{{𝑙}_{𝑗}})$ ways. We have no control over the symbols outside these interiors, but we know that there are at most $|{𝐹}_{𝑛}|-{\sum }_{𝑗=1}^{𝐽}{𝑙}_{𝑗}|({𝑆}_{𝑗}{)}_{𝐷}|$ such symbols. It follows that the number of blocks associated with ${𝒯}^{\prime }$ is at most

$(\prod _{𝑗=1}^{𝐽}{𝑁}_{𝑗}^{{𝑙}_{𝑗}})·|\Lambda {|}^{|{𝐹}_{𝑛}|-\sum _{𝑗=1}^{𝐽}{𝑙}_{𝑗}|({𝑆}_{𝑗}{)}_{𝐷}|}.$

 

Taking logarithms and dividing by $|{𝐹}_{𝑛}|\text{,}$ we obtain

$\frac{1}{|{𝐹}_{𝑛}|}\underset{𝑗=1}{\overset{𝐽}{\sum ︁}}{𝑙}_{𝑗}\mathrm{𝑙𝑜𝑔}{𝑁}_{𝑗}+\frac{|{𝐹}_{𝑛}|-\sum _{𝑗=1}^{𝐽}{𝑙}_{𝑗}|({𝑆}_{𝑗}{)}_{𝐷}|}{|{𝐹}_{𝑛}|}\mathrm{𝑙𝑜𝑔}|\Lambda |.$

 

If ${𝛿}^{\ast }$ was small enough so that the $𝐷$-interiors of the tiles of ${𝒯}^{\prime }$ form a $(1-\frac{𝜂}{2|\Lambda |})$-covering quasitiling (which we can safely assume since $𝐷\text{,}$ $𝜂\text{,}$ and $\Lambda$ were known before we started the construction), then for large enough $𝑛$ the second term will not exceed $\frac{𝜂}{2}\text{.}$ As for the first term, we can further estimate it as follows:

$\frac{1}{|{𝐹}_{𝑛}|}\underset{𝑗=1}{\overset{𝐽}{\sum ︁}}{𝑙}_{𝑗}\mathrm{𝑙𝑜𝑔}{𝑁}_{𝑗}=\frac{1}{|{𝐹}_{𝑛}|}\underset{𝑗=1}{\overset{𝐽}{\sum ︁}}{𝑙}_{𝑗}|{𝑆}_{𝑗}|\frac{1}{|{𝑆}_{𝑗}|}\mathrm{𝑙𝑜𝑔}{𝑁}_{𝑗}<(𝑏-𝜂)\frac{1}{|{𝐹}_{𝑛}|}\underset{𝑗=1}{\overset{𝐽}{\sum ︁}}{𝑙}_{𝑗}|{𝑆}_{𝑗}|<𝑏-𝜂.$

 

It follows that for any right-translate ${𝒯}^{\prime }$ of 𝒯, the number of blocks in $𝑌$ with domain ${𝐹}_{𝑛}$ that have blocks from ${ℬ}_{𝑗}$ in every tile of ${𝒯}^{\prime }$ with shape ${𝑆}_{𝑗}$ does not exceed $2^{(𝑏-\frac{𝜂}{2})|{𝐹}_{𝑛}|}\text{,}$ and thus the cardinality of ${ℬ}_{{𝐹}_{𝑛}}(𝑌)$ does not exceed ${𝐻}_{𝑛}{2}^{(𝑏-\frac{𝜂}{2})|{𝐹}_{𝑛}|}\text{.}$ This lets us estimate that

$\frac{1}{|{𝐹}_{𝑛}|}\mathrm{𝑙𝑜𝑔}|{ℬ}_{{𝐹}_{𝑛}}(𝑌)|\le \frac{1}{|{𝐹}_{𝑛}|}\mathrm{𝑙𝑜𝑔}{𝐻}_{𝑛}+𝑏-\frac{𝜂}{2}<𝑏,$

 

and hence,

$ℎ(𝑌)𝑏.$

 

The lower bound for entropy is much simpler—for any $𝑛$ the number of blocks with domain ${𝐹}_{𝑛}$ which occur in $𝑌$ is equal to at least $({\prod }_{𝑗=1}^{𝐽}{𝑁}_{𝑗}^{{𝑙}_{𝑗}})\text{,}$ where ${𝑙}_{𝑗}$ is the number of tiles of $𝒯$ with shape ${𝑆}_{𝑗}$ which are subsets of ${𝐹}_{𝑛}$ (this time we do not even need to consider possible translates of $𝒯$). Consequently,

$\begin{array}{llll}\hfill \frac{1}{|{𝐹}_{𝑛}|}\mathrm{𝑙𝑜𝑔}|{ℬ}_{{𝐹}_{𝑛}}(𝑌)|\ge \frac{1}{|{𝐹}_{𝑛}|}\underset{𝑗=1}{\overset{𝐽}{\sum ︁}}{𝑙}_{𝑗}\mathrm{𝑙𝑜𝑔}{𝑁}_{𝑗}& =\frac{1}{|{𝐹}_{𝑛}|}\underset{𝑗=1}{\overset{𝐽}{\sum ︁}}{𝑙}_{𝑗}|{𝑆}_{𝑗}|\frac{1}{|{𝑆}_{𝑗}|}\mathrm{𝑙𝑜𝑔}{𝑁}_{𝑗}& \hfill & \\ \hfill & >(𝑎+𝜂)\frac{\sum _{𝑗=1}^{𝐽}{𝑙}_{𝑗}|{𝑆}_{𝑗}|}{|{𝐹}_{𝑛}|},& \hfill & \end{array}$

 

which for large enough $𝑛$ will be greater than $𝑎\text{,}$ and therefore $ℎ(𝑌)>𝑎\text{,}$ which concludes the proof. ■

According to Theorem 2.7, if $ℎ(𝑋)>\mathrm{𝑙𝑜𝑔}𝑁\text{,}$ it is possible to define a surjective block-to-block map from ${ℬ}_{{𝐹}_{𝑛}}(𝑋)$ onto ${\{0,1,\dots ,𝑁-1\}}^{{𝐹}_{𝑛}}\text{,}$ for sufficiently large $𝑛\text{.}$ This allows us, with the use of a tiling, to build a map $\mathrm{\Pi }$ from $𝑋$ onto full $𝐺$-shift over $𝑁$ symbols, by defining it on a tile-by-tile basis. However, without some extra care such a map might not commute with the group action, that is, $\mathrm{\Pi }(𝑔𝑥)$ might not be equal to $𝑔\mathrm{\Pi }(𝑥)\text{.}$ Therefore, we need to work with the subshift ${𝑋}_{𝒯}$ rather than directly with the tiling $𝒯\text{,}$ and we have to find a suitable way of “decoding” a tiling based on the content of $𝑥\in 𝑋\text{.}$ To achieve this, we will use special blocks called marker blocks or, more concisely, markers. Each tile will have its own marker block which we will place inside, for example, in the center, and outside the marker, we will leave enough space to apply the block-to-block map. That way the factor map will involve first locating the marker block to determine the tiling and then using the block-to-block map to determine the content of the image within each tile. The procedure is possible due to strong irreducibility: First, when we put marker blocks in tiles (strong irreducibility gives us a lot of freedom in specifying the content of a tile except within a small “neighborhood” of the marker blocks and near the boundary of the tile) and again, when we “glue” the contents of tiles together to get final point (i.e., we will specify the symbols in the part of the tile that is suitably “far from the boundary,” and strong irreducibility will ensure that any combination of two tiles with such specified content will occur somewhere in $𝑋$). A very important element of the construction (and a major source of difficulties) will be to ensure that we will have only one marker block per tile: Once the marker blocks are placed, a priori there is the risk that the other symbols might accidentally create a marker block in a different position—either within the areas which we will use for the actual block-to-block code or in the “gaps” whose contents we will have to leave mostly uncontrolled in order to take advantage of strong irreducibility. The same problem occurs in the case of ${\mathbb{Z}}^{𝑑}\text{,}$ but the solution is more complicated for general amenable groups, because the lack of commutativity greatly increases the number of possible scenarios we need to consider. Thus, our main goal in this section will be to prove the following theorem, which shows how to build the marker blocks.