A characterization of $\omega$-limit sets in subshifts of Baire Space

For a continuous map $f:X\to X$ on a metric space and a point $x\in X$, the $\omega$-limit set of $x$ is the set

i.e. the set of accumulation points of the forward orbit of $x$ under the action of $f$. It is immediately apparent that understanding the dynamics of the dynamical system $f:X\to X$ requires an understanding of its $\omega$-limit sets.

The $\omega$-limit sets of a dynamical system are well-studied in the context of compact metric domains $X$. Immediately from the definition, it follows that every $\omega$-limit set is closed (and hence compact) and strongly invariant. Hirsch, Smith and Zhao demonstrated that every $\omega$-limit set in a compact dynamical system is internally chain transitive [6]. Barwell, Good, Knight and Raines demonstrated that in a number of systems, the converse also holds, i.e. a closed internally chain transitive set is the $\omega$-limit set of some point $x\in X$ [1]. In particular, they demonstrate this fact for subshifts of finite type and for some maps on the interval. This property has also been verified in a few other settings such as in dendritic Julia sets and in a class of ‘circular’ Julia sets, [2] and [3]. In [9], the authors establish that the internal chain transitivity characterization holds in all compact dynamical systems with the shadowing property (sometimes referred to as the pseudo-orbit tracing property) and this result has been refined by Good and Meddaugh to carry this characterization over to the context of weaker variations of the shadowing property [4].

For systems $f:X\to X$ with $X$ not compact, less is known about $\omega$-limit sets. Again, it is immediate from the definition that the $\omega$-limit sets are closed and invariant, but may fail to be compact. The problem is additionally complicated by fact that $f$ need not be uniformly continous in this setting.

As in [1], we begin the study of $\omega$-limit sets in the noncompact setting by considering shift spaces over countably infinite alphabets. For a countable set $\Lambda$ endowed with the discrete topology, we consider the dynamical system given by the shift map on the space $\Lambda^{\omega}$ of infinite sequences in $\Lambda$ given by $\sigma\langle x_{n}\rangle=\langle x_{n+1}\rangle$. A subshift of $\Lambda^{\omega}$ is then a closed, invariant subsystem. As is the case with finite alphabets, subshifts over countable alphabets serve as useful models for a broad class of dynamical systems including countable state Markov systems [7]. But, unlike shift spaces over finite alphabets which are equivalent to maps on the Cantor middle-thirds set, these systems are not locally compact anywhere. In fact they are equivalent to maps on the irrationals in $\mathbb{R}$.

In the theory of shift spaces over finite alphabets, the subshifts of finite type stand out as one of the most well-studied families. They are well understood in terms of their $\omega$-limit sets [1] and their entropy [8] among many other dynamical properties. Walters showed that these are precisely the subshifts with the shadowing property [11] and Good and Meddaugh have demonstrated that these systems are fundamental to the study of the shadowing property in compact dynamical systems [5].

In this paper, we discuss subshifts of finite type over countable alphabet and characterize the $\omega$-limit sets in these systems. Additionally, we define subshifts of bounded type and demonstrate that these systems are, in a sense, the more natural generalization of subshifts of finite type to the countable alphabet case – they are precisely those subshifts with the shadowing property. This allows us to, in the spirit of [1], characterize a subclass of the $\omega$-limit sets (the attracting $\omega$-limit sets) in terms of internal chain transitivity.

Let $\omega=\mathbb{N}\cup\{0\}$. For any set $\Lambda$, let $\Lambda^{\omega}$ be the set of all infinite words with alphabet $\Lambda$ with topology generated by the product topology taking $\Lambda$ to have the discrete topology. Let $i,j\in\omega$ with $i<j$. Let $x\in\Lambda^{\omega}$. Define $x_{[i,j]}=x_{i}x_{i+1}\dots x_{j}$, and define $x_{(i,j)},x_{[i,j)}$, and $x_{(i,j]}$ similarly. For a finite word $w=w_{0}\cdots w_{n-1}\in\Lambda^{n}$ and $k\in\omega$, define the cylinder set of $w_{0}\cdots w_{n-1}$ centered at $k$ to be the set of all $x\in\Lambda^{\omega}$ with $x_{[k,k+n)}=w$. It is a standard result that the collection of cylinder sets forms a basis for the topology on $\Lambda^{\omega}$, and in fact, it is enough to consider only the cylinder sets centered at $0$.

The space $\Lambda^{\omega}$ is easily seen to be metrizable and it is immediate that the metric given by

is consistent with the topology on $\Lambda^{\omega}$.

If $\Lambda$ is finite and has at least two elements, then $\Lambda^{\omega}$ is a compact metric space which is homeomorphic to the Cantor set. In the event that $\Lambda=\omega$, $\Lambda^{\omega}$ is a non-compact, non-locally compact metric space that is homeomorphic to the irrationals in $\mathbb{R}$. It is traditionally called the Baire space, and it has many uses in descriptive set theory [10].

For a fixed $\Lambda$, we define the full shift on $\Lambda$ to be the dynamical system $(\Lambda^{\omega},\sigma)$ where $\sigma$ is the shift map defined for every $x_{0}x_{1}x_{2}\cdots\in\Lambda^{\omega}$ by $\sigma(x_{0}x_{1}x_{2}\cdots)=x_{1}x_{2}\cdots$. It is easy to check that $\sigma$ is continuous and, in fact, uniformly continuous with respect to the metric $d$.

Let $\Gamma\subseteq\Lambda^{\omega}$. We call $\Gamma$ a subshift of $(\Lambda^{\omega},\sigma)$ provided $\Gamma$ is closed and $\sigma$-invariant, i.e. $\sigma(\Gamma)=\Gamma$. Given a subshift $\Gamma$ of $\Lambda^{\omega}$, let $B_{n}(\Gamma)$ be the collection of all allowed words of length $n$, i.e. $w=w_{0}\cdots w_{n-1}\in B_{n}(\Gamma)$ if, and only if, there is some point $x\in\Gamma$ and some $i\in\omega$ such that $x_{[i,i+n)}=w_{0}\cdots w_{n-1}$, in this case we say that $x$ contains $w$ as a subword or $w$ is a subword of $x$, and if $i=0$ then we call $w$ an initial segment of $x$. Let

We call $B(\Gamma)$ the set of allowed words for $\Gamma$. Notice that, since $\Gamma$ is $\sigma$-invariant, $B(\Gamma)$ is equal to the set of initial segments.

A word $w=w_{0}\cdots w_{n-1}$ is called forbidden in $\Gamma$, or simply forbidden when the context is clear, provided $w\notin B(\Gamma)$. We denote the set of all words forbidden in $\Gamma$ by $F(\Gamma)$, and for each $n\in\mathbb{N}$, we let $F_{n}(\Gamma)$ denote the words of length $n$ that are forbidden in $\Gamma$.

Let $\Gamma\subseteq\Lambda^{\omega}$ be a subshift. A basis for the forbidden words of $\Gamma$ is a subset $\mathcal{F}(\Gamma)$ of $F(\Gamma)$ such that for every $w\in F(\Gamma)$ there is some $w^{\prime}\in\mathcal{F}(\Gamma)$ such that $w$ contains $w^{\prime}$ as a subword. If $\Gamma$ has a basis for its forbidden words that is finite, we call $\Gamma$ a subshift of finite type (SFT). For $\Lambda$ finite, subshifts of finite type are well-studied [8].

For $\Lambda$ infinite, less is known about subshifts of finite type. In fact, it is not immediately clear that subshifts of finite type with infinite alphabet are the most natural analogue to subshifts of finite type with finite alphabet. In particular, the following notion is another possible analogue.

If $\Gamma$ has a basis for its forbidden words, $\mathcal{F}(\Gamma)$, such that there is some $n\in\mathbb{N}$ and every element of $\mathcal{F}(\Gamma)$ has length less than $n$, then we call $\Gamma$ a subshift of bounded type (SBT). The following observation is immediate.

Subshifts display many interesting dynamical properties. This paper focuses primarily on shadowing properties and their relationship with $\omega$-limit sets.

Let $(X,f)$ be a dynamical system on a metric space. For $\delta>0$, a sequence $\langle x_{i}\rangle$ in $X$ is a $\delta$-pseudo-orbit of $f$ provided that $d(f(x_{i}),x_{i+1})<\delta$ for all $i\in\omega$. A sequence $\langle x_{i}\rangle$ is an asymptotic pseudo-orbit of $f$ provided that $d(f(x_{i}),x_{i+1})$ limits to zero.

For a fixed $\epsilon>0$, we say that the sequence $\langle x_{i}\rangle$ $\epsilon$-shadows $\langle y_{i}\rangle$ if $d(x_{i},y_{i})<\epsilon$ for all $i\in\omega$. Similarly, we say that the sequence $\langle x_{i}\rangle$ asymptotically shadows $\langle y_{i}\rangle$ if $d(x_{i},y_{i})$ limits to zero.

We will frequently abuse notation and identify the point $z$ with its orbit sequence $\langle f^{i}(z)\rangle$ and say that the point $z$ shadows the pseudo-orbit $\langle x_{i}\rangle$ in this case.

For an arbitrary sequence $\langle x_{i}\rangle$ in a topological space $X$, the $\omega$-limit set of $\langle x_{i}\rangle$ is the following:

As with shadowing, in the context of a dynamical system, we often associate a point with its orbit sequence, and thus for a dynamical system $(X,f)$ and $x\in X$, the $\omega$-limit set of $x$ is the set $\omega(x)=\omega\langle f^{i}(x)\rangle$. We say that a subset $Z$ of $X$ is an $\omega$-limit set of the system if there is a point $x\in X$ with $\omega(x)=Z$.

The $\omega$-limit sets of points exhibit a variety of dynamical systems that are not shared by $\omega$-limit sets of arbitrary sequences. Among other things, it is well known that $\omega$-limit sets of points are invariant subsets of the dynamical system. In contrast, $\omega$-limit sets of arbitrary sequences are less well studied, and generally less well-behaved. However, the following results concerning the $\omega$-limit sets of sequences are immediate.

An additional concept that we make use of is that of internal chain transitivity. Let $(X,f)$ be a dynamical system with $X$ metric. For $\delta>0$ and $a,b\in X$, a $\delta$-chain from $a$ to $b$ is a finite sequence $x_{0},x_{1},\ldots x_{n}$ such that $d(f(x_{i}),x_{i+1})<\delta$ for $i<n$ with $a=x_{0}$ and $b=x_{n}$.

We begin our exposition of results by exploring the degree to which subshifts of bounded type are the appropriate analogue for subshifts of finite type in the finite alphabet case.

The following characterization of subshifts of bounded type is a generalization of a standard characterization for subshifts of finite type.

In [11], it is demonstrated that the shadowing property completely characterizes subshifts of finite type in the setting of a finite alphabet. If we allow for $\Lambda$ to be (possibly) infinite, we have the following analogous results.

Combining these two results yields the following characterization of shadowing in subshifts over arbitrary $\Lambda$.

It is a standard result that subshifts over finite alphabet are positively expansive, i.e. there exists an expansive constant $c>0$ such that if $x,y\in\Gamma$ with $d(\sigma^{n}(x),\sigma^{n}(y))<c$ for all $n\in\omega$, then $x=y$. In fact, the expansive constant $c$ can be taken to be 1 in this setting, and this carries over to the setting of arbitrary $\Lambda$ perfectly.

As a consequence, we can show that shadowing implies asymptotic shadowing for subshifts of $\Lambda^{\omega}$. In fact, the following more general result holds.

Thus, in the context of shadowing in subshifts over infinite alphabet, subshifts of bounded type are the correct analogue for subshifts of finite type in the finite alphabet case.

It should be noted however, that there are significant differences between these type of dynamical systems, in part due to the lack of compactness.

In a compact dynamical system, an $\omega$-limit set is necessarily internally chain transitive [1]. In the context of shift spaces, the authors have demonstrated that there is a stronger connection between internal chain transitive sets and $\omega$-limit sets. Specifically, they have shown that in a subshift of finite type $\Gamma$, a closed, invariant subset $A$ is an $\omega$-limit set of some point $z\in\Gamma$ if and only if $A$ is internally chain transitive [9]. This connection proves to hold in other classes of dynamical systems exhibiting appropriate forms of shadowing [4].

Strikingly, in the context of shifts over infinite alphabets, even the first result fails.

Perhaps more surprisingly, in subshifts of bounded type over countable alphabets, the converse does hold, i.e. closed invariant internally chain transitive sets are necessarily $\omega$-limit sets.

It is worth noting that there are non-metric notions of shadowing and internal chain transitivity [5], and in these contexts the preceding lemma and proof can easily be modified to accommodate systems which are second countable.

We are now able to prove that in subshifts of bounded type, every closed ICT set is indeed an $\omega$-limit set. In fact, we are able to show that it is an $\omega$-limit set in a particularly tame sense.

With this in mind, we say that a subset $Z$ of $\Lambda^{\omega}$ is an attracting $\omega$-limit set if there is a point $x\in\Lambda^{\omega}$ such that $\omega(x)=Z$ and for all $\epsilon>0$, there exists $N\in\omega$ with $d(\sigma^{i}(x),Z)<\epsilon$ for all $i\geq N$.

Having established this result for subshifts of bounded type, we turn our attention to subshifts of finite type.

This result, combined with Theorem 16 yields the following.

In fact, this is a property unique to countably infinite alphabets.

It is worth noting that there exist subshifts $\Gamma$ of bounded type over countable alphabets which are not internally chain transitive, and indeed, for which $\Gamma$ is not an $\omega$-limit set.

It is interesting to note that the example from Remark 14, while not internally chain transitive, is in fact, a union of closed internally chain transitive sets. While this might lead one to conjecture that in shifts over infinite alphabets, every $\omega$-limit set is a union of closed internally chain transitive sets, we observe that this is false, as the following example demonstrates.

This example demonstrates how significantly the dynamics of shift spaces differ between the finite and infinite alphabet cases. In fact, for subshifts of finite type over infinite alphabet, every closed invariant set is an $\omega$-limit set. Note that this is false for subshifts of bounded type, as witnessed in Remark 21.

Thus, in the context of subshifts of finite type with countable alphabet, the $\omega$-limit sets and the closed invariant sets coincide. Since subshifts of finite type are also subshifts of bounded type, we also see that the attracting $\omega$-limit sets and the closed ICT sets coincide.