### “Guided” and “quantised” dynamical systems

Evegenios Kakariadis and I have recently posted our paper “On operator algebras associated with monomial ideals in noncommuting variables” on the arxiv. The subject of the paper is several operator algebras (at the outset, there are seven algebras, but later we prove that some are isomorphic to others) that one can associate with each monomial ideal, in such a way that these algebras encode various aspects of the relations defining the ideal.

I refer you to the abstract and intro of that paper for more information about we do there. In this post I would like to discuss at some length an issue that came up writing the paper, and the paper itself was not an appropriate place to have this discussion.

#### 1. Dynamical systems in spaces and algebras

For many mathematicians the phrase dynamical system means a set $X$ together with a group $G$ of maps acting on it. In the theory of C*-algebras one usually considers this action as represented by a homomorphism $\alpha : G \rightarrow Aut(A)$, where $Aut(A)$ is the group of *-automorphisms of the C*-algebra $A$ (as in the situation described in Section 6 in this previous post).

I do not adopt this point of view as understanding dynamics as something that is automatically connected to a group – in fact the intuitive picture I have in my head of a dynamical evolution looks nothing like a group. Personally, I have always been more fascinated by dynamical systems which do not involve groups, in other words “irreversible” dynamical systems.

So to me a dynamical system is a pair $(X, \varphi)$ where $X$ is a compact (for the sake of simplicity) and Hausdorff (for the sake of sanity) topological space and $\varphi = (\varphi_1, \ldots, \varphi_d)$ is a finite (as suffices for applicability) sequence of continuous maps $\varphi_i : X \rightarrow X$.

Let us see what we obtain if we consider this from the *-algebraic point of view. Let $A$ the commutative C*-algebra $C(X)$. Every map $\varphi_i : X \rightarrow X$ gives rise to a map $\alpha_i : A \rightarrow A$ by

$\alpha_i(f) = f \circ \varphi_i$.

The maps $\alpha_1, \ldots, \alpha_d$ are all unital *-endomorphisms, and if $\varphi_i$ is a homeomorphism then $\alpha_i$ is a *-automorphism. The dynamical system $(X, \delta)$ can be read from the action of the maps $\alpha_1, \ldots, \alpha_d$ on the algebra $A$.

Since algebras of the form $C(X)$ (where $X$ is compact Hausdorff) constitute all commutative unital C*-algebras, our above discussion motivates (somewhat vaguely) defining a noncommutative dynamical system to be a pair $(A, \alpha)$, where $A$ is a C*-algebra and $\alpha = (\alpha_1, \ldots, \alpha_d)$ is a finite sequence of *-endomorphisms of $A$.

#### 2. Quantised dynamical systems

In the paper we introduce the quantised dynamical systems associated with a monomial ideal. The quantised dynamical system is a certain commutative C*-algebra $A$ and $d$ homomorphisms (*-homomorphisms) that act on it. That is, it is a noncommutative dynamical system in a commutative C*-algebra (the reason for the splashy name is that we had to clearly distinguish between this dynamical and another dynamical system that may be associated to a monomial ideal – a certain natural subshift. The quantised dynamical system is the less obvious dynamical system to study, and deserved a special name, and “quantum” is something you can say when talking about a noncommutative generalisation of something classical, especially when the noncommutative generalisation is commutative…).

The details are not important for my purposes here; the bottom line is that we have a noncommutative dynamical system $(A, \alpha)$ where $\alpha = (\alpha_1, \ldots, \alpha_d)$ is a sequence of *-endomorphisms of a commutative C*-algebra $A$ (this system does play an important role in our analysis; see paper). Now let us check which readers are reading carefully: we have a noncommutative dynamical system on a commutative C*-algebra, say $A = C(X)$; this corresponds to a classical dynamical system in $X$, right?

Not exactly. If $(A, \alpha)$ came from a classical system $(X,\varphi)$, then all maps $\alpha_1, \ldots, \alpha_d$ must be unit preserving maps (because composition of the constant function gives back the constant function). However, in a typical situation our maps are not unit preserving. Then what do we get?

Letting $p_i = \alpha_i(1)$, we find that $p_i$ is a projection in $A = C(X)$, so it is the characteristic function of a clopen set $X_i \subseteq X$. It follows that the map $\alpha_i$ is a unit preserving *-homomorphism of $A$ onto $p_i A p_i = C(X_i)$. Now we get by the usual Gelfand theory a continuous map $\varphi_i : X_i \rightarrow X$.

Thus the noncommutative dynamical system $(A, \alpha)$ gives rise to a partially defined dynamical system, that is $d$ continuous maps $\varphi_i : X_i \rightarrow X$. In the paper we also call $(X, \varphi)$ the quantised dynamical system (associated to a given monomial ideal), and these systems play a key, classifying role in our work on operator algebras associated with monomial ideals. But never mind the application, the concept itself of a partially defined dynamical seems like a very natural and interesting concept to consider. Has it been already considered before?

#### 3. Guided dynamical systems

Funnily enough, I studied partially defined dynamical systems in my masters thesis, which was completed ten years ago. The idea to study such systems is due to my masters thesis advisor, Boris Paneah (in the next section I will explain what for). In my thesis I gave these dynamical systems the name guided dynamical systems, because it is like a dynamical system in which not all orbits are allowed, rather at every point there is a subset of the maps which cannot be used, thus one is guided along some particular orbits (the rough idea will become clearer when I explain how these systems arise in analysis, see next section).

This class of dynamical objects – which sits in between dynamical systems and topological graphs – continued to intrigue me and I had some hopes that it would be pursued further. On the other hand, mathematics is not short in definitions, and I did not want to push this idea artificially. Therefore I was pleasantly surprised that it arose naturally in my joint work with Evgenios (and, to be honest, it was his idea).

Here is a formal definition of a guided dynamical system (technically slightly different from the definition in my thesis).

Definition:guided dynamical system is a triple $(X,\varphi,\Lambda)$ consisting of a space $X$, a sequence of maps $\varphi = (\varphi_1, \ldots, \varphi_d)$ on $X$, and a sequence $\Lambda = (\Lambda_1, \ldots, \Lambda_d)$ of subsets $\Lambda_i \subseteq X$.

The sets $\Lambda$ are called guiding sets. If one is used to thinking of a dynamical system as a space in which points can be moved around by the maps, then one should think of a guided dynamical system as a space in which points can be moved around by the maps,  with the restriction that if a point is in $\Lambda_i$ then one cannot use $\varphi_i$ to move it. One may, if one prefers, consider that the map $\varphi_i$ is only defined on the complement of $\Lambda_i$.

In the notation of the previous section, one would put $\Lambda_i = X \setminus X_i$.

(I briefly discussed this notion and its relation to functional equations in this old post, especially Section 5 there).

This concept leads to “guided” notions of dynamical properties and ideas. For example, there is the notion of $\Lambda$-invariant set : $Y \subseteq X$ is said to be $\Lambda$-invariant (for $\varphi$) if for every $y \in Y$, if $y \notin \Lambda_i$, then $\varphi_i(y) \in Y$ (it is easier to be $\Lambda$ invariant then invariant). Another example: the guided dynamical system $(X,\varphi,\Lambda)$ is said to be $\Lambda$-minimal if the only non-empty $\Lambda$-invariant subset of $X$ is $X$ itself (it is harder to be $\Lambda$-minimal than minimal). Likewise, there are $\Lambda$-attracting sets, and so on.

An important concept is the notion of $\Lambda$-orbit of a point $x \in X$, which is the set of all points that can be reached by a sequence of applications of the maps $\varphi_1, \ldots, \varphi_d$, but where one may apply a map $\varphi_i$ to a point $y$ only if $y \notin \Lambda_i$.

One can study how theorems from classical dynamical systems fare in the guided world. An abstract and systematic study may be in place, on the other hand first things first: one has to show that the concept is rich (lots of examples) and useful (at least one application).

The quantised dynamical systems that arise in our paper are guided dynamical systems where the sets $\Lambda_i$ are the complements of the sets $X_i$ (so, we have a nice example of the phenomena that the noncommutative point of view can lead to some new ideas in the commutative world). But it is better to go back to the original origin of guided dynamical systems.

#### 4. How guided dynamical systems arise in analysis?

Boris Paneah introduced the idea of guided dynamical systems (without using that terminology) in his study of functional equations (which originated from a problem in PDEs). Paneah wrote a series of papers on the subject, and I would recommend this one for starters. I will outline here how guided dynamical systems arise in the theory of functional equations; for an application of guided dynamical system to PDEs (an idea that is also due to Paneah) you can take a look at Chapter 4 of my thesis, which contains nice figures illustrating the dynamics made by Daniel Reem.

Let us consider the interval $I = [-1,1]$. Let $\delta_1, \delta_2 : I \rightarrow I$ be two given continuous maps of the interval. For a given continuous function $h : I \rightarrow \mathbb{R}$ we may consider the following functional equation:

(*) $f(t) - f(\delta_1(t)) - f(\delta_2(t)) = h(t)$  ,  $t \in I$.

The function $f$ is considered as an unknown, and the goal is to study the solvability (or hopefully even solve) the equation (*). We now introduce some more notation to show how a guided dynamical system arises from this problem, and how the (guided) dynamics determine the solvability of the equation.

To get things flowing let’s assume that every function that occurs is sufficiently smooth. Define $\Lambda_i = \{t \in I : \delta_i'(t) = 0\}$. We assume that $\delta_1, \delta_2$ satisfy the following: (a) they are non-decreasing (b) $\delta_1(-1) = -1, \delta_2(1) = 1$ and $\delta_1(1) = \delta_2(-1) = 0$ and $\delta_1(t) + \delta_2(t) = t$ for all $t$. If follows that $0 \leq \delta_i' \leq 1$, and $\delta_1'+\delta_2' = 1$.

We use the sets $\Lambda = (\Lambda_1, \Lambda_2)$ as guiding sets for the dynamical system $(I,\delta)$. We also denote $\Lambda = \Lambda_1 \cup \Lambda_2$. The relation between the dynamical system $(I,\delta, \Lambda)$ and the solvability of the functional equation (*) is summarised in the following theorem (a compilation of things from Section 2.3 in my thesis).

Theorem:  With the above notation, the following are equivalent.

1) The guided dynamical system $(I,\delta, \Lambda)$ is $\Lambda$-minimal.

2) The guided dynamical system $(I,\delta, \Lambda)$ has $\Lambda$-weak attractor in $I \setminus \Lambda$.

3) For every $h \in C^2(I)$ satisfying $h(-1) = h(1)$ and every $c \in \mathbb{R}$, there exists a unique $f \in C^2(I)$ satisfying the functional equation (*) and $f'(0) = c$.

Remarks:

1) A $\Lambda$-weak attractor is a point which is an accumulation point of all the $\Lambda$ orbits (the word “weak” is to distinguish between this notion and another, stronger notion of attractor, which is a point to which all orbits converge). In general the existence of such a weak attractor is strictly weaker than $\Lambda$-minimality, and it is a remarkable feature of these dynamical systems (as well as a surprising byproduct of the analysis of these functional equations) that the notions turn out to be equivalent.

2) It is easy to see that the condition $h(-1) = h(1)$ is necessary for solvability of (*) (just plug in $-1,1$). On the other hand, if $f$ is a solution then so is $f(t) + ct$, for any $c \in \mathbb{R}$. The theorem says that this is the only source of non-uniqueness (this is what the condition on the derivative is about).

In the rest of this section I will show how the existence of a $\Lambda$-weak attractor (and therefore also $\Lambda$-minimality) implies the uniqueness part of the above theorem. (My goal is merely to indicate how the notion of guided dynamical systems naturally arise).

Suppose that

$f(t) - f(\delta_1(t) - f(\delta_2(t)) = 0$.

We need to show that $f(t) = ct$ for some $c \in \mathbb{R}$. Differentiating the equation we find that the continuous function $g := f'$ satisfies the functional equation

$g(t) - \delta_1'(t) g(\delta_1(t)) - \delta_2'(t) g(\delta_2(t)) = 0$ .

Let $t_0$ be a point where $g$ attains its maximum $M$. Then if $t_0 \notin \Lambda_i$ then $0 < \delta_i(t_0) \leq 1$, and then we must have $g(\delta_i(t_0)) = M$. We see inductively that the $g(t) = M$ for every point in the $\Lambda$-orbit of $t_0$. Likewise, if $t_1$ is a point where $g$ attains its minimum in $I$, we conclude that $g$ attains a minimum at very point in the $\Lambda$-orbit of $t_1$. Now if $\hat{t}$ is a $\Lambda$-weak attractor, then it is an accumulation point of the $\Lambda$-orbits of both $t_0$ and $t_1$, so $g$ attains both its minimum and its maximum at $\hat{t}$, whence $g$ is constant. Q.E.D.

(To get the whole theorem is obviously more complicated, and involves among other thing a clever application of Fredholm theory; perhaps I will discuss it some other time. )

Let me remark that such a dynamical system may certainly fail to have a $\Lambda$-weak attractor in $I \setminus \Lambda$, for example if the dynamical system forces a point to go in a $\Lambda$ guided cycle in $\Lambda$ (but it is an open question whether in this setting there must be a guided cycle in $\Lambda$ when $(I,\delta,\Lambda)$ fails to be $\Lambda$-minimal).

#### 5. How guided dynamical systems arise in nature?

Obvious! Everything is a guided dynamical system (Isn’t it clear that this is superior to considering groups?)

#### 6. Final word of warning

Having linked to my masters thesis I cannot resist making fun of my younger self, and say to all the kids out there: beware! the thesis contains some ridiculously unripe examples of how not to write; if you really have nothing else to do, see for example the fifth line in the abstract (which, sorry to say, contains no typo).