Halving dynamical systems

Halving (or doubling) a number is an innocuous operation. Unarguably it changes the size of a number, though for a rational number this change in the real size is exactly compensated by a change in its size as a $2$-adic number. For a real number all the more interesting properties — being rational, algebraic, transcendent, well- or badly-approximable — are unchanged. Doubling (or halving) other objects also makes sense, but with potentially much greater impact on non-trivial properties.

For example, doubling a finite group $G$ may be thought of as forming another group $H$ for which the sequence of groups and homomorphisms

$$\{1\}\longrightarrow C_{2}\longrightarrow H\longrightarrow G\longrightarrow\{1\}$$

is exact, where $C_{2}$ is the group with two elements. Once again this has a simple effect on the size, as $|H|=2|G|$. However, the structure of the group clearly can change, and the property of being abelian or simple is not generally preserved. Halving only makes sense if $|G|$ is even, in which case it might be thought of as follows. If $|G|$ is even then $G$ contains at least one copy of $C_{2}$, and halving means forming the quotient space $G/C_{2}$. This operation does not even preserve the property of being a group, unless the copy of $C_{2}$ happens to be a normal subgroup.

Our purpose here is to discuss doubling and halving in the context of topological dynamical systems, by which we mean pairs $(X,T)$, where $X$ is a compact metric space and $T:X\to X$ is a homeomorphism with a fixed point and with finitely many points of period $n$ for each $n\geqslant 1$. Halving (and doubling) may be thought of as a relationship between pairs $(X,T)$ and $(\overline{X},\overline{T})$ of the following form.

Suppose there is a continuous involution $\imath:X\to X$ that commutes with $T$, and use this to define an equivalence relation on $X$ by saying that $x\sim y$ if and only if $x=\imath(y)$. Writing $[x]_{\mathord{\sim}}$ for the equivalence class $\{x,\imath(x)\}$ of $x$ we define $\overline{X}$ to be the quotient space $X/\mathord{\sim}$ and $\overline{T}$ to be the map defined by $\overline{T}\left([x]_{\sim}\right)=[T(x)]_{\sim}$. The process of passing from $(X,T)$ to $(\overline{X},\overline{T})$ may be thought of as halving, and passing from $(\overline{X},\overline{T})$ to $(X,T)$ as doubling.

The question we address is to ask which structures are preserved and which are not by doubling and halving in this sense. The most important quantity associated to a topological dynamical system is the topological entropy, which will not be of interest here as it is preserved by halving (see [3, Ch. 5] for the definition and details). We are particularly interested in the relationship between closed orbits in the two systems. It turns out that there are some constraints on the relationship between the numbers of closed orbits for $T$ and for $\overline{T}$ but that, within these constraints, essentially everything is possible. The constraints will be described in Lemma 5, and the freedom within those constraints in Corollary 10.

We begin with some notational conventions for a map $T:X\to X$. For $n$ a natural number, we write

$$\mathscr{F}_{T}(n)=\{x\in X\mid T^{n}(x)=x\}$$

for the set of points of period $n$ under iteration of $T$, and $\mathsf{F}_{T}(n)=|\mathscr{F}_{T}(n)|$ for the number of points of period $n$. Similarly, write $\mathscr{O}_{T}(n)$ for the set of closed orbits of length $n$ under iteration of $T$, and $\mathsf{O}_{T}(n)=|\mathscr{O}_{T}(n)|$ for the number of closed orbits of length $n$. The set of points of period $n$ comprises exactly the disjoint union of those points on a closed orbit of length $d$ for each $d$ dividing $n$, and each such orbit consists of $d$ distinct points. Thus

$$\mathsf{F}_{T}(n)=\sum_{d|n}d\mathsf{O}_{T}(d),$$

(1)

and hence, by Möbius inversion,

$$\mathsf{O}_{T}(n)=\textstyle\frac{1}{n}\displaystyle\sum\limits_{d|n}\mu\left(\textstyle\frac{n}{d}\right)\mathsf{F}_{T}(d)$$

for any $n\geqslant 1$.

A convenient generating function for the periodic point data is the dynamical zeta function

$$\zeta_{T}(z)=\exp\left(\sum_{n\geqslant 1}\mathsf{F}_{T}(n)\frac{z^{n}}{n}\right)=\prod_{n\geqslant 1}\left(1-z^{n}\right)^{-\mathsf{O}_{T}(n)},$$

(the second equality is equivalent to the identity (1)) which defines a function under the assumption that $\mathsf{F}_{T}(n)<\infty$ for all $n\geqslant 1$, in which case it has radius of convergence given by

$$\left(\limsup_{n\to\infty}\left(\mathsf{F}_{T}(n)\right)^{1/n}\right)^{-1}.$$

The halving process of the introduction is a special case of forming a topological factor. If $(X,T)$ and $(Y,S)$ are topological dynamical systems with a continuous surjective map $\pi:X\to Y$ satisfying $\pi\circ T=S\circ\pi$, then $(Y,S)$ is called a topological factor of $(X,T)$. In general, closed orbits can behave very badly under a topological factor map, as the following examples illustrate.

Thus we cannot expect to be able to make statements about numbers of closed orbits in factor systems in general. For quotienting by an action of $C_{2}$ (that is, halving), the situation is more restricted.

The following example gives a natural way in which one could double a topological dynamical system, simply by putting together two copies of the system.

Our purpose here is to show how extremely unrepresentative Examples 3 and 4 really are. In general, both the growth rate in the number of periodic points and the arithmetic nature of the zeta function do not survive under doubling or halving. However, in contrast to Examples 1 and 2 we will show that the change of the growth rate in the number of closed orbits is restricted.

We put ourselves back in the halving situation of the introduction. Thus $(X,T)$ is a topological dynamical system, which we recall means a pair with $X$ a compact metric space and $T:X\to X$ a homeomorphism with a fixed point and with finitely many points of period $n$ for each $n\geqslant 1$, and $\imath$ is a continuous involution on $X$ which commutes with $T$. Then $\overline{X}$ is the quotient of $X$ under the equivalence relation induced by $\imath$, with quotient map $\pi:X\mapsto\overline{X}$ given by $\pi(x)=\{x,\imath(x)\}$. Note that $\overline{X}$ is a metric space: if $d_{X}$ is the metric on $X$, then the metric $d_{\overline{X}}$ on $\overline{X}$ is given by

$$d_{\overline{X}}(\overline{x},\overline{y})=\min\{d_{X}(x,y)\mid x\in\pi^{-1}(\overline{x}),y\in\pi^{-1}(\overline{y})\}.$$

The map $\overline{T}:\overline{X}\to\overline{X}$ is (well-)defined by the relation $\pi\circ T=\overline{T}\circ\pi$, and is a homeomorphism.

The factor map $\pi$ maps any closed orbit under $T$ to a closed orbit under $\overline{T}$; conversely, since the fibres of $\pi$ are finite, the inverse image under $\pi$ of a closed orbit under $\overline{T}$ is a finite set closed under $T$ so is a finite union of closed orbits under $T$. In particular, $\pi$ induces a surjective map

$$\bigsqcup_{n=1}^{\infty}\mathscr{O}_{T}(n)\to\bigsqcup_{n=1}^{\infty}\mathscr{O}_{\overline{T}}(n).$$

(4)

In order to analyze this more closely, let $\tau=\{x,T(x),T^{2}(x),\dots,T^{n}(x)=x\}$ be a closed orbit in $(X,T)$ of length $n$. Then $\imath$ and $\tau$ can interact in just three ways.

1. 1.

   It could fix the orbit $\tau$ pointwise, (we say that $\tau$ ‘survives’), that is $x=\imath(x)$ for all $x\in\tau$; we write $\mathscr{O}_{T}^{s}(n)$ for the set of closed orbits of length $n$ under $T$ that are fixed pointwise by $\imath$. Then the factor map $\pi$ induces an injective map $\mathscr{O}_{T}^{s}(n)\to\mathscr{O}_{\overline{T}}(n)$.

2. 2.

   It could map $\tau$ to another closed orbit $\tau^{\prime}$ of the same length (we say that $\tau$ is ‘glued’ to $\tau^{\prime}$), that is $\imath(x)\in\tau^{\prime}$ for all $x\in\tau$; we write $\mathscr{O}_{T}^{g}(n)$ for the set of closed orbits of length $n$ under $T$ that are glued together in pairs. The factor map $\pi$ induces a $2$-to-$1$ map $\mathscr{O}_{T}^{g}(n)\to\mathscr{O}_{\overline{T}}(n)$.

3. 3.

   It could preserve the orbit $\tau$ but not fix it pointwise; in that case, we must have $n=2k$ even and $\imath(x)=T^{k}(x)$, for all $x\in\tau$, so that the orbit $\pi(\tau)$ of $\overline{T}$ has length $k$ (we say that $\tau$ is ‘halved’). we write $\mathscr{O}_{T}^{h}(n)$ for the set of closed orbits of length $n$ under $T$ that are halved in length, and $\pi$ induces an injective map $\mathscr{O}_{T}^{h}(2k)\to\mathscr{O}_{\overline{T}}(k)$.

Clearly

$$\mathscr{O}_{T}(n)=\mathscr{O}_{T}^{h}(n)\sqcup\mathscr{O}_{T}^{g}(n)\sqcup\mathscr{O}_{T}^{s}(n)$$

is a disjoint union, and so

$$\mathsf{O}_{T}(n)=\mathsf{O}_{T}^{h}(n)+\mathsf{O}_{T}^{g}(n)+\mathsf{O}_{T}^{s}(n),$$

(5)

for any $n\geqslant 1$. Similarly, from the surjective map (4) and the three possible behaviours, we get

$$\mathscr{O}_{\overline{T}}(n)=\pi\left(\mathscr{O}_{T}^{h}(2n)\right)\sqcup\pi\left(\mathscr{O}_{T}^{g}(n)\right)\sqcup\pi\left(\mathscr{O}_{T}^{s}(n)\right),$$

and it follows that

$$\mathsf{O}_{\overline{T}}(n)=\mathsf{O}_{T}^{h}(2n)+\textstyle\frac{1}{2}\mathsf{O}_{T}^{g}(n)+\mathsf{O}_{T}^{s}(n),$$

(6)

for any $n\geqslant 1$. Since these numbers are finite and

$$\mathsf{O}_{\overline{T}}(1)\geqslant\mathsf{O}_{T}(1)-\textstyle\frac{1}{2}\mathsf{O}_{T}^{g}(1)\geqslant\textstyle\frac{1}{2}\mathsf{O}_{T}(1)>0,$$

we see that $(\overline{X},\overline{T})$ is also a topological dynamical system.

The way in which the set of orbits of length $n$ under $T$ decomposes into those that halve in length, those that glue together, and those that survive, is not arbitrary. Whatever constraints on $\imath$ arise from having to be a continuous involution on $X$ that commutes with $T$, there are some purely combinatorial constraints as follows:

$$\mathsf{O}_{T}^{h}(n)=0\mbox{ if~$n$ is odd};$$

(7)

$$\mathsf{O}_{T}^{g}(n)\mbox{ is even for all~$n\geqslant 1$}.$$

(8)

These observations already constrain the effect that halving can have on the growth rate of closed orbits.

In the case that $\mathsf{O}_{T}(n)$ grow exponentially, Lemma 5 immediately gives bounds on the logarithmic growth rate of $\mathsf{O}_{\overline{T}}(n)$.

We will see later (see Corollary 10) that every growth rate in the closed interval $[\lambda,2\lambda]$ is obtainable.

Our main observation is that, if we are free to choose the topological dynamical systems, then (7) and (8) are the only constraints on the behaviour of closed orbits under halving. This is a simple extension of an elementary remark in [7]: for any sequence $(a_{n})$ of non-negative integers, there is a topological dynamical system $(X,T)$ with $\mathsf{O}_{T}(n)=a_{n}$ for all $n\geqslant 1$.

Notice in particular that taking $b_{n}^{s}=a_{n}$ and $b_{n}^{g}=b_{n}^{h}=0$, for all $n\geqslant 1$ recovers the basic lemma of [7].

Before giving an algebraic proof, we give a more geometric sketch to give an idea of what is happening. We can construct $X$ as a closed (and hence compact) subset of the triangle $\{(x,y)\mid 0\leqslant x\leqslant 1,0\leqslant y\leqslant x\}\subset\mathbb{R}^{2}$, with the metric inherited from $\mathbb{R}^{2}$. Above each point $(\frac{1}{n},0)$ for $n\geqslant 2$ draw $a_{n}$ disjoint regular $n$-gons in such a way that all of them are disjoint as subsets of the plane. By hypothesis $a_{1}\geqslant 1$, and we draw $a_{1}-1$ points ($1$-gons) above $(1,0)$. Finally locate a single $1$-gon at $(0,0)$ (which may be thought of as a point ‘at infinity’). The space $X$ is now defined to be the union of all the vertices of the polygons. (See Figure 3.) It is closed because all but one point is isolated, and the accumulation point $(0,0)$ is, by construction, a member of $X$. Number the vertices of each $n$-gon with the numbers $1$ to $n$ consecutively clockwise, so that we may speak of the ‘same’ point on two disjoint $n$-gons as being the point with the same symbol. The homeomorphism $T$ is defined to be the map that takes each point on any $n$-gon to the next point in a clockwise orientation around the same $n$-gon (equivalently, adding one using the labels; the action of this map is illustrated by the lines joining vertices of the polygons in Figure 3). This defines a homeomorphism since all but one point is isolated in $X$, and all the points close to the fixed point $(0,0)$ are moved by a very small distance. Then $(X,T)$ is a topological dynamical system, and by construction $\mathsf{O}_{T}(n)=a_{n}$ for all $n\geqslant 1$.

Now we define an action of $C_{2}$ on $X$ using the numbers $a_{n}^{s},a_{n}^{g}$, and $a_{n}^{h}$ as follows.

• •

  For each $n\geqslant 1$ pick $\frac{1}{2}a_{n}^{g}$ pairs of $n$-gons above $(\frac{1}{n},0)$ and define the action of $\imath$ to send a point on any one of them to the same point on the paired $n$-gon (these are the glued orbits).

• •

  For each $n\geqslant 1$ pick $a_{n}^{h}$ of the $n$-gons above $(\frac{1}{n},0)$, chosen from those that have not been chosen already for gluing, and on each polygon (which will by hypothesis have even length) define the action of $\imath$ to be rotation by $\pi$ (these are the halved orbits).

• •

  On all the remaining points that are vertices of polygons that are neither glued nor halved, define $\imath$ to be the identity map (these are the surviving orbits).

It is clear that $\imath$ is continuous (since all points close to the fixed point are moved by a very small distance) and commutes with $T$ and so defines a halved system $(\overline{X},\overline{T})$ which, by construction, has the required numbers of orbits.

Note that, in the proof below, we do not refer to the triangle in the plane and the metric on $X$ we give is not the same one as in this sketch, but it does give the same topology.

Lemma 8 shows that any pair of sequence $(a_{n}),(b_{n})$ for which the combinatorial constraints (5)–(8) are satisfied does in fact arise as the orbit count of a pair of systems related by halving. However, it is not so easy to give conditions directly on the sequences $(a_{n}),(b_{n})$ which guarantee that the combinatorial constraints are indeed satisfied. The following result gives some sufficient conditions.

The conditions on the pair of sequences $(a_{n}),(b_{n})$ here are not necessary for the existence of a suitable halving system, but they are sufficient for our interests and are not so far from being necessary: for $n>N$ odd, the condition

$$\tfrac{1}{2}a_{n}\leqslant b_{n}\leqslant a_{2n}$$

is necessary by Lemma 5.

As a consequence, we get the following result when we consider exponential orbit growth rates.

Bowen and Lanford [1, Th. 2] showed that there are only countably many rational dynamical zeta functions, so Corollary 10 shows in particular that halving and doubling cannot preserve the property of having a rational zeta function. We now discuss some examples that give concrete instances of this phenomenon. The arguments all rely on the following facts: a power series with positive radius of convergence represents a rational function if and only if the coefficients satisfy a linear recurrence (see [4, Sec.1.1]). Moreover, the Skolem–Mahler–Lech Theorem says that, in any linear recurrence sequence $(a_{n})$, the set of zeros, comprising those values of $n\in\mathbb{N}$ for which $a_{n}=0$, is the union of a finite set of arithmetic progressions and a finite set (see [4, Ch. 1] for further details and a proof).

Our next examples use the sum of divisors function $\sigma(n)=\sum_{d|n}d$. There are sophisticated bounds for size of $\sigma(n)$, but for our purposes it is sufficient to note the trivial bounds

$$n\leqslant\sigma(n)\leqslant n^{2}.$$

It follows that the complex power series

$$\theta(z)=\exp\sum_{n\geqslant 1}\sigma(n)\frac{z^{n}}{n}$$

has radius of convergence $1$. It is known that

$$\frac{1}{\theta(z)}=1-z-z^{2}+z^{5}+z^{7}-\cdots,$$

where the powers of $z$ are those of the form $(3k^{2}\pm k)/2$ (see, for example, Pólya and Szegö [6, Sec. VIII, Ex. 75]). This means that $\frac{1}{\theta(z)}$ is a power series with arbitrarily long consecutive sequences of zero coefficients. Thus, by the Skolem–Mahler–Lech Theorem, the coefficients of $\frac{1}{\theta(z)}$ are not a linear recurrence sequence and we deduce that $\theta(z)$ is not a rational function of $z$.

In fact, the zeta function in the previous example is worse than irrational. The function $1/\theta(z)$ has integer coefficients and radius of convergence $1$, but is not rational. Thus, by the Pólya–Carlson Theorem (see [2, 5]) it has the unit circle as natural boundary, and the function $\zeta_{\overline{T}}$ also has natural boundary here. Since the radius of convergence of $\zeta_{\overline{T}}$ is only $\frac{1}{2}$ this is perhaps not so interesting, but our final example shows that it is possible for the circle of convergence and the natural boundary of $\zeta_{\overline{T}}$ to coincide, even when $\zeta_{T}$ is rational.