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Halving dynamical systems

The relationship between two dynamical systems, one of which is obtained from the other by forming the quotient by an action of an involution commuting with the dynamics, is studied. The constraints and the possible extent of freedom in the relationship between the growth of closed orbits in pairs of systems related in this way is explored.


Introduction.
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 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 → X is a homeomorphism with a fixed point and with finitely many points of period n for each n 1. Halving (and doubling) may be thought of as a relationship between pairs (X, T ) and (X, T ) of the following form.
Suppose there is a continuous involution ı : X → X that commutes with T , and use this to define an equivalence relation on X by saying that x ∼ y if and only if x = ı(y). Writing [x] ∼ for the equivalence class {x, ı(x)} of x we define X to be the quotient space X/∼ and T to be the map defined by T ([x] ∼ ) = [T (x)] ∼ . The process of passing from (X, T ) to (X, T ) may be thought of as halving, and passing from (X, 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 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.

Closed orbits
We begin with some notational conventions for a map T : X → X. For n a natural number, we write for the set of points of period n under iteration of T , and F T (n) = |F T (n)| for the number of points of period n. Similarly, write O T (n) for the set of closed orbits of length n under iteration of T , and O T (n) = |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 and hence, by Möbius inversion, for any n 1. A convenient generating function for the periodic point data is the dynamical zeta function (the second equality is equivalent to the identity (1)) which defines a function under the assumption that F T (n) < ∞ for all n 1, in which case it has radius of convergence given by lim sup

Closed orbits and topological factors
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 π : X → Y satisfying π • T = S • π, 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.
Example 1. Closed orbits in (X, T ) may be glued together in a topological factor, as illustrated in Figure 1(a). An extreme instance of this is to take Y = {y} to be a singleton, set S(y) = y, and define the factor map π by π(x) = y for all x ∈ X. Then, whatever the sequence (O T (n)) of numbers of closed orbits under T , the factor system has O S (1) = 1 and O S (n) = 0 for all n 2.
Example 2. Finite pieces of orbits in (X, T ) that are not closed may close up under the factor map, producing closed orbits in (Y, S), as illustrated in Figure 1(b). An extreme instance of this is once again to take any topological dynamical system (Y, S) and form the product system X = Y × T, where T = R/Z is the additive circle, with map T (y, t) = (S(y), t + α (mod 1)) for some fixed α / ∈ Q. Then (Y, S) is a topological factor of (X, T ) via the map Then, whatever the sequence (O S (n)) of numbers of closed orbits under S, the map T has no orbits of finite length. 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.
Example 3. Let X = T = R/Z be the additive circle, and define the map T on X by T (x) = 2x (mod 1). This is not a homeomorphism, but is a convenient familiar map to use as an initial example. The involution ı(x) = 1 − x commutes with T , and so defines a halved system (X, T ). A convenient way to visualize this system is to imagine looking sideways at the unit circle so that points identified by the map ı are seen as a single point, as illustrated in Figure 2. It is straightforward to check that the quotient system (X, T ) is the tent map, [0] = {0} with X = [0, 1 2 ] and For these two systems it is easy to calculate that In this example, a system with approximately 2 n points of period n and a rational zeta function is halved to a system with the same properties.
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.
The involution ı : (y, e) −→ (y, e+1 (mod 2)) commutes with T , giving a halved system (X, T ) which can be identified with the original system (Y, S). Clearly so that lim sup meaning that the zeta functions ζ T and ζ T = ζ S have the same radius of convergence. Moreover, the relation (2) may be written as showing that if ζ S is rational then ζ T is also rational, though we will see later that the converse is not true (see Example 11).
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.

Shortening, Surviving, and Gluing Orbits
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 → X a homeomorphism with a fixed point and with finitely many points of period n for each n 1, and ı is a continuous involution on X which commutes with T . Then X is the quotient of X under the equivalence relation induced by ı, with quotient map π : X → X given by π(x) = {x, ı(x)}. Note that X is a metric space: if d X is the metric on X, then the metric d X on X is given by The map T : X → X is (well-)defined by the relation π • T = T • π, and is a homeomorphism.
The factor map π maps any closed orbit under T to a closed orbit under T ; conversely, since the fibres of π are finite, the inverse image under π of a closed orbit under T is a finite set closed under T so is a finite union of closed orbits under T . In particular, π induces a surjective map In order to analyze this more closely, let τ = {x, T (x), T 2 (x), . . . , T n (x) = x} be a closed orbit in (X, T ) of length n. Then ı and τ can interact in just three ways.
1. It could fix the orbit τ pointwise, (we say that τ 'survives'), that is for all x ∈ τ ; we write O s T (n) for the set of closed orbits of length n under T that are fixed pointwise by ı. Then the factor map π induces an injective map O s T (n) → O T (n). 2. It could map τ to another closed orbit τ ′ of the same length (we say that τ is 'glued' to τ ′ ), that is ı(x) ∈ τ ′ for all x ∈ τ ; we write O g T (n) for the set of closed orbits of length n under T that are glued together in pairs. The factor map π induces a 2-to-1 map O g T (n) → O T (n). 3. It could preserve the orbit τ but not fix it pointwise; in that case, we must have n = 2k even and ı(x) = T k (x), for all x ∈ τ , so that the orbit π(τ ) of T has length k (we say that τ is 'halved'). we write O h T (n) for the set of closed orbits of length n under T that are halved in length, and π induces for any n 1. Similarly, from the surjective map (4) and the three possible behaviours, we get and it follows that for any n 1. Since these numbers are finite and we see that (X, 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 ı arise from having to be a continuous involution on X that commutes with T , there are some purely combinatorial constraints as follows: O g T (n) is even for all n 1.
These observations already constrain the effect that halving can have on the growth rate of closed orbits.
Lemma 5. Suppose (X, T ) is obtained from (X, T ) by halving. Then, for any n 1, The lower bound for F T (n) comes from the fact that the fibres of the factor map π have cardinality at most 2; since π maps F T (n) to F T (n), we deduce that F T (n) The upper bound comes from the containment π −1 (F T (n)) ⊆ F T (2n). If x ∈ F T (2n) \ F T (n), then π(x) ∈ F T (n) if and only if x lies in an orbit which halves in length, in which case x lies in a fibre of cardinality 2. Thus The upper bound is achieved if all orbits of order dividing n survive, while all other orbits of order dividing 2n halve.
by (5) and (6). Again this is achieved when all orbits of order n survive, while all orbits of order 2n halve. For the lower bound, when n is odd we have O h T (n) = 0 so that and, from (6), . This bound is achieved if all orbits of length n are glued in pairs. Remark 6. Note that Lemma 5 does not give a lower bound for O T (n) when n is even. Specifically, when n is even, all orbits of length n might halve in length while orbits of length 2n retain their length. Thus the only possible lower bound is the trivial one O T (n) 0.
In the case that O T (n) grow exponentially, Lemma 5 immediately gives bounds on the logarithmic growth rate of O T (n). We will see later (see Corollary 10) that every growth rate in the closed interval [λ, 2λ] is obtainable.

The basic lemma
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 O T (n) = a n for all n 1. Define a n = a s n + a g n + a h n and b n = b s n + b g n + b h n for all n 1. Then there are a topological dynamical system (X, T ) and a continuous involution ı : X → X that commutes with T , such that O T (n) = a n and O T (n) = b n , for all n 1.
Notice in particular that taking b s n = a n and b g n = b h n = 0, for all n 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) | 0 x 1, 0 y x} ⊂ R 2 , with the metric inherited from R 2 . Above each point ( 1 n , 0) for n 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 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 O T (n) = a n for all n 1. Now we define an action of C 2 on X using the numbers a s n , a g n , and a h n as follows.
• For each n 1 pick 1 2 a g n pairs of n-gons above ( 1 n , 0) and define the action of ı to send a point on any one of them to the same point on the paired ngon (these are the glued orbits).
• For each n 1 pick a h n of the n-gons above ( 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 ı to be rotation by π (these are the halved orbits).
• On all the remaining points that are vertices of polygons that are neither glued nor halved, define ı to be the identity map (these are the surviving orbits).
It is clear that ı 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 (X, 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.
Proof of Lemma 8. We write C 2 = {e, ı}, where e is the identity element, for the cyclic group of order 2. We begin by describing X as a set, before compactifying. It will take the form where each X n will be the union of closed orbits of length n. We set We define T : X → X and ı : X → X by describing their restrictions to each of the sets X s n , X g n , X h n , which will be preserved: • for x = (i, k) ∈ X s n , put T (x) = (i, k + 1 (mod n)) and ı(x) = x; • for x = (i, γ, k) ∈ X g n , put T (x) = (i, γ, k+1 (mod n)) and ı(x) = (i, ıγ, k); • for x = (i, k) ∈ X h n , so that n is even, we put T (x) = (i, k + 1 (mod n)) and ı(x) = (i, k + n 2 (mod n)).
Then ı commutes with T and, by construction, the map T and the induced map T on the quotient set X have the required numbers of orbits. It remains to show that X can be given the structure of a compact metric space, with respect to which T and ı are homeomorphisms. To do this, we pick a point in X s 1 (which is non-empty by hypothesis) and call it ∞ and define a metric as follows: if x ∈ X m and y ∈ X n , with x ∈ y, ∞, then and d(x, x) = 0. It is straightforward to check that this does indeed define a metric and, given any open set U containing ∞, there exists N 1 such that U contains n N X n so that X \ U is finite; hence X is compact. Moreover, since T and ı preserve the sets X n and the point ∞, they are isometries, so homeomorphisms, and we are done.

Growth in closed orbits
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. Proposition 9. Let (a n ) be a sequence of non-negative integers with a 1 1 such that there is an integer N 1 with a 2n 1 2 a n , for all n N . Let (b n ) be any sequence of integers such that b 1 > 1 2 a 1 and 1 2 a n b n a n for n < N, 1 2 a n b n a 2n for n N.
Then there are a topological dynamical system (X, T ) and a continuous involution ı : X → X that commutes with T , such that O T (n) = a n and O T (n) = b n , for all n 1.
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 1 2 a n b n a 2n is necessary by Lemma 5.
Proof. In order to use Lemma 8, we recursively define non-negative integers b s n , b g n and b h n such that b n = b s n + b g n + b h n , a n = b s n + 2b g n + b h n/2 , and b h n a 2n , where we understand b h n/2 = 0 when n/2 ∈ Z. So suppose k 1 and we have defined these for n < k. Then there are two cases.
If b k a k − b h k/2 , which we note is always the case for k < N , then we put On the other hand, if b k > a k − b h k/2 , then we put These are non-negative and, in the latter case, we have b h k b k a 2k , since k N . Note also that, in either case, b s 1 > 0. Now Lemma 8 implies the result. As a consequence, we get the following result when we consider exponential orbit growth rates.
Then there exist a topological dynamical system (X, T ) and an involution ı on X commuting with T , such that O T (n) ∼ λ n and O T (n) ∼ cη n as n → ∞.
Proof. Let N > 1 be any integer such that cη N < λ 2N and define sequences by a n = ⌈λ n ⌉, b n = a n if n < N, ⌈cη n ⌉ if n N, for n 1. This gives a pair of sequences satisfying the hypotheses of Proposition 9, from which the result follows.

Dynamical zeta functions
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 ∈ 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).
Example 11. We revisit Example 4, so that (Y, S) is a topological dynamical system and X = Y × {0, 1} with the map T (y, e) = (S(y), e + 1 (mod 2)). The involution ı : (y, e) = (y, e + 1 (mod 2)) commutes with T , giving the halved system (X, T ) = (Y, S). There is sufficient freedom in the choice of orbits of odd length under S to allow us to find examples with ζ T = ζ S irrational but ζ T rational. In particular, we may take It is a pleasant exercise to verify that these really do arise from a topological dynamical system (that is, 1 n O S (n) is a non-negative integer for each n). Then by (2) is rational. On the other hand We claim that ϕ is irrational, so that ζ S is irrational. To see this, note that the coefficient of z n in ϕ(z) vanishes precisely when n is even or n is an odd prime; since the set of primes is infinite, while any arithmetic progression contains composites, the Skolem-Mahler-Lech Theorem implies the sequence of coefficients cannot be a linear recurrence sequence and hence ϕ cannot be a rational function.
Our next examples use the sum of divisors function σ(n) = d|n d. There are sophisticated bounds for size of σ(n), but for our purposes it is sufficient to note the trivial bounds n σ(n) n 2 .
It follows that the complex power series θ(z) = exp n 1 σ(n) z n n has radius of convergence 1. It is known that where the powers of z are those of the form (3k 2 ± k)/2 (see, for example, Pólya and Szegö [6, Sec. VIII, Ex. 75]). This means that 1 θ(z) is a power series with arbitrarily long consecutive sequences of zero coefficients. Thus, by the Skolem-Mahler-Lech Theorem, the coefficients of 1 θ(z) are not a linear recurrence sequence and we deduce that θ(z) is not a rational function of z.
Example 12. In order to use the irrationality of θ(z), we define b g n = 1 and b h n = 0, for all n 1, and we choose b s n later. Now we define b n , a n as in Lemma 8 and denote by (X, T ), (X, T ) the pair of systems given there. Thus T has one extra orbit in each length, compared to T , and the action of ı on X has the effect of gluing together exactly one pair of orbits of each length. Now we first take b s n = 1 n d|n µ n d 2 d − 1, for n 1, so that b n is the number of orbits of length n in any system with 2 n points of period n. Then By the remarks above, this is not a rational function.
In the reverse direction, we take b s 1 = 1 and b s n = 1 n d|n µ n d 2 d − 2, for n 2, so that a n is the number of orbits of length n in any system with 2 n +1 points of period n. Then which is again irrational.
In fact, the zeta function in the previous example is worse than irrational. The function 1/θ(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 ζ T also has natural boundary here. Since the radius of convergence of ζ T is only 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 ζ T to coincide, even when ζ T is rational. this, we use the following version of Euler's generalization of Fermat's Little Theorem: for any prime p and integers r, m, with p|m, we have r m ≡ r m/p (mod p ordp(m) ).
Thus, for any prime p dividing n, since, for any d dividing n with p ∤ n d , we have ord p (d) = ord p (n) and c d ≡ c d/p (mod p ordp(d) ) by construction.
For non-negativity, when n is 1 or prime we have b n − a n = 0. On the other hand, for n composite, the bounds c n n and c d < 2d n, for d a divisor of n, imply that Finally, for n composite, the same bounds show that b n − a n = 1 n d|n µ n d c d 2 d < (2 n + 2 n 2 +1 ) < 2 n+1 , and, similarly, 1 n (2 n − 2 n 2 +1 ) < a n < 1 n (2 n + 2 n 2 +1 ) < 1 n 2 n+1 .
Finally, one checks that b 4 = 19 < 30 = a 8 and, since b n = a n < a 2n for n prime, the conditions of Proposition 9 are satisfied with N = 2.

Concluding remarks and questions
1. The simple observation in [7] that for any sequence (a n ) of non-negative integers there is a topological dynamical system (X, T ) with O T (n) = a n for all n 1 was extended by Windsor [8], who showed that the map may be required to be an infinitely differentiable map on the 2-torus. Does Lemma 8 also have a smooth version, in which both maps and the involution are differentiable maps on a manifold?