Realizability of Integer Sequences as Differences of Fixed Point Count Sequences

Natascha Neumärker Fakultät für Mathematik Universität Bielefeld Postfach 100131 33501 Bielefeld Germany

            naneumae@math.uni-bielefeld.de

Abstract

A sequence of non-negative integers is exactly realizable as the fixed point counts sequence of a dynamical system if and only if it gives rise to a sequence of non-negative orbit counts. This provides a simple realizability criterion based on the transformation between fixed point and orbit counts. Here, we extend the concept of exact realizability to realizability of integer sequences as differences of the two fixed point counts sequences originating from a dynamical system and a topological factor. A criterion analogous to the one for exact realizability is given and the structure of the resulting set of integer sequences is outlined.

1 Introduction and general setting

A (topological) dynamical system (X, T) is given by a topological space X and a continuous map T : X → X. Associated to each dynamical system are two sequences a = (a_{n})_{n≥1} and c = (c_{n})_{n≥1}, defined by

a_{n} = |{x ∈ X | T^{n}(x) = x}| and c_{n} = |{O | O is a periodic orbit of length n}| . (1)

They count the fixed points and periodic orbits, respectively. In interesting classes of dynamical systems, these numbers are finite for all n, which we assume from now on. Thus, a and c are sequences of non-negative integers.

Clearly, a and c are related by Möbius inversion,

fix(c)_{n} := a_{n} = ∑_{d|n} d c_{d} and orb(a)_{n} := c_{n} = \frac{1}{n} ∑_{d|n} μ (\frac{n}{d}) a_{d}.  (2)

Details on these transformations, considered as linear operators on the space of arithmetic functions, can be found in [6] and references given there. In what follows, we sometimes make use of the operator notation orb(a) and fix(c), referring to the transformations of the sequences a and c according to (2). Since fix and orb are inverses of each other, in the sense that fix ∘ orb = orb ∘ fix = id, it is possible to implicitly define a sequence f by setting orb(f) = g or fix(f) = g for a given sequence g. Note that fix and orb are well-defined for arbitrary sequences of complex numbers. If, however, f is a sequence of integers, then so is fix(f), whereas the converse need not be true.

According to Puri’s terminology [12], a system (X, T) comprises an arbitrary set X and a map T : X → X. An integer sequence (f_{n})_{n≥1} is called exactly realizable if there is a system (X, T) whose fixed point counts are given by (f_{n})_{n≥1}. Membership in the set ER of exactly realizable sequences is characterized by the Basic Lemma [12, Thm. 2.2]:

Theorem 1. A sequence of non-negative integers (f_{n})_{n≥1} is exactly realizable if and only if, for all n ≥ 1, the sum ∑_{d|n} μ (\frac{n}{d}) f_{d} yields a non-negative integer divisible by n, that is, if and only if orb(f) is a sequence of non-negative integers.

In other words, the only restriction for an integer sequence to be exactly realizable is that it gives rise to an orbit counts sequence of non-negative integers. By compactification of ℕ with respect to the discrete topology and the definition of an appropriate permutation of the resulting set, Puri shows that the realizing system can be chosen to be a homeomorphism on a compact space. Windsor [15] even gives a construction of a smooth system realizing an arbitrary sequence from ER. Moss [10] systematically investigates the realization of integer sequences by algebraic dynamical systems. A collection of many exactly realizable sequences from the Online Encyclopedia of Integer Sequences (OEIS) [16] is listed by Puri and Ward [13].

A (topological) factor of a dynamical system (X, T) is a dynamical system (Y, S) for which there is a continuous surjection ϕ : X → Y that makes the diagram

\begin{matrix}X & ⟶^{T} & X \\ ^{ϕ}↓ & & ↓^{ϕ} \\ Y & ⟶^{S} & Y \end{matrix}  (3)

commutative, i.e., ϕ(T (x)) = S(ϕ(x)) for all x ∈ X and, by induction, ϕ(T^{n}(x)) = S^{n}(ϕ(x)).

Being a factor of the dynamical system (X, T) is a much weaker condition than topological conjugacy, for which ϕ is required to be a homeomorphism. While the orbit statistics of both systems coincide in the latter case, a factor can have a completely different orbit structure. Going down to the factor, an arbitrary number of periodic points can be ‘lost’ since, for some arbitrary map U, the map U × S always gives rise to the factor S. An example of the number of periodic points in the factor system exceeding the number of periodic points in the original system is given by the maps dual to x ↦ 2x on ℚ and ℤ, respectively, see example 11.

Since for many dynamical systems it is known that they are a factor of some wellunderstood dynamical system or, vice versa, that some well-studied system is a factor (cf. [5]), the question is raised, in what way their orbit statistics can be related. A natural approach to this question is the classification of integer sequences as relative fixed point counts, that is, as the difference of the fixed point counts sequences associated with (X, T) and (Y, S). It is clear, however, that this is only a very coarse way of relating the dynamics of the two systems. Example 9 later on gives an example of a dynamical system and a factor whose fixed point counts coincide, the factor map being strictly 2 : 1, however.

The purpose of this short note is to derive a realizability criterion for integer sequences as such difference sequences, analogous to the one given in Theorem 1, and to consider a few consequences.

2 Relative realizability

An integer sequence h is called relatively realizable if there is a dynamical system (X, T) and a factor (Y, S) with fixed point counts sequences f and g, respectively, such that h = f − g. We denote the set of relatively realizable sequences by ER_{rel}.

By linearity of the mapping orb, the orbit counts sequences corresponding to f and g satisfy the same relation: orb(h) = orb(f) − orb(g). Since the realization is constructed orbit-wise, it is thus more convenient to consider orbit counts.

If the periodic orbits of (X, T) and (Y, S) partition the respective space, each surjective map on the set of equivalence classes defined by the periodic orbits can be extended to a surjection ϕ : X → Y. More precisely, if a T -orbit O = {x, T x, …, T^{n−1}x} is assigned an S-orbit {y, Sy, …, S^{d−1}y}, the corresponding map ϕ : X → Y is defined via

ϕ(T^{k}(x)) = S^{r_{k}}(y), r_{k} ≤ d − 1, r_{k} ≡ k mod d  (4)

for the elements of O.

Proposition 2. A sequence of integers h is relatively realizable if and only if the formal power series H(x) := ∑_{n=1}^{∞} h_{n}x^{n} ∈ ℤ[[x]] associated with orb(h) admits a decomposition

H(x) = ∑_{n=1}^{∞} b_{n}x^{n} + ∑_{n>d|n} a_{d,n}(−x^{d} + x^{n}),

with a_{d,n}, b_{n} ∈ ℕ_{0}.

Proof. The grouping of the terms in H(x) encodes the definition of an appropriate surjection ϕ. We first note that the n-th term has coefficient

b_{n} + ∑_{n>d|n} a_{d,n} − ∑_{k≥1} a_{n,kn}.  (5)

Let (X, T, Y, S, ϕ) relatively realize h = f − g and set orb(f) = (ν_{1}, ν_{2}, …), orb(g) = (γ_{1}, γ_{2}, …). For each ν_{k} ≥ 0, let C_{1}, …, C_{ν_{k}} denote the T -orbits of length k. For d|k with d < k set

a_{d,k} = |{D ∈ {ϕ(C_{1}), …, ϕ(C_{ν_{k}})} : |D| = d, D has no preimage orbit of length < k}| .

In particular, a_{d,k} = 0 for divisors d of k that do not show up as the length of an image cycle and coinciding image cycles ϕ(C_{i}) = ϕ(C_{j}) for i ≠ j are counted only once. Then, the number of ℓ with a_{d,ℓd} ≠ 0 is finite, and ∑_{ℓ≥1} a_{d,ℓd} = γ_{d}. Define b_{k} := ν_{k} − ∑_{n>d|n} a_{d,k} if ν_{k} > 0, giving b_{k} = 0 otherwise. Thus, the a_{d,k}, b_{k} define a formal power series of the structure indicated above whose n-th coefficient is, according to (5),

b_{n} + ∑_{n>d|n} a_{d,n} − ∑_{k≥1} a_{n,kn} = ν_{n} − γ_{n} = orb(h)_{n}.

Consequently, this is an appropriate decomposition of the power series of orb(h). For the inverse direction, we define

ν_{n} := b_{n} + 1 + ∑_{n>d|n} a_{d,n} and γ_{n} := ∑_{k≥1} a_{n,kn} + 1

and realize the sequences orb(f) := (ν_{1}, ν_{2}, …) and orb(g) := (γ_{1}, γ_{2}, …) on the one-point compactification (ℕ_{∗}, τ_{∗}) of ℕ. If ‘the point at infinity’ ∞, added for the purpose of compactification, is a fixed point of the permutation σ on ℕ_{∗} = ℕ ∪ {∞}, σ is continuous in the obtained topology, see [12] and the reference given therein. In each of the resulting systems (ℕ_{∗}, T) and (ℕ_{∗}, S), there is at least one cycle of length n for all n ∈ ℕ. By construction, (ℕ_{∗}, S) and (ℕ_{∗}, T) have a_{n,kn} corresponding cycles of length n and kn, respectively. ϕ can be defined on these as described by (4). For the b_{n} further T -cycles of length n, one of the n-S-cycles can be chosen as the image cycle. If the 1-orbit {∞} is mapped to its analogue in the factor system, it is straightforward to check that the map ϕ is continuous with respect to the considered topologies. It is surjective by construction, thus turning (ℕ_{∗}, S) into a factor of (ℕ_{∗}, T) and therefore yielding a relative realization of the sequence h.

As a consequence, we obtain that each integer sequence is the difference orbit counts sequence of a dynamical system and a factor.

Theorem 3. A sequence h of integers is relatively realizable if and only if orb(h) is a sequence of integers.

Proof. Since it is clear that the condition is necessary, it suffices to give a decomposition of an arbitrary element H(x) = ∑_{k=1}^{∞} η_{k}x^{k} from ℤ[[x]] as in Proposition 2. A possible approach for doing so is to select a divisor d|n and a multiple ℓn of n and to decompose η_{n} into a sum of the shape η_{n} = b_{n} + a_{d,n} − a_{n,ℓn} with non-negative integers b_{n}, a_{d,n} and a_{n,ℓn}.

Consider the case n odd first. If η_{n} ≥ 0, define b_{n} = η_{n} and a_{n,ℓn} = 0 for all ℓ ≥ 1. If η_{n} < 0, set a_{n,2n} = −η_{n}, a_{n,ℓn} = 0 for all ℓ ≠ 2 and b_{n} = 0. For n even and η_{n} −a_{n/2,n} ≥ 0, set b_{n} = η_{n} − a_{n/2,n}, a_{d,n} = 0 for all d|n; for η_{n} − a_{n/2,n} < 0 set a_{n,2n} = −(η_{n} − a_{n/2,n}), a_{n,ℓn} = 0 for all ℓ ≠ 2 and b_{n} = 0. Thus (5) yields b_{n} + a_{n/2,n} − a_{n,2n} as the n-th coefficient, which, in each of the cases considered above, coincides with η_{n}. Hence, H(x) can be written as in Proposition 2.

Remark 4. Analogous to the proof of the Basic Lemma [12], Proposition 2 and Theorem 3 together give a construction of a realizing permutation system for a given integer sequence.

Remark 5. A combinatorially easier construction can be obtained by defining an infinite preimage-cycle for each cycle of the factor system (Y, S) and, on the other hand, providing (Y, S) with some 1-orbit to which all orbits of (X, T) can be sent. The drawback of such a construction is that, due to the infinite preimages of the periodic orbits in the factor system, there is no direct way of obtaining a result about the existence of a realization by compact dynamical systems as given in the Basic Lemma. In this context, it makes sense to introduce a notion like ‘factor surjective on the set of periodic points’ referring to the restriction of the factor map to the sets of periodic points being surjective. In other words, this defines a subclass of dynamical systems with factors in which every periodic orbit of the factor has a periodic orbit in its preimage. Since the permutation systems in Proposition 2 are of that type, it follows that requiring dynamical systems to be factor surjective on the set of periodic points is not restrictive with regard to relative realizability.

Corollary 6. Every exactly realizable sequence is relatively realizable: ER ⊂ ER_{rel}.

Remark 7. Obviously, the criterion for exact realizability is subsumed by the one for relative realizability, but it is also easy to construct a relative realization for a sequence from ER: for f realized by (X, T), the sequence 2f is realized by the induced mapping on the topological sum X + X whose factor is the original map, resulting in a relative realization of the sequence f. A further construction is given by the exact realization of a sequence f +u, where u_{n} = 1 for all n and the trivial system ({0}, id), being a factor of any dynamical system.

The following result shows that ER_{rel} shares many properties with ER (cf. [12, Section 2]). The proofs are either based on the integrality condition of the orbit counts sequences or on the construction of realizing dynamical systems. They are very similar to those for ER [12] and can be found in [11].

Theorem 8. The set ER_{rel} of relatively realizable sequences satisfies the following properties:

1. There are no zero divisors in ER_{rel}: fg = 0 for f, g ∈ ER_{rel} implies f = 0 or g = 0.

2. ER_{rel} contains the constant sequences over ℤ.

3. ER_{rel} is closed under addition, multiplication und multiplication with elements z ∈ ℤ.

4. The constant sequence u = (1)_{n≥1} is the only completely multiplicative sequence in ER_{rel}. □

Example 9. The Thue-Morse- and Period Doubling-chains are examples of inflation dynamical systems that can be treated symbolically. For background information, consider the articles by Queffelec [1], Allouche and Mendès France [14] and the references given there. Define, over the finite alphabets {a, b} and {A, B}, respectively, the substitution rules

TM : a → ab, b → ba and PD : A → AB, B → AA.

The squares TM^{2} and PD^{2} of each of the two mappings, iterated on the respective oneletter seeds, produce the fixed points {..a|a.., ..b|b..}, {..a|b.., ..b|a..} and {..A|A.., ..B|B..}, {..B.., ..A..}, respectively, i.e., 2-cycles of the original maps. The inflations TM and PD define maps on the corresponding LI classes which coincides with the hulls that are obtained as the orbit closures under the continuous translation action of ℝ.

The factor map ϕ is given by the block map

aa ↦ B, bb ↦ B, ab ↦ A, ba ↦ A.

Since the sequences ..A|A.. and ..B|A.. correspond to the two sequences {..a|b.., ..b|a..} and {..a|a.., ..b|b..}, respectively, the map ϕ is strictly 2 : 1. (The vertical line indicates the reference point of the bi-infinite chain.) Following the method described by Anderson and Putnam [2], the dynamical zeta function of both systems can be calculated as [9]

ζ(z) = \frac{1 − z}{(1 + z)(1 − 2z)},

giving rise to the fixed point counts sequence a_{m} = 2^{m} + (−1)^{m} − 1 (A099430 in the OEIS [16]). Thus, these two dynamical systems share the same fixed point counts and relatively realize the sequence (0, 0, 0, …) (A000004). In fact, this and the property of being 2 : 1 completely determine the combinatorics of the map ϕ. Let

α(n) := # n-orbits of PD whose preimages are two n-orbits of TM and
β(n) := # n-orbits of PD whose preimages are 2n-orbits of TM.

Clearly, c_{n} = α(n) + β(n). Furthermore, β(n) = \frac{c_{n}}{2} if n is odd and β(n) = (c_{n} + β(n/2))/2 if n is even. Since a_{n} and c_{n} are related via (2), the calculation of β(n) yields

β(n) = \frac{1}{2n} ∑_{d|n, d odd} μ(d) · 2^{n/d} − δ_{n,1},

where δ_{n,1} = 1 for n = 1 and 0 otherwise. Except for β(1) = 0, this is A000048.

Example 10. The torus parametrization of substitution tilings [5] yields a large class of dynamical systems with a torus automorphism as a topological factor. The one-dimensional Fibonacci chain, obtained by the standard projection method, gives rise to the relative fixed point counts sequence h_{n} = (−1)^{n} (arising from the fixed point counts A001610 and A001350), which corresponds to an orbit counts sequence of orb(h) = (−1, 1, 0, 0, …). More complicated examples are provided by higher dimensional projections. The Penrose tiling and its torus parametrization, for instance, lead to the relative fixed point and orbit counts sequences

(−1, 9, −16, 29, −51, 84, −141, …) and (−1, 5, −5, 5, −10, 15, −20, …)

which can be calculated from the corresponding dynamical zeta functions stated explicitly by Baake and Grimm [4]. The more complicated difference sequences in the last case reflect the phenomenon of a large (though always finite) number of singular tilings being sent to the same torus parameter occurring in higher dimensional systems, whereas the first case illustrates that the one-dimensional Fibonacci torus parametrization is ‘nearly one-to-one’.

Another large group of dynamical systems with non-trivial factors is provided by Sinteger dynamical systems, cf. [7], [8] and references given therein. Let Q, P be subsets of the set of all primes with Q ⊂ P. Then the map S, dual to x ↦ 2x on the ring of Q-integers, is a factor of T, the map dual to x ↦ 2x on the ring of P -integers via the dual of the inclusion map from the ring of P -integers to the ring of Q-integers.

Example 11. Choose P to be the set of all rational primes, Q = ∅ and α : ℚ → ℚ, x ↦ 2x. The resulting commutative diagramme is obtained by setting X = ℚ̂ and Y = ℤ̂ in (3). The dual map α̂ : ℚ̂ → ℚ̂ is characterized by χ ↦ χ ∘ α for all χ ∈ ℚ̂. Thus, the fixed point equation α̂^{k}(χ) = χ is equivalent with χ(2^{k}x) = χ(x) for all x ∈ ℚ or, by the properties of characters, χ((2^{k} − 1)x) = 1 for all x ∈ ℚ. An appropriate choice of x shows that only the trivial character satisfies this condition, yielding a fixed point counts sequence of (1, 1, 1, 1, …) (A000012). On Y, each k ∈ ℤ gives rise to the element e^{2πik} ∈ ℤ̂, such that the (2^{n} − 1)-th roots of unity constitute the 2^{n} − 1 fixed points of S^{n} (A000225).

For P chosen as above and Q = P \ {3} we obtain X = ℚ̂ and Y = ℤ̂_{(3)}, where ℤ_{(3)} = ℤ[\frac{1}{p} : p prime, p ≠ 3]. According to [8, Example 4.1], the n-periodic points of S are given by |2^{n} − 1|_{3}^{−1}, i.e., (1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 9, 1, 3, 1, …), yielding a relative fixed point sequence of (0, −2, 0, −2, 0, −8, 0, −2, 0, −2, 0, −8, 0, −2, 0, …).

3 Acknowledgements

It is my pleasure to thank Michael Baake, Franz Gähler, Christian Huck and Tom Ward for discussions and helpful suggestions. This work was supported by DFG, within the CRC 701.

References

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[9] F. Gähler, private communication.

[10] P. Moss, Algebraic realizability problems, PhD thesis, University of East Anglia, Norwich, 2003.

[11] N. Neumärker, Orbitstatistik und relative Realisierbarkeit, Diplomarbeit, Universität Bielefeld, 2007.

[12] Y. Puri, Arithmetic and growth of periodic orbits, PhD thesis, University of East Anglia, Norwich, 200.

[13] Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Sequences 4 (2001), Article 01.2.1.

[14] M. Queffellec, Spectral study of automatic and substitutive sequences, in F. Axel and D. Gratias, eds., Beyond Quasicrystals, Springer, Berlin and Les Editions de Physique, Les Ulis, 1995, pp. 369–414.

[15] A.J. Windsor, Smoothness is not an obstruction to realizability, Ergodic Th. & Dynam. Syst., 28, (2008), 1037–1041.

[16] N. J. A. Sloane, The Online Encyclopedia of Integer Sequences,

    http://www.research.att.com/∼njas/sequences/.

2000 Mathematics Subject Classification: Primary 37A45; Secondary 11A25. Keywords: dynamical systems, integer sequences, realizability.

(Concerned with sequences \underline{A000004}, \underline{A000007}, \underline{A000012}, \underline{A000048}, \underline{A000225}, \underline{A001350}, \underline{A001610}, \underline{A060280}, and \underline{A099430}.)

Received April 22 2009; revised version received May 8 2009. Published in Journal of Integer Sequences, May 12 2009.

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