- Email: [email protected]
Integer sequences and dynamics
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 70 (2018) 83–88
www.elsevier.com/locate/endm
Integer sequences and dynamics Tom Ward 1,2 Ziff Building, Room 13.01 University of Leeds Leeds LS2 9JT, U.K.
Abstract Integer sequences express and capture many important concepts in number theory. They also arise naturally in some parts of dynamical systems, and we will explain some of the questions and relationships that arise in looking at integer sequences from these two perspectives. One of these connections occurs between prime numbers and closed orbits. In algebraic and geometric settings, there are hints of a quite widespread P´ olya–Carlson dichotomy. Keywords: Integer sequences, dynamical systems, natural boundary, Lehmer–Pierce sequence, elliptic divisibility sequence, linear recurrence sequence
1
Introduction
Dynamical systems generate integer sequences in many ways, and these sometimes throw up questions or observations of a number-theoretic sort. Our starting point is a fragment of a Greek poem attributed to Archilochus (c. 680– c. 645 BC), which is usually translated as ‘a fox knows many things, but a 1
Dedicated to the memory of Graham Everest (1957–2010), with whom many of these ideas were discussed. 2 Email: [email protected] https://doi.org/10.1016/j.endm.2018.11.014 1571-0653/© 2018 Elsevier B.V. All rights reserved.
84
T. Ward / Electronic Notes in Discrete Mathematics 70 (2018) 83–88
hedgehog one important thing’. Number theory (the fox) has a huge diversity of perspectives and motivations, while arguably dynamical systems (the hedgehog) begins with one simple idea, and that is iteration. Perhaps after taking a Poincar´e section in a dynamical system evolving in time, one obtains a map T : X → X. The space X may inherit (or be born with) some structure: a manifold describing some conserved quantity like energy or momentum, a natural volume measure, a notion of distance, and so on. The map T may reflect the structure, by being a smooth map, or a measure-preserving map, or continuous, and so on. In all cases we arrive at the following question: how can properties of the orbit x, T (x), T 2 (x), . . . of a point x under the action of iterating T be described? Our purpose is to illustrate some very specific aspects of this via quite elementary examples that lie particularly close to the topic of integer sequences in number theory.
2
Entropy and Mahler measures
Initially, we let X be a compact metric space and T : X → X a continuous map. For an open cover U, write H(U) for the least cardinality of a subcover of U. Then it may be shown (see [3]) that the integer sequence whose nth term is H(U ∨ T −1 U ∨ · · · ∨ T −n+1 U) grows like enh(T ) for some constant h(T ) 0, and this growth rate is the topological entropy of T . Because of the connection between entropy and the structure of dynamical systems, two questions are of specific interest: Under what conditions does h(T ) vanish? What values can h(T ) take on? A relatively easy construction shows that for any h ∈ [0, ∞] there is a continuous map T with h(T ) = h. Example 2.1 Automorphisms of the torus provide an important family of algebraic examples. Let X = Td , the d-torus, and let T = TA be the group automorphism given by the linear action of a matrix A ∈ SLd (Z). Then h(TA ) =
log+ |λ|,
λ∈σ(A)
where σ(A) denotes the spectrum (set of eigenvalues) of A, and log+ t denotes max{0, log t} (we refer to the introduction of [10] for an account of the history of this result). The quantity h(TA ) is well-known in number theory, as the logarithmic Mahler measure of the characteristic polynomial χA of the matrix A. This was introduced by Mahler [11] in connection with inequalities
T. Ward / Electronic Notes in Discrete Mathematics 70 (2018) 83–88
85
for integer polynomials, and we can also write h(TA ) =
1
log |χA (e2πit | dt
0
by Jensen’s formula. Kronecker’s lemma shows that h(TA ) = 0 exactly when χA is a product of a monomial and some cyclotomic polynomials, answering the first question for this context. The second question, asking for the possible values of h(TA ), is much more involved. Lehmer [8] found a polynomial of degree 10 with a particularly small positive logarithmic Mahler measure, and asked if arbitrarily small positive examples could be found. This remains open: in our terminology, this amounts to asking if there is an automorphism of a torus with entropy lying in the interval (0, 0.1623569 . . . ). By some deep results in ergodic theory, this question turns out to be fundamental to finding out whether the collection of automorphisms of ergodic compact groups modulo the equivalence of isomorphism as measure-preserving dynamical systems is countable or uncountable (we refer to Lind [9] for an explanation of this).
3
Counting closed orbits
Returning to an abstract map (which for convenience we now assume to be a bijection) T : X → X, an individual orbit x, T (x), T 2 (x), . . . either comprises an infinite set or returns to itself, giving a closed orbit of length k in the case that |{T n (x) | n ∈ N}| = k. Under the assumption that it only has OT (n) < ∞ closed orbits of each length, a bijection T : X → X then has two associated integer sequences: • its ‘count of periodic points’ FT (n) , where FT (n) = |{x ∈ X | T n x = x}; • its ‘count of closed orbits’, OT (n) . Windsor [14] proved that any sequence of non-negative integers arises as the ∞ count of closed orbits of a C map of the 2-torus. However, FT (n) has to obey the rules arising from the relation FT (n) = d|n dOT (d) which holds for all n 1. For example, the Fibonacci sequence (1, 1, 2, 3, . . . ) cannot count periodic points for a map, as the single fixed point requires that the number of points of period 3 must be equal to 1 modulo 3. On the other hand, the Lucas sequence (1, 3, 4, 7, . . . ) does turn out to be a count of periodic points.
T. Ward / Electronic Notes in Discrete Mathematics 70 (2018) 83–88
86
3.1 Linear recurrences A natural problem is to characterize those linear recurrence sequences that count periodic points. A sample result is that (1, a, 1 + a, 1 + 2a, 2 + 3a, 3 + 5a, . . . ) counts periodic points if and only if a = 3 (see [13]). Partial results are known [7], but the full picture of which linear recurrences have this property is not complete. 3.2 Analytic questions The relation FT (n) = d|n dOT (d) can be expressed in multiple ways using the ‘dynamical zeta function’ ζT and the ‘orbit Dirichlet series’ dT : OT (n) = n1 μ(n/d)FT (d) d|n
ζT (z) := exp
zn F (n) n T
n1
dT (z) :=
O n1
T (n) nz
=
(1 − z n )−OT (n)
n1
F (n)/n 1 T = . z ζ(z + 1) n≥1 n
Analytic properties of ζT or dT can give asymptotics for orbit counting functions, much as analytic properties of the classical Riemann zeta function give the prime number theorem. Example 3.1 A baby case illustrates this, and also shows up some surprises. If T : X → X has exactly one orbit of each length, then dT (z) = ζ(z) (the Riemann zeta function). The natureof the singularity at z = 1 shows, via Perron’s theorem, the trivial result nN OT (n) ∼ N. Here the left-hand side is exactly N, but the point is the right-hand side is arrivedat using only b (n) the analytic information. The Hardy–Ramanujan formula for n≥1 σa (n)σ nz 2
shows that for the Cartesian square T × T we have dT ×T (z) = ζ (z)ζ(z−1) , and ζ(2z) π2 2 Perron’s theorem then gives nN OT ×T (n) ∼ 12 N . One might expect the pattern to continue, however, dT ×T ×T shows new phenomena: this has abscissa of convergence at 3, a meromorphic extension to (z) > 1 — but a natural boundary at (z) = 1. We find 2 ζ(3) OT ×T ×T (n) ∼ π 18 N3 × (1 + p−5 + 2p−2 + 2p−3 ) nN
p
T. Ward / Electronic Notes in Discrete Mathematics 70 (2018) 83–88
87
where the latter — unexpected — factor is a product taken over all primes. 3.3 Group automorphisms Another natural problem is to characterize those integer sequences that count periodic points for some group automorphism. There are partial results in the thesis of Moss [12]: clearly such a sequence must be a divisibility sequence, but Moss develops ‘local’ criteria at each prime. Example 3.2 The periodic sequence (1, 1, 1, 1, 6, 1, 1, 1, 1, 6, . . . ) is a linear recurrence sequence and a divisibility sequence, but it cannot count periodic points for a group automorphism because no group with six elements can have an automorphism with one fixed point and a cycle of length five. We refer to [1,4,5] for the main results and some examples of orbit-growth for compact group automorphisms. 3.4 A P´olya–Carlson dichotomy? Toral automorphisms have rational zeta functions, but the simplest non-toral compact abelian group automorphisms have periodic point counts like (2n − 1)
×
expected formula
|2n − 1|3
.
some orbits killed
1−z A periodic point sequence 2n − 1 has dynamical zeta function 1−2z , but it n n is easy to show that the periodic point sequence (2 − 1) × |2 − 1|3 gives a dynamical zeta function with a natural boundary at |z| = 12 (see [6]). It is conjectured that the zeta function of a compact group automorphism is either rational or admits a natural boundary. If true (we refer to [2] for what is known), this is related to the elusive interaction between the group structure, dynamics, and the structure of linear recurrence sequences.
References [1] Baier, S., S. Jaidee, S. Stevens and T. Ward, Automorphisms with exotic orbit growth, Acta Arith. 158 (2013), pp. 173–197. URL https://doi.org/10.4064/aa158-2-5 [2] Bell, J., R. Miles and T. Ward, Towards a P´ olya-Carlson dichotomy for algebraic dynamics, Indag. Math. (N.S.) 25 (2014), pp. 652–668. URL https://doi.org/10.1016/j.indag.2014.04.005
88
T. Ward / Electronic Notes in Discrete Mathematics 70 (2018) 83–88
[3] Einsiedler, M., E. Lindenstrauss and T. Ward, Entropy in ergodic theory and homogeneous dynamics, to appear. URL https://tbward0.wixsite.com/books/entropy [4] Everest, G., R. Miles, S. Stevens and T. Ward, Orbit-counting in non-hyperbolic dynamical systems, J. Reine Angew. Math. 608 (2007), pp. 155–182. URL https://doi.org/10.1515/CRELLE.2007.056 [5] Everest, G., R. Miles, S. Stevens and T. Ward, Dirichlet series for finite combinatorial rank dynamics, Trans. Amer. Math. Soc. 362 (2010), pp. 199– 227. URL https://doi.org/10.1090/S0002-9947-09-04962-9 [6] Everest, G., V. Stangoe and T. Ward, Orbit counting with an isometric direction, in: Algebraic and topological dynamics, Contemp. Math. 385, Amer. Math. Soc., Providence, RI, 2005 pp. 293–302. URL https://doi.org/10.1090/conm/385/07202 [7] Everest, G., A. J. van der Poorten, Y. Puri and T. Ward, Integer sequences and periodic points, J. Integer Seq. 5 (2002), pp. Article 02.2.3, 10. [8] Lehmer, D. H., Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), pp. 461–479. URL https://doi.org/10.2307/1968172 [9] Lind, D. A., The structure of skew products with ergodic group automorphisms, Israel J. Math. 28 (1977), pp. 205–248. URL https://doi.org/10.1007/BF02759810 [10] Lind, D. A. and T. Ward, Automorphisms of solenoids and p-adic entropy, Ergodic Theory Dynam. Systems 8 (1988), pp. 411–419. URL https://doi.org/10.1017/S0143385700004545 [11] Mahler, K., An application of Jensen’s formula to polynomials, Mathematika 7 (1960), pp. 98–100. URL https://doi.org/10.1112/S0025579300001637 [12] Moss, P., “The arithmetic of realizable sequences,” Ph.D. thesis, University of East Anglia (2003). [13] Puri, Y. and T. Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quart. 39 (2001), pp. 398–402. [14] Windsor, A. J., Smoothness is not an obstruction to realizability, Ergodic Theory Dynam. Systems 28 (2008), pp. 1037–1041. URL https://doi.org/10.1017/S0143385707000715
Electronic Notes in Discrete Mathematics 70 (2018) 83–88
www.elsevier.com/locate/endm
Integer sequences and dynamics Tom Ward 1,2 Ziff Building, Room 13.01 University of Leeds Leeds LS2 9JT, U.K.
Abstract Integer sequences express and capture many important concepts in number theory. They also arise naturally in some parts of dynamical systems, and we will explain some of the questions and relationships that arise in looking at integer sequences from these two perspectives. One of these connections occurs between prime numbers and closed orbits. In algebraic and geometric settings, there are hints of a quite widespread P´ olya–Carlson dichotomy. Keywords: Integer sequences, dynamical systems, natural boundary, Lehmer–Pierce sequence, elliptic divisibility sequence, linear recurrence sequence
1
Introduction
Dynamical systems generate integer sequences in many ways, and these sometimes throw up questions or observations of a number-theoretic sort. Our starting point is a fragment of a Greek poem attributed to Archilochus (c. 680– c. 645 BC), which is usually translated as ‘a fox knows many things, but a 1
Dedicated to the memory of Graham Everest (1957–2010), with whom many of these ideas were discussed. 2 Email: [email protected] https://doi.org/10.1016/j.endm.2018.11.014 1571-0653/© 2018 Elsevier B.V. All rights reserved.
84
T. Ward / Electronic Notes in Discrete Mathematics 70 (2018) 83–88
hedgehog one important thing’. Number theory (the fox) has a huge diversity of perspectives and motivations, while arguably dynamical systems (the hedgehog) begins with one simple idea, and that is iteration. Perhaps after taking a Poincar´e section in a dynamical system evolving in time, one obtains a map T : X → X. The space X may inherit (or be born with) some structure: a manifold describing some conserved quantity like energy or momentum, a natural volume measure, a notion of distance, and so on. The map T may reflect the structure, by being a smooth map, or a measure-preserving map, or continuous, and so on. In all cases we arrive at the following question: how can properties of the orbit x, T (x), T 2 (x), . . . of a point x under the action of iterating T be described? Our purpose is to illustrate some very specific aspects of this via quite elementary examples that lie particularly close to the topic of integer sequences in number theory.
2
Entropy and Mahler measures
Initially, we let X be a compact metric space and T : X → X a continuous map. For an open cover U, write H(U) for the least cardinality of a subcover of U. Then it may be shown (see [3]) that the integer sequence whose nth term is H(U ∨ T −1 U ∨ · · · ∨ T −n+1 U) grows like enh(T ) for some constant h(T ) 0, and this growth rate is the topological entropy of T . Because of the connection between entropy and the structure of dynamical systems, two questions are of specific interest: Under what conditions does h(T ) vanish? What values can h(T ) take on? A relatively easy construction shows that for any h ∈ [0, ∞] there is a continuous map T with h(T ) = h. Example 2.1 Automorphisms of the torus provide an important family of algebraic examples. Let X = Td , the d-torus, and let T = TA be the group automorphism given by the linear action of a matrix A ∈ SLd (Z). Then h(TA ) =
log+ |λ|,
λ∈σ(A)
where σ(A) denotes the spectrum (set of eigenvalues) of A, and log+ t denotes max{0, log t} (we refer to the introduction of [10] for an account of the history of this result). The quantity h(TA ) is well-known in number theory, as the logarithmic Mahler measure of the characteristic polynomial χA of the matrix A. This was introduced by Mahler [11] in connection with inequalities
T. Ward / Electronic Notes in Discrete Mathematics 70 (2018) 83–88
85
for integer polynomials, and we can also write h(TA ) =
1
log |χA (e2πit | dt
0
by Jensen’s formula. Kronecker’s lemma shows that h(TA ) = 0 exactly when χA is a product of a monomial and some cyclotomic polynomials, answering the first question for this context. The second question, asking for the possible values of h(TA ), is much more involved. Lehmer [8] found a polynomial of degree 10 with a particularly small positive logarithmic Mahler measure, and asked if arbitrarily small positive examples could be found. This remains open: in our terminology, this amounts to asking if there is an automorphism of a torus with entropy lying in the interval (0, 0.1623569 . . . ). By some deep results in ergodic theory, this question turns out to be fundamental to finding out whether the collection of automorphisms of ergodic compact groups modulo the equivalence of isomorphism as measure-preserving dynamical systems is countable or uncountable (we refer to Lind [9] for an explanation of this).
3
Counting closed orbits
Returning to an abstract map (which for convenience we now assume to be a bijection) T : X → X, an individual orbit x, T (x), T 2 (x), . . . either comprises an infinite set or returns to itself, giving a closed orbit of length k in the case that |{T n (x) | n ∈ N}| = k. Under the assumption that it only has OT (n) < ∞ closed orbits of each length, a bijection T : X → X then has two associated integer sequences: • its ‘count of periodic points’ FT (n) , where FT (n) = |{x ∈ X | T n x = x}; • its ‘count of closed orbits’, OT (n) . Windsor [14] proved that any sequence of non-negative integers arises as the ∞ count of closed orbits of a C map of the 2-torus. However, FT (n) has to obey the rules arising from the relation FT (n) = d|n dOT (d) which holds for all n 1. For example, the Fibonacci sequence (1, 1, 2, 3, . . . ) cannot count periodic points for a map, as the single fixed point requires that the number of points of period 3 must be equal to 1 modulo 3. On the other hand, the Lucas sequence (1, 3, 4, 7, . . . ) does turn out to be a count of periodic points.
T. Ward / Electronic Notes in Discrete Mathematics 70 (2018) 83–88
86
3.1 Linear recurrences A natural problem is to characterize those linear recurrence sequences that count periodic points. A sample result is that (1, a, 1 + a, 1 + 2a, 2 + 3a, 3 + 5a, . . . ) counts periodic points if and only if a = 3 (see [13]). Partial results are known [7], but the full picture of which linear recurrences have this property is not complete. 3.2 Analytic questions The relation FT (n) = d|n dOT (d) can be expressed in multiple ways using the ‘dynamical zeta function’ ζT and the ‘orbit Dirichlet series’ dT : OT (n) = n1 μ(n/d)FT (d) d|n
ζT (z) := exp
zn F (n) n T
n1
dT (z) :=
O n1
T (n) nz
=
(1 − z n )−OT (n)
n1
F (n)/n 1 T = . z ζ(z + 1) n≥1 n
Analytic properties of ζT or dT can give asymptotics for orbit counting functions, much as analytic properties of the classical Riemann zeta function give the prime number theorem. Example 3.1 A baby case illustrates this, and also shows up some surprises. If T : X → X has exactly one orbit of each length, then dT (z) = ζ(z) (the Riemann zeta function). The natureof the singularity at z = 1 shows, via Perron’s theorem, the trivial result nN OT (n) ∼ N. Here the left-hand side is exactly N, but the point is the right-hand side is arrivedat using only b (n) the analytic information. The Hardy–Ramanujan formula for n≥1 σa (n)σ nz 2
shows that for the Cartesian square T × T we have dT ×T (z) = ζ (z)ζ(z−1) , and ζ(2z) π2 2 Perron’s theorem then gives nN OT ×T (n) ∼ 12 N . One might expect the pattern to continue, however, dT ×T ×T shows new phenomena: this has abscissa of convergence at 3, a meromorphic extension to (z) > 1 — but a natural boundary at (z) = 1. We find 2 ζ(3) OT ×T ×T (n) ∼ π 18 N3 × (1 + p−5 + 2p−2 + 2p−3 ) nN
p
T. Ward / Electronic Notes in Discrete Mathematics 70 (2018) 83–88
87
where the latter — unexpected — factor is a product taken over all primes. 3.3 Group automorphisms Another natural problem is to characterize those integer sequences that count periodic points for some group automorphism. There are partial results in the thesis of Moss [12]: clearly such a sequence must be a divisibility sequence, but Moss develops ‘local’ criteria at each prime. Example 3.2 The periodic sequence (1, 1, 1, 1, 6, 1, 1, 1, 1, 6, . . . ) is a linear recurrence sequence and a divisibility sequence, but it cannot count periodic points for a group automorphism because no group with six elements can have an automorphism with one fixed point and a cycle of length five. We refer to [1,4,5] for the main results and some examples of orbit-growth for compact group automorphisms. 3.4 A P´olya–Carlson dichotomy? Toral automorphisms have rational zeta functions, but the simplest non-toral compact abelian group automorphisms have periodic point counts like (2n − 1)
×
expected formula
|2n − 1|3
.
some orbits killed
1−z A periodic point sequence 2n − 1 has dynamical zeta function 1−2z , but it n n is easy to show that the periodic point sequence (2 − 1) × |2 − 1|3 gives a dynamical zeta function with a natural boundary at |z| = 12 (see [6]). It is conjectured that the zeta function of a compact group automorphism is either rational or admits a natural boundary. If true (we refer to [2] for what is known), this is related to the elusive interaction between the group structure, dynamics, and the structure of linear recurrence sequences.
References [1] Baier, S., S. Jaidee, S. Stevens and T. Ward, Automorphisms with exotic orbit growth, Acta Arith. 158 (2013), pp. 173–197. URL https://doi.org/10.4064/aa158-2-5 [2] Bell, J., R. Miles and T. Ward, Towards a P´ olya-Carlson dichotomy for algebraic dynamics, Indag. Math. (N.S.) 25 (2014), pp. 652–668. URL https://doi.org/10.1016/j.indag.2014.04.005
88
T. Ward / Electronic Notes in Discrete Mathematics 70 (2018) 83–88
[3] Einsiedler, M., E. Lindenstrauss and T. Ward, Entropy in ergodic theory and homogeneous dynamics, to appear. URL https://tbward0.wixsite.com/books/entropy [4] Everest, G., R. Miles, S. Stevens and T. Ward, Orbit-counting in non-hyperbolic dynamical systems, J. Reine Angew. Math. 608 (2007), pp. 155–182. URL https://doi.org/10.1515/CRELLE.2007.056 [5] Everest, G., R. Miles, S. Stevens and T. Ward, Dirichlet series for finite combinatorial rank dynamics, Trans. Amer. Math. Soc. 362 (2010), pp. 199– 227. URL https://doi.org/10.1090/S0002-9947-09-04962-9 [6] Everest, G., V. Stangoe and T. Ward, Orbit counting with an isometric direction, in: Algebraic and topological dynamics, Contemp. Math. 385, Amer. Math. Soc., Providence, RI, 2005 pp. 293–302. URL https://doi.org/10.1090/conm/385/07202 [7] Everest, G., A. J. van der Poorten, Y. Puri and T. Ward, Integer sequences and periodic points, J. Integer Seq. 5 (2002), pp. Article 02.2.3, 10. [8] Lehmer, D. H., Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), pp. 461–479. URL https://doi.org/10.2307/1968172 [9] Lind, D. A., The structure of skew products with ergodic group automorphisms, Israel J. Math. 28 (1977), pp. 205–248. URL https://doi.org/10.1007/BF02759810 [10] Lind, D. A. and T. Ward, Automorphisms of solenoids and p-adic entropy, Ergodic Theory Dynam. Systems 8 (1988), pp. 411–419. URL https://doi.org/10.1017/S0143385700004545 [11] Mahler, K., An application of Jensen’s formula to polynomials, Mathematika 7 (1960), pp. 98–100. URL https://doi.org/10.1112/S0025579300001637 [12] Moss, P., “The arithmetic of realizable sequences,” Ph.D. thesis, University of East Anglia (2003). [13] Puri, Y. and T. Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quart. 39 (2001), pp. 398–402. [14] Windsor, A. J., Smoothness is not an obstruction to realizability, Ergodic Theory Dynam. Systems 28 (2008), pp. 1037–1041. URL https://doi.org/10.1017/S0143385707000715