Mertens' theorem for toral automorphisms
aa r X i v : . [ m a t h . D S ] M a y MERTENS’ THEOREM FOR TORAL AUTOMORPHISMS
SAWIAN JAIDEE, SHAUN STEVENS, AND THOMAS WARD
Abstract.
A dynamical Mertens’ theorem for ergodic toral automorphismswith error term O( N − ) is found, and the influence of resonances among theeigenvalues of unit modulus is examined. Examples are found with many more,and with many fewer, periodic orbits than expected. Introduction
Discrete dynamical analogs of Mertens’ theorem concern a map T : X → X , andare motivated by work of Sharp [7] on Axiom A flows. A set of the form τ = { x, T ( x ) , . . . , T k ( x ) = x } with cardinality k is called a closed orbit of length | τ | = k , and the results pro-vide asymptotics for a weighted sum over closed orbits. For the discrete case of ahyperbolic diffeomorphism T , we always have M T ( N ) := X | τ | N h | τ | ∼ log( N ) , where h is the topological entropy, with more explicit additional terms in manycases. The main term log( N ) is not really related to the dynamical system, but is aconsequence of the fact that the number of orbits of length n is n e hn + O(e h ′ n ) forsome h ′ < h (see [6]). Without the assumption of hyperbolicity, the asymptoticschange significantly, and in particular depend on the dynamical system. For quasi-hyperbolic (ergodic but not hyperbolic) toral automorphisms, Noorani [5] finds ananalogue of Mertens’ theorem in the form M T ( N ) = m log( N ) + C m ∈ N . The constant C N − ) is recov-ered using elementary arguments, and the coefficient m of the main term in (1)is expressed as an integral over a sub-torus. This reveals the effect of resonancesbetween the eigenvalues of unit modulus, and examples show that the value of m may be very different to the generic value given in [5]. Date : October 29, 2018.2000
Mathematics Subject Classification. Toral automorphisms
Let T : T d → T d be a toral automorphism corresponding to a matrix A T in GL d ( Z ) with eigenvalues { λ i | i d } , arranged so that | λ | > · · · > | λ s | > | λ s +1 | = · · · = | λ s +2 t | > | λ s +2 t +1 | > · · · > | λ d | . The map T is ergodic with respect to Lebesgue measure if no eigenvalue is a rootof unity, is hyperbolic if in addition t = 0 (that is, there are no eigenvalues of unitmodulus), and is quasihyperbolic if it is ergodic and t >
0. The topological entropyof T is given by h = h ( T ) = P sj =1 log | λ j | . Theorem 1.
Let T be a quasihyperbolic toral automorphism with topological en-tropy h . Then there are constants C m > with X | τ | N h | τ | = m log N + C + O (cid:0) N − (cid:1) . The coefficient m in the main term is given by m = Z X t Y i =1 (2 − πx i )) d x . . . d x t , where X ⊂ T d is the closure of { ( nθ , . . . , nθ t ) | n ∈ Z } , and e ± π i θ , . . . , e ± π i θ t are the eigenvalues with unit modulus of the matrix defining T . As we will see in Example 3, the quantity m appearing in Theorem 1 takes on awide range of values. In particular, m may be much larger, or much smaller, thanits generic value 2 t . Proof.
Since T is ergodic, F T ( n ) = |{ x ∈ T d | T n ( x ) = x }| = | Z d / ( A nT − I ) Z d | = d Y i =1 | λ ni − | , so O T ( n ) = 1 n X m | n µ ( n/m ) d Y i =1 | λ mi − | . Write Λ = Q si =1 λ i (so the topological entropy of T is log | Λ | ) and κ = min {| λ s | , | λ s +2 t +1 | − } > . The eigenvalues of unit modulus contribute nothing to the topological entropy, butmultiply the approximation | Λ | n to F T ( n ) by an almost-periodic factor boundedabove by 2 t and bounded below by A/n B for some A, B >
0, by Baker’s theorem(see [3, Ch. 3] for this argument).
Lemma 2. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F T ( n ) − | Λ | n s +2 t Y i = s +1 | λ ni − | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · | Λ | − n = O( κ − n ) . Proof.
We have d Y i =1 ( λ ni −
1) = s Y i =1 ( λ ni − | {z } U n s +2 t Y i = s +1 ( λ ni − | {z } V n d Y i =2 t + s +1 ( λ ni − | {z } W n , (2) ERTENS’ THEOREM FOR TORAL AUTOMORPHISMS 3 where U n is equal to the sum of Λ n and (2 s −
1) terms comprising products of eigen-values, each no larger than κ − n | Λ | n in modulus, W n is equal to the sum of ( − d − s and 2 d − t − s − κ − n , and | V n | t . Itfollows that (cid:12)(cid:12)(cid:12)Q di =1 ( λ ni − − ( − d − s Λ n Q s +2 ti = s +1 ( λ ni − (cid:12)(cid:12)(cid:12) | Λ | n = (cid:12)(cid:12) V n (cid:0) U n W n − ( − d − s Λ n (cid:1)(cid:12)(cid:12) | Λ | n = | V n (Λ n + O (Λ n /κ n ) − Λ n ) || Λ | n = O( κ − n ) . The statement of the lemma follows by the reverse triangle inequality. (cid:3)
Now M T ( N ) = N X n =1 n | Λ | n F T ( n ) + X d | n,d
11 0 0 80 1 0 −
60 0 1 8 . (5)Here X is a diagonally embedded circle, and m = Z Z { x = x } Y j =1 (2 − πjx j )) d x d x = Z (2 − πx )) d x = 6 > . Extending this example, let T n be the automorphism of T n defined by the ma-trix A ⊕ · · · ⊕ A ( n terms). The matrix corresponding to T n has 2 n eigenvalues withmodulus one (comprising two conjugate eigenvalues with multiplicity n ). Then X is again a diagonally embedded circle, and m = Z (2 − πx )) t d x = (2 t )!( t !) ∼ t √ πt by Stirling’s formula. This is much larger than 2 t , reflecting the density of thesyndetic set on which the almost-periodic factor is close to 2 t . Indeed, this exampleshows that m t may be arbitrarily large.(c) A simple example with m < t is the following. Let S be the automorphismof T defined by the matrix A ⊕ A ⊕ A , with A as in (5). Again X is a diagonallyembedded circle, and m = Z Z Z { x = x = x } Y j =1 (2 − πjx j )) d x d x d x = Z (2 − πx ))(2 − πx ))(2 − πx )) d x = 6 < . Extending this example, the value of m for the automorphism of T t defined by thematrix A ⊕ A ⊕ · · · ⊕ A t as t varies gives the sequence2 , , , , , , , , , , , , , , , , , , , , . . . (we thank Paul Hammerton for computing these numbers). This sequence, en-try A133871 in the Encyclopedia of Integer Sequences [8], does not seem to bereadily related to other combinatorial sequences.(d) Generalizing the example in (c), for any sequence ( a n ) of natural numbers, wecould look at the automorphisms S n of T n defined by the matrices L nk =1 A a k ,with A as in (5). In order to make m small, we need a “sum-heavy” sequence,that is, one with many three-term linear relations of the form a i + a j = a k . Moreprecisely, one would like many linear relations with an odd number of terms, andfew with an even number of terms. Constructing such sequences, and understandinghow dense they may be, seems to be difficult.Taking ( a n ) to be the sequence whose first eight terms are 1 , , , , , , , a n +8 = 100 a n , we find SAWIAN JAIDEE, SHAUN STEVENS, AND THOMAS WARD that the automorphism S n of T n has m = 2 n = 2 t/ . Thus m t may be arbitrarilysmall.We close with some remarks.(a) In the quasihyperbolic case the O(1 /N ) term is oscillatory, so no improvementof the asymptotic in terms of a monotonic function is possible. The extent to whichthe exponential dominance of the entropy term fails in this setting is revealed by thefollowing. Let F T ( n ) denote the number of points fixed by the automorphism T n .On the one hand, Baker’s theorem implies that F T ( n ) /n → e h as n → ∞ . Onthe other hand Dirichlet’s theorem shows that F T ( n + 1) /F T ( n ) does not converge(see [1, Th. 6.3]).(b) The formula for m in the statement of [5, Th. 1] is incorrect in a minor way; asstated in [5, Rem. 2] and as illustrated in the examples above, m should be K (1),which is not necessarily the same as 2 t .(c) The proof of Theorem 1 also gives an elementary proof of the asymptotics inthe hyperbolic case: in the notation of the proof, V n = 1 so m = 1. Applying nowthe Euler-MacLaurin summation formula (see Ram Murty [4, Th. 2.1.9]) we get anasymptotic of the shape X | τ | N h | τ | = log N + C k − X r =0 B r +1 ( r + 1) N r +1 + O (cid:16) N − ( k +1) (cid:17) , where B = − , B = ,. . . are the Bernoulli numbers, for any k > References [1] V. Chothi, G. Everest, and T. Ward. S -integer dynamical systems: periodic points. J. ReineAngew. Math. , 99–132, 1997.[2] G. Everest, R. Miles, S. Stevens, and T. Ward. Orbit-counting in non-hyperbolic dynamicalsystems.
J. Reine Angew. Math. , 155–182, 2007.[3] G. Everest and T. Ward.
Heights of polynomials and entropy in algebraic dynamics.
Springer-Verlag London (1999).[4] M. Ram Murty.
Problems in analytic number theory , volume 206 of
Graduate Texts in Math-ematics . Springer-Verlag, New York, 2001.[5] Mohd. Salmi Md. Noorani. Mertens theorem and closed orbits of ergodic toral automorphisms.
Bull. Malaysian Math. Soc. (2) (2), 127–133, 1999.[6] A. Pakapongpun and T. Ward. Functorial orbit counting J. Integer Sequences , , Article09.2.4, 2009.[7] R. Sharp. An analogue of Mertens’ theorem for closed orbits of Axiom A flows. Bol. Soc.Brasil. Mat. (N.S.) , (2), 205–229, 1991.[8] N. J. A. Sloane. An on-line version of the encyclopedia of integer sequences. Electron. J.Combin. :Feature 1, approx. 5 pp., 1994. ..