The Nielsen numbers of iterations of maps on infra-solvmanifolds of type (R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {(R)}$$\end{document} and periodic orbits

We study the asymptotic behavior of the sequence of the Nielsen numbers {N(fk)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{N(f^k)\}$$\end{document}, the essential periodic orbits of f and the homotopy minimal periods of f using the Nielsen theory of maps f on infra-solvmanifolds of type (R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {(R)}$$\end{document}. We give a linear lower bound for the number of essential periodic orbits of such a map, which sharpens well-known results of Shub and Sullivan for periodic points and of Babenko and Bogatyǐ for periodic orbits. We also verify that a constant multiple of infinitely many prime numbers occur as homotopy minimal periods of such a map.


Introduction
Let f : X → X be a map on a finite complex X. A point x ∈ X is a fixed point of f if f (x) = x; a periodic point of f with period n if f n (x) = x. The smallest period of x is called the minimal period. We will use the following notations: The number of essential fixed point classes is called the Nielsen number of f , denoted by N (f ) [22]. The Nielsen number is always finite and is a homotopy invariant lower bound for the number of fixed points of f . In the category of compact, connected polyhedra the Nielsen number of a map is, apart from in certain exceptional cases, equal to the least number of fixed points of maps with the same homotopy type as f .
From the dynamical point of view, it is natural to consider the Nielsen numbers N (f k ) of all iterations of f simultaneously. For example, N. Ivanov [17] introduced the notion of the asymptotic Nielsen number, measuring the growth of the sequence N (f k ) and found the basic relation between the topological entropy of f and the asymptotic Nielsen number. Later on, it was suggested in [9][10][11]36] to arrange the Nielsen numbers N (f k ) of all iterations of f into the Nielsen zeta function The Nielsen zeta function N f (z) is a nonabelian analogue of the Lefschetz zeta function where L(f n ) := dim X k=0 (−1) k tr {f n * k : H k (X; Q) → H k (X; Q)} is the Lefschetz number of the iterate f n of f . Nice analytical properties of N f (z) [11] indicate that the numbers N (f k ) are closely interconnected. The manifestation of this is Gauss congruences for any k > 0, where f is a map on an infra-solvmanifold of type (R) [12].
The fundamental invariants of f used in the study of periodic points are the Lefschetz numbers L(f k ), and their algebraic combinations, the Nielsen numbers N (f k ) and the Nielsen-Jiang periodic numbers NP n (f ) and NΦ n (f ).
The study of periodic points using the Lefschetz theory has been done extensively by many authors in the literatures such as [2,8,19,22,35]. A natural question is to know how much information we can get about the set of essential periodic points of f or about the set of (homotopy) minimal periods of f from the study of the sequence {N (f k )} of the Nielsen numbers of iterations of f . Even though the Lefschetz numbers L(f k ) and the Nielsen numbers N (f k ) are different from the nature and not equal for maps f on infra-solvmanifolds of type (R), we can utilize the arguments employed the averaging formulas for the Lefschetz and the Nielsen numbers L(f k ) and N (f k ) [29,Theorem 4.2] and [13], we have Concerning the Nielsen numbers N (f k ) of all iterates of f , we recall the following results. These results about N (f k ) are crucial in our discussion. for all k > 0.
Indeed, we have shown in [12,Theorem 11.4] that the left-hand side is non-negative because it is equal to the number of isolated periodic points of f with least period k. By [ is a rational function.
In fact, it is well known that Hence, L f (z) is a rational function with coefficients in Q. In [7,Theorem 4.5], it is shown that N f (z) is either Vol. 20 (2018) The Nielsen numbers of iterations of maps Page 5 of 31 62 where p is the number of real eigenvalues of D * which are > 1 and n is the number of real eigenvalues of D * which are < −1. Here, f + is a lift of f to a certain twofold covering of M which has the same affine homotopy lift (d, D) as f . Consequently, N f (z) is a rational function with coefficients in Q.
On the other hand, since N f (0) = 1 by definition, z = 0 is not a zero nor a pole of the rational function N f (z). Thus, we can write with all λ i distinct nonzero algebraic integers (see for example [3] or [2, Theorem 2.1]) and ρ i nonzero integers. Taking log on both sides of the above identity, we obtain This induces Note that r(f ) is the number of zeros and poles of N f (z). Since N f (z) is a homotopy invariant, so is r(f ). This argument tells us that whenever we have a rational expression of N f (z), we can write down all N (f k ) directly from the expression. However, even though we can compute all N (f k ) using the averaging formula, it can be rather complicated to write down the rational expression of N f (z). We consider another generating function associated with the sequence {N (f k )}: Then it is easy to see that which is a rational function with simple poles and integral residues, and 0 at infinity. The rational function S f (z) can be written as where the polynomials u(z) and v(z) are of the form with a i and b j integers, see (3) ⇒ (5), Theorem 2.1 in [2] or [19,Lemma 3.1.31]. Letṽ(z) be the conjugate polynomial of v(z), i.e.,ṽ(z) = z t v(1/z). Then, the numbers {λ i } are the roots ofṽ(z), and r(f ) = t.
The following can be found in the proof of (3) ⇒ (5), Theorem 2.1 in [2]. Lemma 2.4. If λ i and λ j are roots of the rational polynomialṽ(z) which are algebraically conjugate (i.e., λ i and λ j are roots of the same irreducible polynomial), then ρ i = ρ j .
Proof. Let Σ = Q(λ 1 , · · · , λ r ) ⊂ C be the field of the rational polynomial v(z) and let σ be an automorphism of Σ over Q, i.e., σ is the identity on Q. The group of all such automorphisms is called the Galois group of Σ. Since the σ(λ i ) are again the roots ofṽ(z), we have σ( Since the N (f k ) are integers, σ(N (f k )) = N (f k ) and consequently As a matrix form, we can write ⎡ (2) . . .
Since the λ i are distinct, the matrices in this equation are nonsingular (the Vandermonde determinant). Thus, ρ i = ρ σ −1 (i) for all i = 1, . . . , r. On the other hand, it is known that the Galois group acts transitively on the set of algebraically conjugate roots. Since λ i and λ j are conjugate roots ofṽ(z), we can choose σ in the Galois group so that σ(λ i ) = λ j . Hence, σ(i) = j and so ρ i = ρ j .
Letṽ(z) = s α=1ṽ α (z) be the decomposition of the monic integral polynomialṽ(z) into irreducible polynomialsṽ α (z) of degree r α . Of course, Vol. 20 (2018) The Nielsen numbers of iterations of maps Page 7 of 31 62 If {λ (α) i } are the roots ofṽ α (z), then the associated ρ's are the same ρ α . Consequently, we can rewrite (N1) as: Consider the r α × r α -integral square matrices The characteristic polynomial is det(zI − M α ) =ṽ α (z) and therefore {λ Then Remark that from Theorem 2.3 (the rationality of the zeta functions on infra-solvmanifolds of type (R)), it was possible to derive the identities (N2) above. Thus, we can reprove the Dold for all r > 0 and all prime numbers p.
We remark also that the rationality of the Nielsen zeta function N f (z) on an infra-solvmanifold of type (R) is equivalent to the existence of a selfmap g of some topological space X such that N (f k ) = L(g k ) for all k ≥ 1. In addition, due to the Gauss congruences (DN) in Theorem 2.2 we can choose X to be a compact polyhedron, see [2, Theorem 2.1 and Theorem (Dold)]. Thus, we can say that Nielsen theory on infra-solvmanifolds of type (R) is simpler than we have expected and is reduced to Lefschetz theory but on different spaces that are not necessarily closed nor aspherical manifolds.
We will show in Proposition 5.4 that if A k (f ) = 0 then N (f k ) = 0 and hence f has an essential periodic point of period k. In the following, we investigate some other necessary conditions under which N (f k ) = 0. Recall that N (f k ) = the number of essential fixed point classes of f k .
If F is a fixed point class of f k , then f k (F) = F and the length of F is the smallest number p for which f p (F) = F, written p(F). We denote by F the f -orbit of F, i.e., F = {F, f(F), . . . , f p−1 (F)} where p = p(F). If F is essential, so is every f i (F) and F is an essential periodic orbit of f with length p(F) and p(F) | k. These are variations of Corollaries 2.3, 2.4 and 2.5 of [2].
In particular, f has at least N (f i ) essential periodic points of period i and an essential periodic orbit with the length p | i, i ≤ r(f ).

Proof. Recall from (2.4) that
Assume that N (f ) = · · · = N (f r(f ) ) = 0. For simplicity, write the above identity as S f (z) = u(z)/v(z) = s(z) where v(z) is a polynomial of degree r(f ) and s(z) is the series of the right-hand side. Then, u(z) = v(z)s(z). A simple calculation shows that the higher order derivative of v(z)s(z) up to order r(f ) − 1 at 0 are all zero. Since u(z) is a polynomial of degree r(f ) − 1, it shows that u(z) = 0, a contradiction. Proof. The conditions ρ(f ) = 0 and r(f ) ≥ 1 immediately imply that r(f ) ≥ 2.
Since r(f ) = 0 by the previous corollary there As in the proof of the above corollary, we consider In particular, f has at least N (f i ) essential periodic points of period i and an essential periodic orbit with the length Then, the matrix M + has size m(f ) and M − has size M (f ). We write the eigenvalues of M + and M − , respectively, as: Hence, m(f ) = M (f ) and the λ i 's associated with ρ i > 0 and the λ j 's associated with ρ j < 0 are the same. This implies that the rational function N f (z) has the same poles and zeros of equal multiplicity and hence N f (z) ≡ 1, contradicting that r(f ) > 0.

Radius of convergence of N f (z)
From the Cauchy-Hadamard formula, we can see that the radii R of convergence of the infinite series N f (z) and S f (z) are the same and given by We will understand the radius R of convergence from the identity N (f k ) = are the poles or the zeros of the rational function N f (z).
If r(f ) = 0, i.e., if N (f k ) = 0 for all k > 0, then N f (z) ≡ 1 and 1/R = 0. In this case, we define customarily λ(f ) = 0. When r(f ) = 0, we define We shall assume now that r(f ) = 0 or λ(f ) > 0. First, we can observe easily the following: It follows from the above observations that 1/R = lim sup( i =j ρ i λ k i ) 1/k . Consequently, we may assume that N (f k ) = j ρ j λ k j with all |λ j | = λ(f ) and then we have and so the sequence of integers are eventually zero, i.e., N (f k ) = 0 for all k sufficiently large. This shows that 1/R = 0 and, furthermore, N f (z) is the exponential of a polynomial. Hence, the rational function N f (z) has no poles and zeros. If

Let f be a map on an infra-solvmanifold of type (R). Let R denote the radius of convergence of the Nielsen zeta function
In particular, In this paper, the number λ(f ) will play a similar role as the "essential spectral radius" in [19] or the "reduced spectral radius" in [2]. Theorem 3.2 below shows that 1/λ(f ) is the "radius" of the Nielsen zeta function N f (z). Note also that λ(f ) is a homotopy invariant.
By Theorem 3.2, we see that the sequence N (f k ) is either bounded or exponentially unbounded.

Remark 3.4. Recall from (2.3) and (2.4) that
These show that all the 1/λ i are the poles of S f (z), whereas the 1/λ i with corresponding ρ i > 0 are the poles of N f (z). The radius of convergence of a power series centered at a point a is equal to the distance from a to the nearest point where the power series cannot be defined in a way that makes it holomorphic. Hence, the radius of convergence of S f (z) is 1/λ(f ) and the radius of convergence of N f (z) is 1/ max{|λ i | | ρ i > 0}. In particular, we have shown that Notice this identity in Example 3.7.
On the other hand, we can understand the radius R of convergence using the averaging formula. Compare our result with [12, Theorem 7.10]. Let {μ 1 , · · · , μ m } be the eigenvalues of D * , counted with multiplicities, where m is the dimension of the manifold M . We denote by sp(A) the spectral radius of the matrix A which is the largest modulus of an eigenvalue of A. From the definition, we have when sp(D * ) ≤ 1.
Now we ascertain that if D * has no eigenvalue 1, then and hence from the averaging formula This finishes the proof of our assertion. Following from (R1) and (R2), we immediately have We recall that the asymptotic Nielsen number of f is defined to be We also recall that the most widely used measure for the complexity of a dynamical system is the topological entropy h(f ). A basic relation between these two numbers is h(f ) ≥ log N ∞ (f ), which was found by Ivanov in [17].
There is a conjectural inequality h(f ) ≥ log(sp(f )) raised by Shub [25,26,38]. This conjecture was proven for all maps on infra-solvmanifolds of type (R), see [12,33,34]. Now, we are able to state about relations between N ∞ (f ), Vol. 20 (2018) The Nielsen numbers of iterations of maps Proof. From [12,Theorem 4.3] and Theorem 3.5, we have that The following example shows that the assumption in Theorem 3.5 and its Corollary that 1 is not in the spectrum of D * is essential. and its proof that r is odd or q = 0, and r 0 0 q when r is odd; * * , r 0 2 0 when r is even and q = 0, when r is odd and qr > 0 (−1) k q k (r k − 1) when r is odd and qr < 0 when q = r = 0 r k − 1 when r > 0 and q = 0 (−1) k (r k − 1) when r < 0 and q = 0.
A simple calculation shows that These observations show that when one of the eigenvalues is 1, the invariants N f (z), sp( D * ) and λ(f ) still strongly depend on the other eigenvalue. Remark also in this example that the identity λ(f ) = max{|λ i | | ρ i > 0} holds.

Asymptotic behavior of the sequence {N (f k )}
In this section, we study the asymptotic behavior of the Nielsen numbers of iterates of maps on infra-solvmanifolds of type (R). We can write N (f k ) = (2) The sequence {N (f k )/λ(f ) k } has the same limit points as a periodic sequence { j α j k j } where α j ∈ Z, j ∈ C and q j = 1 for some integer q > 0. Proof. For simplicity, we denote λ(f ) by λ 0 . Recall that λ 0 = 0 if and only if all N (f k ) = 0 and otherwise, λ 0 ≥ 1. Suppose that λ 0 ≥ 1. We may assume that are all the λ i of modulus λ 0 (see Definition 3.1 for n(f )). From (Λ), we see that the sequence {N (f k )/λ k 0 } has the same asymptotic behavior as the sequence We consider the continuous function For any subset S of {1, · · · , n(f )}, we have a sub-torus The restriction of the above continuous function to the sub-torus T |S| is continuous and has its maximum m S because T |S| is compact. Now we show that either Vol. 20 (2018) The Nielsen numbers of iterations of maps Page 15 of 31 62 If dim Z {θ 1 , · · · , θ n(f ) , 1} = 1, then all θ j are rational p j /q j . Every λ j /λ 0 = e 2iπθj is a q j th root of unity, and thus all λ j /λ 0 = e 2iπθj are roots of unity of degree q = lcm(q 1 , · · · , q n(f ) ), and hence the sequence { n(f ) j=1 ρ j e 2iπ(kθj ) } is periodic of period q. This proves (2).
In Theorem 3.5, we showed that if D * has no eigenvalue 1 then λ(f ) = sp( D * ). In Example 3.7, we have seen that when D * has an eigenvalue 1, there are maps f for which λ(f ) = sp( D * ) with λ(f ) = 0, and λ(f ) = sp( D * ) with λ(f ) ≥ 1. In fact, we prove in the following that the latter case is always true.
It is important to know not only the rate of growth of the sequence {N (f k )} but also the frequency with which the largest Nielsen number is encountered. The following theorem shows that this sequence grows relatively dense. The following are variations of Theorem 2.7, Proposition 2.8 and Corollary 2.9 of [2]. If λ(f ) ≥ 1, then there exist γ > 0 and a natural number N such that for any m > N there is an ∈ {0, 1, · · · , n(f ) − 1} such that N (f m+ )/λ(f ) m+ > γ.
Proof. From the definition of Dold multiplicity I k (f ), we have Let C be any number such that 2M (f ) ≤ C. Then, for any d > 0 Thus, we have [19,Ex 3.2.17], and since λ(f ) > 1, we have lim k→∞ τ (k)/λ(f ) k/2 = 0, and so there exists an integer N such that Cτ (k)/λ(f ) k/2 < /2 for all k > N. Let m > N such that N (f m )/λ(f ) m ≥ . The above inequality induces the required inequality Theorem 4.5 and Proposition 4.7 imply immediately the following:

Essential periodic orbits
In this section, we shall give an estimate from below the number of essential periodic orbits of maps on infra-solvmanifolds of type (R). First of all, we recall the following: This theorem is not true for continuous maps. Consider the one-point compactification of the map of the complex plane f (z) = 2z 2 /||z||. This is a continuous degree two map of S 2 with only two periodic points. But L(f k ) = 2 k+1 .
However, when M is an infra-solvmanifold of type (R), the theorem is true for all continuous maps f on M . In fact, using the averaging formula ([29, Theorem 4.3], [13]), we obtain If L(f k ) is unbounded, then so is N (f k ) and hence the number of essential fixed point classes of all f k is infinite. In fact, the inequality |L(f )| ≤ N (f ) for any map f on an infra-solvmanifold was proved in [41].

Corollary 5.2. Let f be a map on an infra-solvmanifold of type (R).
Suppose {N (f k )} is unbounded. If every periodic point of f is isolated, then the set of minimal periods of f is infinite.
Proof. By assumption, each Fix(f m ) consists of isolated points, so Fix(f m ) and hence P m (f ) are finite. If, in addition, f has finitely many minimal periods then f must have finitely many periodic points. This implies that {N (f k )} is bounded, a contradiction.
Recall that any map f on an infra-solvmanifold of type (R) is homotopic to a mapf induced by an affine map (d, D). By [12, Proposition 9.3], every essential fixed point class off consists of a single element x with index sign det(I − df x ). Hence, N (f ) = N (f ) is the number of essential fixed point classes off . It is a classical fact that a homotopy between f andf induces a one-one correspondence between the fixed point classes of f and those off , which is index preserving. Consequently, we obtain This induces the following conjectural inequality (see [38,39]) for infrasolvmanifolds of type (R): We denote by O(f, k) the set of all essential periodic orbits of f with length ≤ k. Thus Proof. As mentioned earlier, we may assume that every essential fixed point class F of any f k consists of a single element F = {x}. Denote by Fix e (f k ) the set of essential fixed point (class) of f k . Thus, N (f k ) = #Fix e (f k ).
Recalling also that f acts on the set Fix e (f k ) from the proof of [12, Theorem 11.4], we have Observe further that if x is an essential periodic point of f with minimal period p, then x ∈ Fix e (f q ) if and only if p | q. The length of the orbit x of x is p, and (d,d ) ).

Then, we have
A m (f ) = then there is i with n ≤ i ≤ n + n(f ) − 1 such that A i (f ) = 0. This leads to the estimate Assume that x has minimal period p. Then, we have Thus, if m is not a multiple of p then by definition which is 0 when and only when r > 1. Consequently, In all, we obtain the required inequality We consider the set of periodic points of f with minimal period k It is clear that Fix(f ) ⊂ Fix(f 2 ), i.e., any fixed point class of f is naturally contained in a unique fixed point class of f 2 . It is also known that Fix e (f ) ⊂ Fix e (f 2 ). We define Fix e (f d ), the set of essential periodic points of f with minimal period k. Because we have Proposition 5.4. For every k > 0, we have In particular, if I k (f ) = 0 then N (f k ) = 0.
Proof. We apply the Möbius inversion formula to the above identity and then we obtain #EP k (f ) = d|k μ k d N (f d ), which is exactly I k (f ) by its definition. Theorem 5.7. Let f be a map on an infra-solvmanifold of type (R). Let k > 0 be an odd number. Suppose that α (2) Proof. By Proposition 5.4, #EP k (f ) = I k (f ). Hence, it is sufficient to show that I k (f ) is even. Let α = α (2) (f ). Consider the case where α 2 | k. If d | k and μ(k/d) = 0, then it follows that α | d. By the definition of α, N (2) Assume p is a prime such that p | k and p ≡ 2 i mod α for some i ≥ 0. Write k = p j r where (p, r) = 1. Then Since α is a N (2) -period of f , it follows that the sequence {I r (f i )} i is αperiodic in its mod 2 reduction, i.e., I r (f j+α ) ≡ I r (f i ) mod 2 for all j ≥ 1.
Since p ≡ 2 i mod α, we have I r (f p s ) ≡ I r (f 2 is ) mod 2 for all s ≥ 0. Recall [5,Proposition 5]: For any square matrix B with entries in the field F p and for any j ≥ 0, we have tr B p j = tr B. Due to this result, we obtain Vol. 20 (2018) The Nielsen numbers of iterations of maps Page 25 of 31 62 and it follows that I r (f 2 is ) ≡ I r (f ) mod 2. Consequently, we have This finishes the proof.

Homotopy minimal periods
In this section, we study (homotopy) minimal periods of maps f on infrasolvmanifolds of type (R). We want to determine HPer(f ) only from the knowledge of the sequence {N (f k )}. This approach was used in [1,14,23] for maps on tori, in [18][19][20][21]31,32] for maps on nilmanifolds and some solvmanifolds, and in [28,30] for expanding maps on infra-nilmanifolds. According to Dirichlet prime number theorem, since (1, q) = 1, there are infinitely many primes p of the form 1 + q. Consider all primes p i satisfying |I pi (h)| ≥ (γ/2)λ(h) pi . Remark that for any prime number p, where the last identity follows from that fact that Fix e (h) ⊂ Fix e (h p ). Since p is a prime, the set Fix e (h p ) − Fix e (h) consists of essential periodic points of h with minimal period p.
Because  We next consider the condition that the sequence {Ñ (f k )} contains an interval. This means by Theorem 4.1 that the set of limit points of the sequence {N (f k )/λ(f ) k } contains an interval.
The following example shows that the condition λ(f ) > 1 in Theorem 6.1 is essential. This condition is equivalent to the unboundedness of the sequence {N (f k )} by Theorem 3.2.

Example 6.2.
Consider the map f on T 2 induced by the matrix We have observed in Example 4.2 that Observe also that since f 3 = id we have Per(f ) ⊂ {1, 2, 3}. In fact, we can see that Per(f ) = {1, 3}.
In the proof of Theorem 6.1, we have shown the following, which proves that the algebraic period is a homotopy minimal period when it is a prime number.  Proof. Since λ(f ) > 1, by Theorem 4.5 there exist γ > 0 and N such that if k > N then there exists = (k) < r(f ) such that N (f k− )/λ(f ) k− > γ. Then for all sufficiently large k, the monotonicity induces Applying Proposition 4.7 with = γ/λ(f ) r(f ) , we see that I k (f ) = 0 and so A k (f ) = 0 for all k sufficiently large. Now our assertion follows from Corollary 6.3.
We next recall the following: We can not only extend but also strengthen Corollary 6.4 as follows: Proposition 6.8. Let f be a map on an infra-solvmanifold of type (R). Suppose that the sequence {N (f k )} is strictly monotone increasing. Then: (1) All primes belong to HPer(f ). We can consider as well the lower densities of Per(f ) and HPer(f ), see also [16]: Since I k (f ) = #EP k (f ) by Proposition 5.4, it follows that A(f ) ⊂ HPer(f ) ⊂ Per(f ). Hence, we have DA(f ) ≤ DH(f ) ≤ DP(f ). By Corollary 4.8, when λ(f ) > 1, we have a natural number N such that if m ≥ N then there is with 0 ≤ < n(f ) such that A m+ (f ) = 0. This shows that DA(f ) ≥ 1/n(f ).
On the other hand, by Theorem 4.1, we can obtain the following: If λ(f ) = 0 then N (f k ) = 0 and A k (f ) = 0 for all k, which shows that DA(f ) = 0. Consider first Case (2), i.e., the sequence {N (f k )/λ(f ) k } is asymptotically a periodic and nonzero sequence { n(f ) j=1 ρ j e 2iπ(kθj ) } of some period q. Now from the identity (2.2), it follows that DA(f ) ≥ 1/q. Finally consider Case (3). Then, the sequence {N (f k )/λ(f ) k } asymptotically has a subsequence { j∈S ρ j e 2iπ(kθj ) } where S = {j 1 , · · · , j s } and {θ j1 , · · · , θ js , 1} is linearly independent over the integers. Therefore by [6, Theorem 6, p. 91], the sequence (kθ j1 , · · · , kθ js ) is uniformly distributed in T |S| . From the identity (2.2), it follows that DA(f ) = 1 (see [19,  Proof. Under the same assumption, we have shown in the proof of Corollary 6.4 that there exists N such that if k > N then I k (f ) > 0. This means EP k (f ) is nonempty by Proposition 5.4 and hence k ∈ HPer(f ). Now we can prove the main result of [30]. Proof Since f is expanding, we have that λ(f ) = sp( D * ) > 1. For any k > 0, we can write as before N (f k ) = Γ(f k ) + Ω(f k ) so that Ω(f k ) → 0 and Γ(f k ) → ∞ as k → ∞. This implies that N (f k ) is eventually monotone increasing. The assertion follows from Corollary 6.10.
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