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On the density of Birkhoff sums for Anosov diffeomorphisms

Let f be an Anosov diffeomorphism on a nilmanifold. We consider Birkhoff sums for a Holder continuous observation along periodic orbits. We show that if there are two Birkhoff sums distributed at both sides of zero, then the set of Birkhoff sums of all periodic points is dense in the whole set of real numbers.


Introduction
Dynamics is aimed to describe the long term evolution of systems under the known "infinitesimal" evolution rule. The hyperbolicity exactly plays one of the most important roles in the fields of differentiable dynamical systems, for the reason that it can bring some sort of chaotic behaviors which are the deep nature for many amazing phenomenons. Especially when the dynamical system is driven by a strong hyperbolic diffeomorphism, we are supposed to expect some interesting results. In this paper, we will try to make this philosophical principle come true from a topological perspective. Precisely, in the following, we will focus on a uniformly hyperbolic diffeomorphism f : M → M , namely Anosov system (see Definition 2.1). In Throughout this paper, we denote by S φ f (z) = π(z)−1 i =0 φ( f i (z)) the Birkhoff sum of an observation φ along z, for z ∈ Per( f ) with the period π(z).
The density of empirical measures of periodic orbits is a crucial property in studying uniformly hyperbolic systems. Therefore, it is natural to ask: what are the distributions of all Birkhoff sums along periodic orbits? Under a proper assumption on the observation, we get a dense distribution of all Birkhoff sums along periodic orbits. This reveals the magically "infinitesimal" evolution rule of the hyperbolicity of the dynamical system.
More precisely, we focus on a Hölder continuous observation φ : M → R. Here is our main result.
Remark 1.1. In particular, this theorem holds for Anosov diffeomorphisms on the torus T n . In [GS19], it is shown that the density of Birkhoff sums is closely related to the transitivity of Kan's map, and for expanding maps on the circle S 1 = R 1 /Z 1 , this theorem has been proved by using direct calculations for a C 1 observation, which is different from this paper.

Anosov Diffeomorphisms
In this subsection, we collect some properties of Anosov diffeomorphisms on nilmanifolds, which will be used in our proof later.
for every unit vector v * ∈ E * , for * = s, u.
Remark 2.2. Equivalently, this definition indeed means that the entire manifold M is a hyperbolic set.
The first property we need is the shadowing property for Anosov diffeomorphisms. Recall that a sequence {x n } n∈Z of points in M is a δ-pseudo-orbit, if for any n ∈ Z, d ( f (x n ), x n+1 ) ≤ δ; if there exists m ∈ N such that for any n ∈ Z, x n+m = x n , then it is called a periodic pseudo-orbit. A point z ∈ M ε-shadows the pseudo-orbit {x n } n∈Z if for any n ∈ Z, d ( f n (z), x n ) ≤ ε. Anosov diffeomorphisms were systematically discussed since 1960s. For instance, see [Ano62], [Sma67], [Fra70] and [Man73] in some detail. One of the most typical examples is the Anosov toral automorphism that is also called Thom toral automorphism. Precisely, any hyperbolic linear map A : R n → R n with A ∈ GL(n, Z) will induce a quotient map on the n-torus T n = R n /Z n , and the induced map is called an Anosov toral automorphism. More general Anosov diffeomorphisms are hyperbolic nilmanifold automorphisms.
Let G be a simply connected nilpotent Lie group and Γ a uniform lattice of G, i.e., Γ is a discrete subgroup of G and G/Γ is compact, then M G/Γ is called a nilmanifold. Assume that φ : G → G is a continuous automorphism with φ(Γ) = Γ, φ will naturally induce a diffeomorphism f : M → M .
Denote by g = T e M the Lie algebra of G and exp : g → G the exponential map, where T e M is the tangent space of G at the identity element e ∈ G. Let Φ = D e φ : g → g be the Lie algebraic automorphism with x ∈ W s (p) W u (q) and y ∈ W s (q) W u (p).

By [Rag72, Theorem 2.11]
, the automorphism f * : π 1 (M ) = Γ can be uniquely extended as an automorphism (still denoted by) f * : G . Theorem 2.2 claims that its induced diffeomorphism (still denoted by) f * : G/Γ is a hyperbolic nilmanifold automorphism, and the original Anosov diffeomorphism f is topologically conjugate to f * .
3. Smale in [Sma67] poses the problem of classifying all the Anosov diffeomorphisms on compact manifolds up to the topological conjugacy. The positive answer by Manning that extended the result of Franks in [Fra69] is actually based on nilmanifolds. In fact, it should be noticed that the analogical conclusion may not be true for Anosov diffeomorphisms on infra-nilmanifolds, which is a more general algebraic class of manifolds. And there is some discussion and progress made by Dekimpe and Hammerlindl (see [Dek12] and [Ham14] for commentary).

The asymptotically rational independence
The final preparation is related to an algebraic notion. Recall that we say two real numbers a and b are rationally independent if k · a + l · b = 0 for any k, l ∈ Z with |k| + |l | > 0. It is clear that if a and b are rationally independent, then the set k · a + l · b | k, l ∈ Z is dense near 0 (hence dense in R). Here is an asymptotic version of the rational independence introduced in [GS19].
Definition 2.5. Let {a n } n∈N be a sequence of real numbers and b a real number. The sequence {a n } n∈N is called asymptotically rationally independent of b, if there exist 0 < ε n → 0 and k n , l n ∈ Z, such that 0 < k n · a n + l n · b < ε n .
The following simple properties about the asymptotic rational independence will be helpful later.
Lemma 2.6. Let {a n } n∈N be a sequence of real numbers, and b ∈ R \ {0}, such that a n /b = l n /k n with l n , k n ∈ Z and gcd(l n , k n ) = 1.
1. If the sequence {a n } n∈N is asymptotically rationally independent of b, then |k n | → ∞ as n → ∞. Moreover, 2. If any subsequence of {a n } n∈N is not asymptotically rationally independent of b, then there exists c > 0 such that for every n ∈ N, a n = s n · c and b = t · c with s n , t ∈ Z.
Proof. It is easily seen that {a n } is asymptotically rationally independent of b if and only if Since a n /b = l n /k n with gcd(l n , k n ) = 1, we have This directly leads to inf k,l ∈Z and lim n→∞ |k n | = +∞.
If any subsequence of the sequence {a n } is not asymptotically rationally independent of b, there exists an upper bound K ∈ N for the related sequence |k n | n∈N . Direct calculations show that the positive number c = |b/K!| will satisfy the need of the desired conclusion. Namely, there are s n , t ∈ Z such that a n = s n · c and b = t · c. This ends the proof.

Proof of Theorem 1.1
Proof of Theorem 1.1. For any K 0 ∈ R and ε > 0, we are aimed to show that there exists a periodic point Since f is a transitive Anosov diffeomorphism, by the item 1 of Remark 2.4, for these two given periodic points p and q, we have two points x ∈ W s (p) W u (q) and y ∈ W s (q) W u (p).
Note that φ : M → R is a Hölder continuous function, i.e., there exist two constants θ ∈ (0, 1) and C > 0 such that for any x, y ∈ M , where d is the metric induced by the Riemannian structure on the manifold M . Figure 1: Periodic pseudo-orbit Take L i ∈ N, i = 1, 2, 3, 4, such that L 1 , L 4 are multiples of π(p) and L 2 , L 3 are multiples of π(q). Consider the periodic pseudo-orbit Q as shown in Figure 1, where d s (·, ·) and d u (·, ·) are distances induced by the Riemannian metric restricted in the stable and unstable manifolds, respectively. Then the period of pseudo-orbit Q is L. Moreover, Q is a δ-pseudoorbit, where δ = 2δ 0 λ L 0 .
In the following, we will always choose L i , i = 1, 2, 3, 4, such that there exists α ∈ (0, 1) satisfying By the Lipschitz pseudo-orbit shadowing property in Proposition 2.3, we are going to obtain a proper L-periodic point z that µδ-shadows the periodic δ-pseudo-orbit Q. Without loss of generality, we denote by z the point shadowing exactly the point y ∈ W s (q) W u (p). Furthermore, in order to obtain the desired L-periodic point z satisfying S φ f (z) ∈ (K 0 − ε, K 0 + ε), we will take care of these iterations L i 's with great patience later.
In the light of Lemma 2.6, we now divide our proof into the following lemmas by studying the relationships between S φ f (p) and S φ f (q) in the viewpoint of the rational independence.
Proof of Lemma 3.1. Fix K 0 ∈ R and ε > 0, we want to show there exists a periodic point z ∈ Per( f ) such that S φ f (z) ∈ (K 0 − ε, K 0 + ε). For simplicity, we consider p and q are fixed points at first.
(3.2) Therefore, combining (3.1) and (3.2), we get the estimation as follow: Moreover, in the same style, we conclude that there existsL i large enough such that It should be noticed that this constant K is closely related to the periodic points p, q ∈ Per( f ).
Since φ(p) and φ(q) are rationally independent, so are 2φ(p) and 2φ(q). For the constant K 0 − K , there exist m, n ∈ N large enough such that (3.5) It is worth mentioning here that there are infinitely many m, n ∈ N satisfying (3.5), i.e., for any k ∈ N, there exist m k , n k ≥ k such that (3.6) Moreover, from the inequality (3.6) divided by n k , we can get m k /n k → −φ(q)/φ(p) as k → ∞. Thus, for k large enough, we take L 1 = L 4 = m k and L 2 = L 3 = n k , then we can get α ∈ (0, 1) satisfying L 0 ≥ αL i (i = 1, 2, 3, 4), and L i ≥ k (i = 1, 2, 3, 4) satisfying all these above estimations contributed to the estimation (3.4) in advance.
Hence, combining (3.4) and (3.6), we obtain the desired L-periodic point z such that Now we assume that p and q are general periodic points, and we just need deal with this situation under some slight modifications. Analogously, we can denote by K = K 1 + K 2 + K 3 + K 4 , where Since S φ f (p) and S φ f (q) are rationally independent now, so are 2S φ f (p) and 2S φ f (q). For the constant K 0 − K , for any k ∈ N, there exist m k , n k ≥ k, such that (3.8) For k large enough, let L 1 = L 4 = m k · π(p) and L 2 = L 3 = n k · π(q), then L i 's satisfy all the following estimations in advance, Combining these estimations, we have (3.9) Then there exists an L-periodic point z ∈ Per( f ) that µδ-shadows the δ-periodic pseudo-orbit Q, and the same calculations as contributed to (3.2) show that (3.10) Finally, combining (3.8), (3.9) and (3.10), we obtain the desired L-periodic point z ∈ Per( f ) such that S φ f (z) ∈ (K 0 − ε, K 0 + ε). Consequently, the set S φ f (z) | z ∈ Per( f ) is dense in R. This ends the proof of Lemma 3.1.
Remark 3.2. We actually proved a more general result. Let f : M → M be a C 1 diffeomorphism, and φ : M → R be a Hölder continuous observation. If there exist two hyperbolic periodic points p, q with the same index satisfying • p and q are homoclinic related, • S φ f (p) and S φ f (q) are rationally independent, The idea for proving this claim is the following. Take x ∈ W s (p) W u (q) and y ∈ W s (q) W u (p), then the set is a hyperbolic set. There exists η > 0, such that the maximal invariant set Λ that is contained in the η-neighborhood of Γ is also hyperbolic. Since the Lipschitz pseudo-orbit shadowing property (Proposition 2.3) still works on Λ, we can apply the same argument as proving Lemma 3.1.
From now on, we assume that S φ f (p) and S φ f (q) are rationally dependent. Given a sequence {p n } n∈N of periodic points, if the sequence S φ f (p n ) n∈N is asymptotically rationally independent of S φ f (q), then S φ f (p n ) n∈N is also asymptotically rationally independent of S φ f (p). Moreover, if there exists p n ∈ Per( f ) which is rationally independent of S φ f (q), we are done by Lemma 3.1. Therefore, we just come to deal with the following situation.
are rationally dependent, moreover, there exists a sequence {p n } n∈N of periodic points such that • the sequence S φ f (p n ) n∈N is asymptotically rationally independent of S φ f (q), • for any n ∈ N, Proof of Lemma 3.3. Fix K 0 ∈ R and ε > 0, we want to show there exists a periodic point z ∈ Per( f ) such that S φ f (z) ∈ (K 0 − ε, K 0 + ε). By the idea of proving Lemma 3.1, for the slightly different situation now, we can get the same estimation as (3.8).
From Definition 2.5, we may assume that the sequence S φ f (p n ) n∈N never contains 0. By taking subsequence if necessary, we assume S φ f (p n ) n∈N is a negative sequence with respect to the fact that S φ f (q) > 0. Otherwise, we just need to consider that the positive sequence S φ f (p n ) n∈N is asymptotically rationally independent of S φ f (p).
Let S φ f (p n ) S φ f (q) = l n k n with −l n , k n > 0, and gcd(l n , k n ) = 1. From the item 1 of Lemma 2.6, we have Therefore, for above ε > 0, there exists N > 0 such that n ≥ N implies For the choice of n (hence p n ), we take x ∈ W s (p n ) W u (q) and y ∈ W s (q) W u (p n ), and denote by K p n =K 1 +K 2 +K 3 +K 4 , wherẽ φ( f i ·π(q)+ j (y)) − φ( f j (q)) , Here K p n is closely related to the above choice of p n (and q, x, y naturally). Note that gcd(l n , k n ) = 1 with −l n , k n > 0, there exist m 0 , n 0 ∈ N large enough such that m 0 · l n + n 0 · k n = 1, if K 0 − K p n ≥ 0 −1, if K 0 − K p n < 0 , This implies 2m 0 · S φ f (p n ) + 2n 0 · S φ f (q) = 2 S φ f (q) k n (m 0 · l n + n 0 · k n ) ∈ − ε 9 , ε 9 .
Then we lift Ψ : M → S 1 = R/cZ to a real-valued function ψ : M → R, and corresponding to the equation (3.14), we get m ∈ Z such that φ = ψ • f − ψ + mc.
Above all, these three lemmas complete the whole proof of Theorem 1.1.