¹

1 Introduction↩︎

We consider a discrete dynamical system \((\mathcal{X},\sigma)\) given by a set \(\mathcal{X}\) and iteration of a self-map \(\sigma \colon \mathcal{X} \rightarrow \mathcal{X}\). Just like the positive integers under multiplication are the free multiplicative abelian monoid generated by the primes, prime (i.e., finite closed) orbits of a point under iteration of \(\sigma\) span a free additive monoid, whose elements, that we call general orbits, correspond bijectively to orbits of finite multisets in \(\mathcal{X}\); the cardinality of a prime orbit, its length, extends additively to general orbits. Thus, it makes sense to study analogues for such dynamical systems of statistical properties of prime decomposition, such as the expected number of prime factors of an integer (Hardy–Ramanujan [1], 1917), convergence to the normal distribution (Erdős–Kac [2], 1940), speed of convergence (Rényi–Turán [3], 1958) and Large Deviation Principles (Mehrdad–Zhu [4], 2016).

Some such generalisations exist in the setting of normed abelian monoids with axiomatic assumptions on the corresponding prime counting function akin the prime number theorem (a research area known as ‘abstract analytic number theory’ [5] or the theory of ‘Beurling primes’ [6]), see, e.g., [7] for Erdős–Kac and [8] for Large Deviations. This theory is not applicable to most of the discrete dynamical systems we will consider.

Results like the Central Limit Theorem (of which the Erdős–Kac theorem is a number-theoretical analogue) and Large Deviation Principles (LPD’s for short, see, e.g.[9]) take place on different scales, and there is no reason to expect that their generalisations to other contexts should depend on the same assumptions or should be treated in historically chronological order. In this paper, we focus solely on deducing LPD’s for orbit decomposition in discrete dynamical systems, depending on minimalistic conditions: only the convergence properties of the (Artin–Mazur) dynamical zeta function and the existence of a Cesàro mean for the rescaled fixed point count need to be assumed. From this, an asymptotic count of general orbits, as well as a dynamical analogue of Mertens’s second theorem from prime number theory can be deduced (other examples of such analogues can be found in [10], [11]). These results are then sufficient to apply the Gärtner–Ellis theorem and get a LDP. This typically answers questions of the form ‘what is the probability that a general orbit has a given fraction more than the expected number of components in its decomposition, depending on its length?’. The analogue of the Erdős–Kac theorem seems to require stronger hypotheses, related to the error terms in the asymptotics, and we hope to return to this in a subsequent paper. Finally, note that for dynamical systems, one may also consider the statistics of orbit length itself instead of orbit decomposition, see, e.g., [12], [13]; this is different from what we do here (our method only handles strongly additive functions \(g\), for which \(g(P^k)=g(P)\) for prime orbits \(P\)).

Notation↩︎

To describe our results in a precise way, we first fix some notation. Let \(\sigma_k\) denote the number of fixed points of the \(k\)-th iterate \(\sigma^k \colon \mathcal{X} \rightarrow \mathcal{X}\) of a self-map \(\sigma \colon \mathcal{X} \rightarrow \mathcal{X}\) on a set \(\mathcal{X}\). Assume that \(\sigma\) is confined, i.e., \(\sigma_k\) is finite for all \(k\). The growth rate of \((\mathcal{X},\sigma)\) is \[\Lambda \mathrel{\rlap{ \raisebox{0.24ex}{\m@th\cdot}} \raisebox{-0.24ex}{\m@th\cdot}} =\exp \limsup \log(\sigma_k)/k.\] A prime (or primitive, or finite) orbit for \(\sigma\) of length \(\ell\) is a set \(O = O(x) = \{x, \sigma(x),\sigma^2(x),\dots,\sigma^{\ell}(x)\}\) of exact cardinality \(\ell=\ell(O)\) for some \(x \in \mathcal{X}\). A general orbit is an element of the free additive abelian monoid generated by the prime orbits, with the length function extended additively. Equivalently, a general orbit is a finite orbit of a finite multiset in \(\mathcal{X}\): if \(S\) is a finite multiset in \(\mathcal{X}\) consisting of elements \(x_i\) with multiplicity \(m_i\) then the corresponding general orbit is the sum \(\sum m_i O(x_i)\) in the monoid generated by prime orbits.

Let \(\mathcal{P}\) denote the set of prime orbits, and \(\mathcal{N}\) the set of general orbits, and for \(X \geqslant 1\), let \(\mathcal{P}(X), \mathcal{N}(X)\) denote the subset of all elements of length less than or equal to \(X\) in the respectively sets. Let \(N_n\) denote the total number of general orbits of length \(n\), and \(P_\ell\) the total number of prime orbits of length \(\ell\). Define the corresponding cumulative counting functions as \[N_\sigma(X) \mathrel{\rlap{ \raisebox{0.24ex}{\m@th\cdot}} \raisebox{-0.24ex}{\m@th\cdot}} =\# \mathcal{N}(X), \;P_\sigma(X) \mathrel{\rlap{ \raisebox{0.24ex}{\m@th\cdot}} \raisebox{-0.24ex}{\m@th\cdot}} =\# \mathcal{P}(X)and M_\sigma(X) \mathrel{\rlap{ \raisebox{0.24ex}{\m@th\cdot}} \raisebox{-0.24ex}{\m@th\cdot}} =\sum_{P \in \mathcal{P}(X)} \Lambda^{-\ell(P)} = \sum_{\ell \leqslant X} P_\ell/\Lambda^\ell.\] This last function \(M_\sigma(X)\) is the analogue of Mertens’s counting function \(\sum_{p \leqslant X} 1/p\) (\(p\) prime), where we count every prime orbit \(O\) with ‘norm’ \(\Lambda^{\ell(O)}\).

Asymptotic counting↩︎

The crucial quantity is a certain Cesàro mean. Recall that the Cesàro mean \(\mathcal{C}(a_k)\) of a sequence \(\{a_k\}_{k \geqslant 1}\) of real numbers is defined as \[\mathcal{C}(a_k) \mathrel{\rlap{ \raisebox{0.24ex}{\m@th\cdot}} \raisebox{-0.24ex}{\m@th\cdot}} =\lim_{X \rightarrow + \infty} \frac{1}{X} \sum_{k \leqslant X} a_k.\]

To write the leading constant in ?? as \(C/\Gamma(B)\) is natural from the proof and useful in the formulation of further results below. The proof of Theorem 1 hinges on the relations between \(\sigma_k, N_n\) and \(P_\ell\) expressed via the Artin–Mazur dynamical zeta function \(\zeta_\sigma(z) = \exp \sum \sigma_k z^k/k\) of \(\sigma\), which is at the same time the exponential generating function for \(\sigma_k\), the Taylor series for \(N_n\) and the Euler product for \(P_\ell\). The generating series has radius of convergence \(1/\Lambda\) and in general has a natural boundary along \(|z|=1/\Lambda\), but we can apply a logarithmic version of the Hardy–Littlewood real Tauberian theorem, taking the —weak, but generally optimal— error term from work of Freud [14]. The rest of the proof involves standard applications of partial summation. The total count in ?? can have \(B-1\) positive or negative (as seen in the Examples listed below), meaning that there may be polynomial excess or deficiency in the total count compared to dominant growth \(\Lambda^X\). For the quite different case where \(\Lambda=1\), see Propositions 1 and 12. The case \(\Lambda<1\) is not interesting, see Remark 2.

An LDP↩︎

As a consequence of Theorem 1, we prove a large deviation principle for the number of prime orbits occurring in the decomposition of a general orbit.

The expected value of \(W(X)\) equals \(B \log X+o(\log X)\), which is a very weak analogue of the Hardy–Ramanujan theorem. About the higher moments, we can say nothing. Note that the speed is dependent on the Cesàro mean but the rate function \(I\) in ?? is universal; it is the rate function of the standard Poisson distribution with unit mean; we do not know whether there is a deeper reason for why exactly the Poisson distribution shows up here. We prove a more technical Large Deviation Theorem for suitable strongly additive functions \(g\) on the monoid of general orbits, see Theorem 4. In this case, we need an assumption of convergence of the Laplace transform of counting measures related to \(g\), and the resulting rate function depends on \(g\). The structure of the proof is very close to that of a similar result in [8], which in turn follows the template from [4], and is ultimately based on using the Gärtner–Ellis theorem. A crucial difference between existing proofs and our arguments is found in the proof of Lemma 7. As an application, we establish large deviation principles for a family of strongly additive functions whose value is zero over a positive density subset of \(\mathcal{P}\), see Example 4. We also construct functions for which the assumption of convergence of their related Laplace transform does not hold (see Example 1), as well as functions whose associated rate function does not give a meaningful LDP (see Examples 4 and 5).

Applicability: algebraic group endomorphisms and beyond↩︎

We consider the applicability of Theorem 1 and 2 to concrete dynamical systems.

To show the existence of the Cesàro mean for such systems, we use a Fourier analytic method. The somewhat exotic looking concept of FAD-system in fact embraces many natural examples: endomorphisms \(\sigma\) of algebraic groups \(G\) (abelian varieties, linear algebraic groups, …) over \(\overline{\mathbf{F}}_p\), with \(\mathcal{X}=G(\overline{\mathbf{F}}_p)\) [15]; additive cellular automata \(\sigma\) with \(\mathcal{X}=(\mathbf{F}_p^r)^{\mathbf{Z}}\) (these turn out to be equivalent to vector group endomorphisms, see [16]); as well as solenoids given as \(S\)-integer dynamical systems for global fields in the sense of Chothi–Everest–Ward [17] (see also [18]). Below are some concrete examples, each illustrating a different aspect.

For an algebraic group, the nature of the Cesàro limit \(B\) depends on the underlying geometry. We will in fact compute \(B\) for all integer multiplications on ordinary elliptic curves in characteristic \(p\geqslant 3\) (Proposition 13), and for all additive cellular automata (Formula 19 ). We also prove a general result, using the following concept. In [15] it is shown that for any algebraic group \(G\) over \(\overline{\mathbf{F}}_p\), there exists a descending series of fully characteristic subgroups (meaning subgroups \(N\) for which \(\sigma(N) \subseteq N\) for all \(\sigma \in \mathop{\mathrm{\mathrm{End}}}(G)\)) in which all factor groups are either finite, a torus, a semisimple group, an abelian variety or a vector group. Call any such series an endoseries for \(G\).

The algebraic and rational case are shown by direct calculation, and the complementary transcendence statement is shown using the method of Mahler functions. The theorem does not deal with groups whose endoseries has several occurrences of vector groups with non-rational zeta functions.

2 Asymptotic results for counting functions↩︎

The goal of this section is to prove Theorem 1, namely, that existence and positivity of the Cesàro mean imply ?? and ?? .

We fix \((\mathcal{X},\sigma)\) and leave out the subscript \(\sigma\) from the quantities involved. We also set \(N_0=1\). Consider the Artin–Mazur dynamical zeta function of the system \((\mathcal{X},\sigma)\) given by \[\label{AM} \zeta(z) \mathrel{\rlap{ \raisebox{0.24ex}{\m@th\cdot}} \raisebox{-0.24ex}{\m@th\cdot}} =\exp ( \sum_{k \geqslant 1} \sigma_k z^k/k) = \sum_{n \geqslant 0} N_n z^n = \prod_{\ell \geqslant 1} (1-z^\ell)^{-P_\ell},\tag{1}\] see [20] or [21]. By the Cauchy–Hadamard criterion and the definition of \(\Lambda\), the function \(\zeta(z)\) is holomorphic in the disk \(|z|<1/\Lambda\), but in general may have a natural boundary on \(|z|=1/\Lambda\), i.e, cannot analytically be extended to any open set intersecting the boundary; see, e.g., [15] for examples. The last equality in 1 is a formal consequence of the factorisation of general orbits into prime orbits. The equality between first and last term follows since every one of the \(\ell\) points in a prime orbit of length \(\ell\) is a fixed point of \(\sigma^k\) precisely if \(\ell\) divides \(k\), i.e., we have the first equation below (and the second one follows by Möbius inversion of the first): \[\label{swapsp} \sigma_k = \sum_{\ell \mid k} \ell P_\ellandP_\ell = \frac{1}{\ell} \sum_{n \mid \ell} \mu(l/n) \sigma_n.\tag{2}\]

We add the corresponding quite different result for \(\Lambda=1\) (i.e., the case of subexponential growth rate of the number of fixed points).

For the zeta function to have a natural boundary along \(|z|=1\) means that it cannot be analytically continued to any open set intersecting the unit circle (see, e.g.[24]).

3 Large deviation principles↩︎

In this section, we show how Theorem 1 implies Theorem 2. More generally, we will show large deviation principles for a wider class of strongly additive functions. We consider the following condition analogous to [8]. To simplify notations in this section, we fix \(\sigma\) and leave it out of the notation.

We prove the theorem by adapting the arguments presented in [8]; only Lemma 7 requires an entirely new argument. For each prime orbit \(P\), we define the independent random variable \(Y_P\) for which \[Y_P = \begin{cases} 1 &\text{ with probability } {\Lambda^{-\ell(P)}}, \\ 0 &\text{ otherwise }. \end{cases}\] For each \(X \in \mathbf{Z}_{>0}\), we denote by \(V(X)\) a uniformly chosen element from the set \(\mathcal{N}(X).\) For each prime orbit \(P\), we also define a random variable \(Z_P\) such that \[Z_P = \begin{cases} 1 &\text{ if } P \mid V(X), \\ 0 &\text{ otherwise }. \end{cases}\]

We state the exponential Chebyshev/Markov Inequality, which we will use in upcoming lemmas.

We first show that it suffices to consider the problem over subset of orbits \(O\) such that all its prime components \(P\) satisfy \(|g(P)| \leqslant C\) for some sufficiently large \(C\).

Given \(C>0\), we define the following two sets: \[\begin{align} & \mathcal{A}(X,C) \mathrel{\rlap{ \raisebox{0.24ex}{\m@th\cdot}} \raisebox{-0.24ex}{\m@th\cdot}} =\left\{ P \in \mathcal{P} \colon k_X \leqslant\ell(P) \leqslant X \text{ or } \ell(P) < \tilde{k}_X, \text{ and } |g(P)| \leqslant C \right\}, \\ & \mathcal{B}(X,C) \mathrel{\rlap{ \raisebox{0.24ex}{\m@th\cdot}} \raisebox{-0.24ex}{\m@th\cdot}} =\left\{ P \in \mathcal{P} \colon \tilde{k}_X \leqslant\ell(P) \leqslant k_X, \text{ and } |g(P)| \leqslant C \right\}. \end{align}\] We first show that, for proving large deviation principles, contributions from prime orbits lying in the set \(\mathcal{A}(X,C)\) can be ignored.

Next, we show that the contributions from prime orbits lying in the sets \(\mathcal{B}(X,C)\) can be understood by using the random variables \(Y_P\). At this point, our arguments are completely different from the ones used in [8], that are not applicable to our setup, because the error term in ?? is too weak.

We state the Gärtner-Ellis theorem, which we will use the prove Theorem 4.

4 The Cesàro mean for FAD-systems↩︎

We recall the general context of FAD-systems from [15], and prove Theorem 3, i.e., that such systems satisfy the conditions of Theorem 1.

This somewhat exotic looking concept in fact embraces the following natural examples:

1. Confined endomorphisms \(\sigma\) of algebraic groups \(G\) (abelian varieties, linear algebraic groups, …) over \(\overline{\mathbf{F}}_p\), then \(\mathcal{X} = G(\overline{\mathbf{F}}_p)\) and \(S=\{p\}\); see [15].

2. Additive cellular automata \(\sigma\) with \(\mathcal{X} = V^{\mathbf{Z}}\) and \(V=\mathbf{F}_p^r\) (i.e., continuous—for the product topology of the discrete topology on \(V\)—such maps that are additive and commute with the operation of shifting the bi-infinite sequence one to the right); again with \(S=\{p\}\), \(c=p^a\) for some integer \(a \geqslant 0\), the determinant factor is absent, \(r_n=1\), \(s_n=0\); see [15]. In fact, this collection of examples corresponds in a precise sense to endomorphisms defined over \(\mathbf{F}_p\) of vector groups \(\mathbf{G}_a^r\), see [16].

3. Solenoids given as \(S\)-integer dynamical systems for global fields in the sense of Chothi–Everest–Ward [17] with finite \(S\), where \(S\) is indeed \(S\) for a number field, and \(t_{p,n}=0\); and \(S=\{p\}\) for a function field of characteristic \(p>0\); see [15].

5 Examples of explicit asymptotic results↩︎

In this section, we find the exact values of \(B\) and \(C\) in several cases of interest, more general than the examples presented in the introduction. We focus on the Cesàro limit \(B\). For some cases, we also find an explicit formula for \(C\), which requires knowing the entire asymptotics of \(\zeta(u/\Lambda)\) near \(u=1\) and using 4 and 6 . After that, we discuss the nature of the constants \(B\) and \(C\) (rationality, algebraicity, transcendence) and the relation of their nature to the geometry of the system, in case it arises from an algebraic group. In Example (FF), it is clear that \(B=C=1\), so we now deal with the other examples. If \(S\) is a set of integers, we will use the suggestive notation \(\mathbb{P}(n \colon n \in S)\) for the natural density of \(S\) (if it exists).

Multiplication on elliptic curves↩︎

In this subsection, we generalise the Example (E) to the following situation: suppose \(E\) is an ordinary elliptic curve over \(\overline{\mathbf{F}}_p\) with \(p\) an odd prime, and suppose that \(\sigma(P)=nP\) is multiplication by a positive integer \(n\) on \(E\). The number of fixed points of \(\sigma^k\) is the number of elements of the torsion subgroup \(E[n^k-1](\overline{\mathbf{F}}_p)\), and, decomposing into primary subgroups using that \(E\) is ordinary, so \(\# E[p^r](\overline{\mathbf{F}}_p) = p^r\), we find that \(\sigma_k = (n^k-1)^2 \cdot |n^k-1|_p\). Thus, \(\Lambda=n^2\) and \(B=\mathcal{C} (|n^k-1|_p)\).

If \(p \mid n\), \(|n^k-1|_p=1\) for all \(k\), and \(B=1\). We also find a rational zeta function \[\zeta(u/n^2) = \exp \sum_{k \geqslant 0} \frac{1}{k} \left( u^k - 2 (u/n)^k + (u/n^2)^k\right) = \frac{(1-u/n)^2}{(1-u)(1-u/n^2)} \underset{u \rightarrow 1^-}{\sim} (1-u)^{-1} \frac{n-1}{n+1},\] from which we infer that \(C=(n/(n+1))^2\).

In the more interesting case where \(n\) is coprime to \(p\), let \(d\) denote the order of \(n\) in \((\mathbf{Z}/p\mathbf{Z})^*\) and \(e \mathrel{\rlap{ \raisebox{0.24ex}{\m@th\cdot}} \raisebox{-0.24ex}{\m@th\cdot}} =v_p(n^d-1)\). The ‘lifting the exponent lemma’ says that for all \(k\) not divisible by \(d\), \(v_p(n^k-1)=0\), whereas if \(d \mid k\), we have \(v_p(n^k-1) = e + v_p(k/d)\). This implies \[B =\mathbb{P}(d \nmid k) + \lim_{X \rightarrow + \infty} \frac{1}{X} \sum_{m \leqslant X/d} \frac{1}{p^{e+v_p(m)}} = 1 - \frac{1}{d} + \frac{1}{dp^e} \mathcal{C}(|m|_p).\] Now splitting summands by their exact \(p\)-valuations, we get \[\mathcal{C} (|m|_p) = \sum_{j \geqslant 0} \frac{1}{p^j} \mathbb{P}(m \colon p^j \mid m \wedge p^{j+1} \nmid m) = \sum_{j \geqslant 0} p^{-j} \left( \frac{1}{p^j} - \frac{1}{p^{j+1}} \right) = \frac{1-p^{-1}}{1-p^{-2}} = \frac{p}{p+1}.\]

To compute \(C\), we use the exact asymptotics of \(\zeta(u/n^2)\) near \(u=1\). From the definition, we get \(\log \zeta(u/n^2) = F(u)-2F(u/n)+F(u/n^2)\) where \[\begin{align} F(u) &\mathrel{\rlap{ \raisebox{0.24ex}{\m@th\cdot}} \raisebox{-0.24ex}{\m@th\cdot}} =\sum_{k \geqslant 1} |n^k-1|_p \frac{u^k}{k} =\sum_{\substack{k \geqslant 1 \\d \nmid k}} \frac{u^k}{k} + \frac{1}{dp^e} \sum_{m \geqslant 0} \frac{1}{p^{v_p(m)}} \frac{(u^d)^m}{m} \\ & = \log \frac{\sqrt[d]{1-u^d}}{1-u} + \frac{1}{dp^e} \sum_{j \geqslant 0} \frac{1}{p^{2j}} \sum_{\substack{m \geqslant 1 \\ p \nmid m}} \frac{(u^{dp^j})^m}{m} = \log \Big( \frac{\sqrt[d]{1-u^d}}{1-u} \Big( \prod_{j \geqslant 0} \frac{(1-u^{dp^{j+1}})^{1/p^{2j+1}}}{ (1-u^{dp^{j}})^{1/p^{2j}}} \Big)^{\frac{1}{dp^e}} \Big) \end{align}\] Combining factors in the product, this simplifies further to give \[\label{expF} \exp F(u) = (1-u)^{-1} (1-u^d)^{\frac{1}{d}(1-\frac{1}{p^e})} \prod_{j \geqslant 1} (1-u^{dp^j})^{\frac{1}{p^{2j-1}} \cdot \frac{1}{dp^e}(1-\frac{1}{p})}.\tag{18}\] To find the asymptotics near \(u=1\), we divide out a factor \(u-1\) for each factor in the product, notice that \((1-u^m)/(1-u) \sim m\) to get (with the sanity check that the exponent of \(1-u\) is indeed \(-B\)), \[\exp F(u) \sim (1-u)^{-B} \cdot d^{\frac{1}{d} ( 1-\frac{1}{p^{e-1}(p+1)} ) } \cdot p^{\frac{1}{dp^{e-2}(p-1)(p+1)^2}}.\] On the other hand, \(\exp F(1/n)\) and \(\exp F(1/n^2)\) can be computed directly from 18 , and if we collect all terms, we find the following.

Example (E) from the introduction is the case \(p=3\), \(n=2\), \(d=2\), \(e=1\).

Additive cellular automata and vector group endomorphisms↩︎

For a confined endomorphism \(\sigma\) of \(\mathbf{G}_a^r\), we have \(\sigma_k = p^{kc-t_k|k|_p^{-1}}\) for non-negative integers \(c\) and \(t_k\), where \(t_k\) is periodic of period \(\varpi\) coprime to \(p\), see [16]. Here, \(\Lambda=p^c\), and, writing \(\varpi\) for the period of \(t_k\), and using that this period is coprime to \(p\), we find \[\begin{align} \label{BforCA} B & = \lim_{X \rightarrow \infty} \frac{1}{X} \sum_{a=0}^{\varpi-1} \sum_{j \geqslant 0} \sum_{\substack{k \leqslant X \\ v_p(k)=j \\ k \equiv a \text{ mod } \varpi}} \frac{1}{p^{t_a p^j}} = \lim_{X \rightarrow \infty} \sum_{a=0}^{\varpi-1} \sum_{j \geqslant 0} \underbrace{\mathbb{P}(k \colon v_p(k)=j \wedge k \equiv a \text{ mod } \varpi)}_{\frac{1}{\varphi(\varpi)} p^{-j} \left( 1 - \frac{1}{p} \right)} \frac{1}{p^{t_a p^j}} \nonumber \\ & = \frac{p-1}{\varphi(\varpi)} \sum_{a=0}^{\varpi-1} \mathcal{C} (p^{-1-j-t_a p^j}). \end{align}\tag{19}\] This number is similar to the so-called ‘Kempner number’ \(\mathcal{C}(1/2^{2^j})\) (see, e.g., [28]) and we can prove that \(B\) is transcendental if and only if there is a \(t_k \neq 0\), using the method of Mahler (e.g., from Nishioka [27]), as follows. Define the function \[F(x) \mathrel{\rlap{ \raisebox{0.24ex}{\m@th\cdot}} \raisebox{-0.24ex}{\m@th\cdot}} =\sum_{a=0}^{\varpi-1} \sum_{j \geqslant 0} \frac{x^{t_a p^j}}{p^j},\] and notice that \(F\) satisfies a functional equation in Mahler form \[\frac{1}{p} F(x^p) = F(x) - \sum_{a=0}^{\varpi-1} x^{t_a}.\] We can conclude that \(B=F(1/p)\) is transcendental, by checking the following conditions required by [27]:

1. \(1/q< \min \{1,R\}\), where \(R\) is the convergence radius of \(F(x)\) as power series, which is true since \(R=1\);

2. the degree \(m\) of the Mahler equation in \(F(x)\) is \(1\), the exponent \(d\) in \(F(x^p)\) is \(p\), and the exponent of \(F(x^p)\) in the equation is \(n=1\), and we require \(\max\{d,m\} n^2 < d^2\), which holds;

3. finding suitable functions \(g_0(x,u)\) and \(g_1(x,u)\) such that \(g(x) = g_0(x,u)/p + g_1(x,u) (-u+\sum x^{t_a})\) is independent of \(u\), namely, \(g_0=u\) and \(g_1=1/p\); then \(g(x) = \frac{1}{p} \sum_{a=0}^{\varpi-1} x^{t_a}\), we require \(g(1/p^d) \neq 0\) for all \(d\), which holds since there exists a least one \(a\) with \(t_a\neq 0\);

4. the function \(F(x)\) is not algebraic over \(\mathbf{C}(x)\), which holds, since, if it were algebraic, it would be rational by [27]; but then the functional equation implies that, if it is rational of degree \(d\), \(dp=d-\max\{t_a\}\), and this happens precisely if all \(t_a\) are zero.

Finally, if all \(t_a\) are zero, summing the geometric series in 19 shows that \(B=1\).

As we have remarked before, such \(\sigma\) defined over \(\mathbf{F}_p\) correspond bijectively to the class of additive cellular automata \(\tau\) with alphabet \(\mathbf{F}_p^r\). The fixed point count for iterates of \(\tau\) is identical to that of the corresponding \(\mathbf{F}_p\)-rational endomorphism of the vector group \(\mathbf{G}_a^r\) over \(\mathbf{F}_p\) [15]. Hence the result also applies to this class of examples.

We conclude the following:

In the specific example (GA) from the introduction, we have \(p=2\), \(c=1\), \(t_n=1\) for all \(n\), and so \(B=\mathcal{C} (p^{-1-j-p^j}) = \sum_{j \geqslant 0} 2^{-1-j-2^j}\), as indicated. To find \(C\) in this case, we again revert to finding the precise asymptotics of \(\zeta(u/2)\) around \(u=1\). Substituting the value of \(\sigma_k\) and writing \(k=2^j m\) with \(m\) odd, we find \[\log \zeta(u/2) = \sum_{k \geqslant 1} \frac{1}{2^{2^{v_2(k)}}} \frac{u^k}{k} = \sum_{j \geqslant 0} \frac{1}{2^{2^j+j}} \sum_{m \text{ odd } \geqslant 1} \frac{(u^{2^j})^m}{m}.\] The inner sum is \(1/2 \log((1+u^{2^j})/(1-u^{2^j}))\), and hence for \(u \rightarrow 1^-\), we find \[\zeta(u/2) = \prod_{j \geqslant 0} \Big( \frac{1+u^{2^j}}{1-u^{2^j}} \Big)^{1/2^{2^j+j+1}} \sim (1-u)^{-B} \prod_{j \geqslant 0} (2/2^j)^{1/2^{2^j+j+1}} \sim (1-u)^{-B} 2^{B-A}\] with \(A = \sum_{j \geqslant 0} j/2^{2^j+j+1}\). As in 6 , with \(\Lambda=2\), we thus find \(C=2^{1+B-A}.\)

A torus endomorphism without unique dominant root↩︎

As final example, we consider the torus endomorphism given in Example (GM) from the introduction. This is an example where in Formula 15 , \(m \neq 0\), so that there is a sinusoidal fluctuation in the fix point count sequence \(\sigma_k\); this is the case where the linear recurrence for the factor \(\det(A^n-1)\) in the expression for \(\sigma_k\) does not have a unique so-called dominant root, i.e., \(A\) has several eigenvalues of maximal absolute value (\(=\Lambda/c\)).

In the Example (GM) at hand, \[\sigma_k = 4\Lambda^k \sin^2(k\theta/2) r_k |k|_5^{s_k}with(r_k,s_k) = \left\{ \begin{array}{ll} (5^{-2},4) &if3 \mid n; \\ (1,0) &otherwise , \end{array} \right.\] where \(\Lambda = \frac{1}{4} \left(3+\sqrt{5} + \sqrt{6 \sqrt{5}-2}\right) \approx 2.15372\) and \(8 \cos \theta = 3-\sqrt{5}\) (so \(\theta \approx 1.37863\dots\)), see [15]. Now \(\theta\) is not a rational multiple of \(\pi\): if this were the case, \(2 \cos \theta\) would be an algebraic integer, but it has minimal polynomial \(4x^2-6x+1\). Hence by Remark 11, \(B= 2 \mathcal{C}(r_k |k|_5^{s_k}).\) This can be calculated by methods similar to the ones above: \[\begin{align} B &= \frac{2}{X} \Big( \sum_{\substack{k \leqslant X \\ 3 \nmid k}} 1 + \sum_{\substack{k \leqslant X \\ 3 \mid k}} \frac{1}{5^2} |k|_5^4 \Big) = \frac{4}{3} + 2 \sum_{j \geqslant 0} \underbrace{\mathbb{P}(3 \mid k \wedge v_5(k)=j)}_{1/3 \cdot (1/5^j-1/5^{j+1})} \frac{1}{5^2} \frac{1}{5^{4j}} \\ & = \frac{4}{3} + \frac{8}{3\cdot 5^3} \frac{1}{1-1/5^5} = \frac{2 \cdot 23^2}{11 \cdot 71}. \end{align}\]

We refrain from computing the constant \(C\) in the asymptotic formula ?? for this case, but it can be computed with some effort from the explicit representation if \(\sigma_k\) is given as above.

The Cesàro mean and the geometry of the algebraic group↩︎

In this subsection, we prove Theorem 4 which relates the geometry of a general algebraic group \(G\), more specifically its endoseries, to the possible rationality/algebraicity/transcendence of the corresponding Cesàro mean \(B\).

References↩︎