Dynamically affine maps in positive characteristic

Dynamically affine maps in positive characteristic

Jakub Byszewski (JB) Wydział Matematyki i Informatyki Uniwersytetu Jagiellońskiego, ul. S. Łojasiewicza 6, 30-348 Kraków, Polska jakub.byszewski@uj.edu.pl Gunther Cornelissen (GC, MH, LvdM) Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, Nederland g.cornelissen@uu.nl, loisvandermeijden@gmail.com  and  Marc Houben (MH) Current address: Departement Wiskunde, KU Leuven, Celestijnenlaan 200b—box 2400, 3001 Leuven, België marc.houben@kuleuven.be
Abstract.

We study fixed points of iterates of dynamically affine maps (a generalisation of Lattès maps) over algebraically closed fields of positive characteristic . We present and study certain hypotheses that imply a dichotomy for the Artin–Mazur zeta function of the dynamical system: it is either rational or non-holonomic, depending on specific characteristics of the map. We also study the algebraicity of the so-called tame zeta function, the generating function for periodic points of order coprime to . We then verify these hypotheses for dynamically affine maps on the projective line, generalising previous work of Bridy, and, in arbitrary dimension, for maps on Kummer varieties arising from multiplication by integers on abelian varieties.

Key words and phrases:
Fixed points, Artin–Mazur zeta function, dynamically affine map, recurrence sequence, holonomic sequence, natural boundary
2010 Mathematics Subject Classification:
37P55, 37C30, 14L10, 37C25, 11B37, 14G17
JB gratefully acknowledges the support of National Science Center, Poland under grant no. 2016/23/D/ST1/01124. We thank Martin Bright, Stefano Marseglia and Sławomir Rams for some illuminating discussions.
copyright: ©:

with an appendix by the authors and Lois van der Meijden

1. Introduction

We consider so-called dynamically affine maps, a concept in algebraic dynamics introduced by Silverman [43, §6.8] in order to unify various interesting examples, such as Chebyshev and Lattès maps, cousins of which occur in complex dynamics under the name of “finite quotients of affine maps” or “rational maps with flat orbifold metric” [35]. We will only consider the case of a ground field of positive characteristic . (Most of our methods would simplify considerably in characteristic zero and lead to results of a rather different flavour.) Before we present the definition, we will illustrate by approximative pictures (constructed in Mathematica [51], using the function RandomInteger for randomisation) what distinguishes the dynamics of such maps from that of other polynomial maps and random maps.

1.1. A compilation of (restrictions of) maps

Let denote a map from a finite set to itself. It can be represented by a directed graph (sometimes called the “function digraph” of ), with vertices labelled by elements of and an arrow from a vertex to a vertex occurring precisely if . In Figure 1, we plotted the graphs corresponding to two random such maps where is a set with elements.

Now consider a rational function defined over (in this subsection we assume for convenience that ). To represent pictorially, consider the restrictions for various . In Figure 2, we plotted the graph of the polynomial function for various and , and in Figure 3, we did the same for . At first sight, the graph for a random map looks similar to the graph for , but the graph for looks much more structured. This is no coincidence; Figure 3 represents the graph of restrictions of a dynamically affine map, whereas Figure 2 does not.

Figure 1. Graph of two random maps on a set with elements

Figure 2. Graphs of on over a field with and elements (left to right)

A common feature of all function digraphs is that their connected components are cycles (consisting of periodic points) with attached trees. What is different in Figure 3 is the symmetry in the attached trees; this is well-understood for the polynomial , which relates to the Lucas–Lehmer test and failure of the Pollard rho method of factorisation, see, e.g. [50, 38]. Let us mention one further result [29, Thm. 1.5 & Example 7.2]: for the graph of a quadratic polynomial with integer coefficients, the value of

is for but for .

Figure 3. Graphs of on over a field with and elements (left to right)

Figure 4. Graphs of the Lattès map arising from doubling modulo inversion on the elliptic curve over a field with and elements (left to right)

To explain what is special about the dynamically affine map as opposed to the polynomial map , notice that , where is the normalised Chebyshev polynomial of the second kind, defined by . This reveals a hidden group structure: the map arises from the group endomorphism on the multiplicative group after quotienting on both sides by the automorphism group generated by the inversion that commutes with . That (for ) is not special in this sense follows from the classification of dynamically affine maps on [9].

We perform a similar construction using another algebraic group, the elliptic curve , and the doubling map . After taking the quotient by , we find a so-called Lattès map which we have graphed over various finite fields in Figure 4. Again, we see a very structured picture, rather different from Figure 1 and Figure 2.

We will not dwell any longer on the study of iterations of maps on finite sets, both random and “polynomial over finite fields”—a rich subject in itself—but rather switch to our main object of study: dynamically affine maps over algebraically closed fields of positive characteristic.

1.2. What is a dynamically affine map?

Let be an algebraic variety over an algebraically closed field of characteristic and a morphism. We make the following assumption throughout:

  1. The map is confined, i.e. the number of fixed points of the -th iterate of is finite for all .

Definition.

A morphism of an algebraic variety over is called dynamically affine if there exist:

  1. a connected commutative algebraic group ;

  2. an affine morphism , that is, a map of the form

    where is a confined isogeny (i.e. a surjective homomorphism with finite kernel) and ;

  3. a finite subgroup ; and

  4. a morphism that identifies with a Zariski-dense open subset of

such that the following diagram commutes:

Remark.

In this paper, we adhere to the convention that a dynamically affine map consists of all the given (fixed) data in the definition, so that we can refer to the constituents directly. The same map might arise from different sets of data, and in our sense be a different dynamically affine map despite being the same map on .

Example.

As explained above, the map is dynamically affine for the data (written multiplicatively); its restrictions (to certain finite fields) were represented in Figure 3.

The map is dynamically affine for the data , where is the elliptic curve ; its restrictions were represented in Figure 4.

Remark.

We have slightly modified Silverman’s definition [43, §6.8] of a dynamically affine map. Instead of assuming confinedness of , Silverman imposes the condition (as in Erëmenko’s classification theorem [20]). As long as is one-dimensional and , the definitions are equivalent.

In a general setup one could assume merely that is an isogeny and only require to be confined. This reduces, after some case distinctions, to the case where is a confined isogeny, so we choose to put the latter property in the definition.

1.3. Counting fixed points, orbits, and the dynamical Artin–Mazur zeta function

A natural way to begin a quantitative analysis of a discrete dynamical system such as iteration of a map is to consider the sequence given by the number of fixed points of the -th interate of . Confinedness implies that this is a well-defined sequence of integers, and we can form the (full) Artin–Mazur dynamical zeta function ([2], [45, §4]) defined as

(1)

We consider this a priori as a formal power series, but the question of convergence in a neighbourhood of (equivalent to growing at most exponentially in ) is interesting, and we study this in Appendix A.

Counting fixed points and closed orbits is related: if denotes the number of closed orbits of length , then and there is an “Euler product”

(2)

where the product runs over the closed orbits .

It is interesting to understand the nature of the function (Smale [45, Problem 4.5]); Artin and Mazur [2, Question 2 on p. 84]). For example, rationality or algebraicity of means that there is an easy recipe to compute all from a finite amount of data (in the rational case, it implies that is linearly recurrent). Zeta functions of more general dynamical systems can:

be rational:

e.g. for “Axiom A” diffeomorphisms by Manning [32, Cor. 2], for rational functions of degree on the Riemann sphere by Hinkkanen [27, Thm. 1], for the Weil zeta function (when is the Frobenius map on a variety defined over a finite field) by Dwork [19] and Grothendieck [26, Cor. 5.2], for endomorphisms of real tori [4, Thm. 1], and when replaced by the Lefschetz number of [45];

be algebraic but not rational:

e.g. when is an orientation preserving surface homeomorphism and is replaced by the Nielsen number of by Pilyugina and Fel’shtyn [36], [21, Thm. 36];

be transcendental:

e.g. for restrictions of shifts by Bowen and Lanford [7, §3–4] and for separable dynamically affine maps on by Bridy [8, Thm. 1], [9, Thm. 1.2 & 1.3];

have an essential singularity:

e.g. for some flows by Gallavotti [23, §4];

have a natural boundary:

e.g. for certain beta-transformations by Flatto, Lagarias and Poonen [22, Thm. 2.4], for some -actions () by Lind [31], for some flows by Pollicott [37, §4] and Ruelle [40], for a “random” such zeta function by Buzzi [10], for some explicit automorphisms of solenoids by Bell, Miles, and Ward [6], and for most endomorphisms of abelian varieties in characteristic by the first two authors [11].

Following the philosophy of [11], we will also study “tame dynamics” via the so-called tame zeta function defined by

(3)

summing only over that are not divisible by . Tame and “full” dynamics are related by the formulae in (4) below, but the tame zeta function tends to be better behaved. In Appendix B, we give some explicit expressions for the tame zeta function of several dynamically affine maps on .

1.4. Main results

Bridy studied the zeta function for dynamically affine maps on . The main results in [9, Thm. 1.2 & 1.3] imply that if is dynamically affine for and , then is transcendental over if and only if is separable; otherwise is rational. Bridy’s full result applies to all ; the proof uses a case-by-case analysis (see Table 1 in Appendix B below) and is based on the relation between transcendence and automata theory. This starkly contrasts with the fact that in characteristic zero all dynamically affine maps have a rational zeta function (a much more general result by Hinkkanen was quoted above).

In this paper, we prove a strengthening of Bridy’s result. For this, we need some further concepts. Let be a dynamically affine map.

Definition.

An endomorphism is said to be coseparable if is a separable isogeny for all . A dynamically affine map is called coseparable if the associated isogeny is coseparable.

Remark.

In [11], we called a coseparable endomorphism of an abelian variety “very inseparable” and showed that this implies inseparability [11, 6.5(ii)]. However, it is not true that coseparable dynamically affine maps are inseparable in general. For example, if is the map for transcendental over , then is both coseparable and separable (a more general statement is given in [9, Thm. 1.3]).

Definition.

A holomorphic function on a connected open subset is said to have a natural boundary along if it has no holomorphic continuation to any larger such [41, §6]. We call a function root-rational if for some . We call holonomic if it satisfies a nontrivial linear differential equation with coefficients in .

Since algebraic functions are holonomic [49, Thm. 6.4.6], the following is indeed a strengthening of Bridy’s result. At the same time, it shows that “tame” dynamics is better behaved.

Theorem A.

Assume is a dynamically affine map.

  1. If is coseparable, is a rational function; otherwise, is not holonomic; more precisely, it is a product of a root-rational function and a function admitting a natural boundary along its circle of convergence.

  2. For all , is root-rational; equivalently, it is algebraic and satisfies a first order differential equation over .

We mention an amusing corollary of Theorem A: although is in general not holonomic, the pair always satisfies a simple differential equation; see Corollary 2.4 for a precise statement.

Rather than using results from automata theory, we prove Theorem A essentially relying on a method of Mahler (see [5]). We structure the proof abstractly, showing the result for dynamically affine maps (in any dimension) that satisfy certain hypotheses (H1)(H4) (see Section 3), and then verify these for .

We give a more general discussion of when the hypotheses hold or fail, in this way producing the first higher-dimensional examples of dynamically affine maps in positive characteristic with nontrivial where we understand the nature of the dynamical zeta function. Recall that the quotient of an abelian variety by the group is called a Kummer variety.

Theorem B.

Let denote a Kummer variety arising from an abelian variety , and let denote the dynamically affine map induced by the multiplication-by- map for some integer . Then is root-rational. The function is not holonomic if is coprime to and rational otherwise.

Remark.

We use the word “Kummer variety” for the variety that, for , is singular at points in the finite subset of , but the name is sometimes used for the minimal resolution of . Since the set of singular points is finite and stable by , the map can be seen as a birational map with locus of indeterminacy stable by , and the above theorem can be interpreted as a statement about the periodic points of this birational map outside the preimage of the singular points.

Remark.

The non-holonomicity shows that the sequence of number of fixed points of the iterates of is somewhat “complex”, but it does not mean that is “uncomputable”. As a matter of fact, the results in [12] say that for an endomorphism of an algebraic group there exists a formula expressing in terms of a linear recurrent sequence and two specific periodic sequences of integers that control a -adic deviation of from being linearly recurrent. These data can in principle be computed by breaking up the algebraic group into abelian varieties, tori, vector groups, and semisimple groups. Similarly, one can in principle trace through our proofs to compute for dynamically affine maps satisfying our hypotheses.

We finish the introduction by mentioning a few possibilities for future research.

  • The relation between fixed points and closed orbits may be used to study the distribution of closed orbit lengths (analogously to the prime number theorem). Because of the analytic nature of the function revealed by our results, one cannot in general use standard Tauberian methods. We have studied this question via a different route for maps on abelian varieties [11] and for maps on general algebraic groups [12] (which covers the case of dynamically affine maps with trivial , , and , but is more general, since we do not require the group to be commutative). It would be interesting to extend this to general dynamically affine maps.

  • We have no good understanding of the dynamical zeta function of general rational functions on that are not dynamically affine, e.g.  in characteristic (see [8, Question 2]). It would be interesting to investigate the nature of the (tame) zeta function for such examples.

  • Inhowfar the hypotheses (H1)(H4) are necessary to reach the conclusion of the main theorem merits attention, since they are extracted from a “method of proof” rather than intrinsic.

  • In general, may be singular. It is interesting to study whether admits a resolution to which extends as a morphism, and the relation between the zeta function of that extended morphism and the zeta function of . This is nontrivial already for Kummer surfaces (where, for , the minimal resolution is a K3 surface, and hence has trivial étale fundamental group [28, pp. 3–6]).

The structure of the paper is as follows: After some generalities, we introduce the hypotheses in Section 3 and prove the main result, conditional on the hypotheses, in the following section. Then, in Section 5 we discuss the validity of the hypotheses in various settings (giving examples and counterexamples). The main theorems then follow immediately from these results. In the first appendix, we consider the radius of convergence of , and in the second appendix, we compute a collection of examples of tame zeta functions of dynamically affine maps.

2. Generalities

Relations between zeta functions

Proposition 2.1.

The tame and full dynamical zeta function are related by the following equalities of formal power series:

(4)
Proof.

For the first equality, note that

The second equality follows by applying the first one to the functions for . ∎

Remark 2.2.

A useful computational fact is the following: if is a map and decomposes as a union with and , then

and similarly for .

Recurrences

We recall some well-known facts (see e.g. [11, §1]). If is a sequence of complex numbers, then the ordinary generating function is rational if and only if the sequence is linear recurrent, and if and only if there exist and polynomials such that

(5)

for sufficiently large . The statement that the zeta function

(6)

is rational is stronger: this happens if and only if Equation (5) holds for all with the replaced by integers independent of . The occurring in (5) are called the roots of the recurrence, the polynomials their multiplicities. We say that satisfies the dominant root assumption if there is a unique root of maximal absolute value, possibly with multiplicity .

For a zeta function in (6), we may consider its tame variant

It follows from the formula

(7)

that if is rational, then is root-rational.

Algebraicity properties and differential equations

If a formal power series satisfies a nontrivial linear differential equation over , it is said to be holonomic. If is algebraic over , it is holonomic [49, Thm. 6.4.6]. On the other hand, if converges on some nontrivial open disc and has natural boundary along , then it cannot be holonomic, since a holonomic function has only finitely many singularities (for a precise statement, see [48, 4(a)]).

The equivalence statement in Theorem A2 is implied by the following lemma, which is certainly well-known, but for which we were unable to find a convenient reference. (A more general result can be found in [49, Exercise 6.62] together with an argument attributed to B. Dwork and M. F. Singer.)

Lemma 2.3.

An algebraic function is root-rational if and only if satisfies a first order homogeneous differential equation with .

Proof.

First assume that is root-rational, i.e.  with , . We may assume that , and then satisfies the equation with .

The converse direction can be proven by direct integration and partial fraction expansion of , but we give a somewhat different argument. Assume that satisfies the equation with , where we may assume . Let be the minimal polynomial of over . Write with . Differentiating the equation gives

(8)

where is obtained from by differentiating the coefficients and is the usual derivative of . Substituting into (8), we see that is a root of the polynomial , which is a polynomial of degree with leading coefficient , and hence

Comparing the coefficients at for , we see that each satisfies the equation

which differs from the equation satisfied by only by a multiplicative constant. Comparing these solutions gives for some . If for all , we get . Otherwise, for some we have , and is root-rational. ∎

Thus, Theorem A2 immediately implies the result alluded to in the introduction:

Corollary 2.4.

If is a dynamically affine map, then the pair of zeta functions satisfies a nonlinear first order differential equation

for some rational function , regardless of whether of not is coseparable.

Proof.

The root-rationality of implies that it satisfies a differential equation of the form for some rational function . The result follows by taking derivatives in the first identity in (4). ∎

3. Introduction of the general hypotheses

Let be a dynamically affine map with data as in diagram (1.2). Denote by the forward orbit of under . For an isogeny , we denote by and the degree and inseparable degree of the field extension , respectively. Then we have

(9)

The following lemma, taken from [9, Lemma 2.4] (cf. Remark 4.2), will be crucial to control the sequence , as it allows us to express in terms of kernels of isogenies on the algebraic group . The proof will be given in Section 4.

Lemma 3.1.

Let be a dynamically affine map. Consider the set

Then

(10)

Combining Lemma 3.1 with (9), we see that in order to understand the sequence it suffices to control, for every ,

  1. the sequence ;

  2. the “inseparable degree sequence” ;

  3. the “degree sequence” .

Notice that the translation parameter no longer occurs in (10).

We now introduce the four hypotheses that we require in order to prove the main theorems. The first three hypotheses (H1), (H2) and (H3) are employed to control the sequences 1, 2 and 3, respectively, while (H4) is a technical hypothesis that we require to avoid an unexpected cancellation of singularities in our proof of the existence of a natural boundary.

We use the following

convention: If a hypothesis is assumed in an environment (definition, lemma, theorem, hypothesis, …), we label the environment by this hypothesis in square brackets.

Hypothesis (H1).

The zeta function corresponding to is rational.

For the second hypothesis, we recall the following notion: a discrete valuation on a (not necessarily commutative) ring is a map such that for all we have if and only if , , and . It follows from these properties that whenever .

Hypothesis (H2).

Both and belong to a subring of all of whose nonzero elements are isogenies, and such that there exists a discrete valuation satisfying for all isogenies .

Note that the valuation considered in (H2) takes only nonnegative values.

Before introducing the last two hypotheses, we set up some notation.

Notation 3.2.

Let be as in (H2). For , we let

This defines a descending filtration of normal subgroups of

where

For we define to be the smallest integer such that for some ; in general, might not exist, but certainly does, and we will show in Lemma 4.11 that for either none of the exist or all do depending on whether or not is coseparable. Write and .

Hypothesis (H3).

[(H2)] Let . If exists, then

Remark 3.3.

The statement of Hypothesis (H3) a priori depends on the choice of the elements . However, it will follow from Lemma 4.122 below that it is independent of such a choice.

Hypothesis (H4).

[(H2)] The number exists and the sequence

(11)

is a linear recurrent sequence satisfying the dominant root assumption.

Remark 3.4.

If exists and (H3) holds, then the sequence (11) is automatically linear recurrent. Moreover, by Lemma 4.11, exists if and only if is not coseparable, and the element is then unique.

We then have the following results:

Theorem 3.5.

Assume is a dynamically affine map satisfying the hypotheses (H1)(H4). Then is not holonomic. More precisely, it is a product of a root-rational function and a function admitting a natural boundary along its circle of convergence.

Theorem 3.6.

Assume is a dynamically affine map satisfying the hypotheses (H1)(H3). Then is root-rational.

The proofs of these theorems will be given in the next section.

4. Proofs of Theorems 3.5 and 3.6

Preliminary results on the action of

Lemma 4.1.

Let be a dynamically affine map.

  1. There exists a group automorphism such that for any , and .

  2. The map is an isogeny for all and .

  3. for all and .

Proof.

1 That exists as a map of sets follows from [43, Prop. 6.77(a)(b)]. Recall that, by assumption, is surjective and has finite kernel. Now, for all we have

which implies that is a group homomorphism. For , we have , and so . Since is finite and is connected, we must have , and so . This shows that is injective, and hence bijective.

2 Let and . We will show that has finite kernel. Suppose that is such that . Put . Then

(12)

Since is injective and is finite, there exists for which

so that . Since by assumption is confined, we have that is finite, and the desired result follows.

3 For every , there exists such that for all . We then have

where in the last equality we use the fact that is an isogeny. ∎

Proof of Lemma 3.1.

The proof of [9, Lemma 2.4] shows that

The desired result now follows from Lemma 4.13. ∎

Remark 4.2.

The claim in [9, Lemma 2.4] that (10) holds for dynamically affine maps using Silverman’s definition (under the additional assumption that is surjective), is incorrect. For example, when for an elliptic curve , , , and with a non-torsion point, then , but is infinite for all . The mistake in the proof is that under the assumptions in Silverman’s definition, Lemma 4.13 does not need to hold (for this one needs part 2 of the lemma, which is equivalent to being confined). Nevertheless, in [9] the result is only applied for , where Silverman’s definition implies confinedness of , hence none of the other results are affected.

Preliminary results on valuations

Proposition 4.3.

Let denote a (not necessarily commutative) ring with a valuation . Then the following statements hold for all and :

  1. has no nontrivial zero divisors.

  2. The characteristic of is either zero or prime.

  3. We have .

  4. We have .

  5. Assume that and commute, , and . Then:

    1. if and , then ;

    2. if and for some prime , then if , we have ;

    3. if , then .

  6. In cases 5b and 5c above, if satisfies , then is unbounded as ranges over .

Proof.

1 Follows directly from the fact that the valuation of is infinite if and only if .

2 Follows from 1.

3 Follows from the formula .

4 We have , where lies in the two-sided ideal of generated by , and hence .

5 Since and commute, we have

(13)

If , then the first term has strictly smaller valuation than the second one, and hence , proving case 5a, as well as cases 5b and 5c for . It now suffices to consider 5b and 5c for ; the general result will then follow by induction on . For 5b, the assumption on implies that

(14)

for all . This shows that again in (13) the first term has strictly smaller valuation than the second one, which yields . For 5c, note that .

6 Follows from the formula (13) with , , and an arbitrarily large power of . ∎

Remark 4.4.

[(H2)] If as above is the endomorphism ring of a connected commutative algebraic group over and is a valuation on satisfying [(H2)], then Proposition 4.32 can be made slightly more explicit: the characteristic of will then be either zero or equal to . In fact, if is a prime and , then the multiplication-by- map is either zero or an inseparable isogeny, and hence its differential, which on the tangent space at is given by multiplication by , is not an isomorphism. Since the tangent space at is a -vector space, we must have and . This also implies that the prime found in 55b is equal to .

Remark 4.5.

The assumption that and commute is necessary in Proposition 4.355b. Consider the quaternion algebra generated over by with and , and let be the ring of Hurwitz quaternions, which is a maximal order in . Consider the valuation on corresponding to the prime element . Put and . Then , but . The assumption that and commute is missing from [9, Lemma 6.2], but the result is only applied for , and so other results in that reference are not affected.

Recall that is a dynamically affine map with associated data as in diagram (1.2). Assume that satisfies (H2). In order to obtain more information about , we will apply Proposition 4.3 to the ring and the valuation supplied by (H2).

Lemma 4.6.

[(H2)] If is coseparable, then is a separable isogeny for all and .

Proof.

Let and . By Lemma 4.12, is an isogeny, so it remains to show that it is separable. Applying Proposition 4.34, we see that

where in the last equality we use that is coseparable. ∎

Proposition 4.7.

[(H1)(H3)] If is coseparable, then is rational and is root-rational.

Proof.

If is coseparable, then by Lemma 4.6, for all and . The desired result for then follows by applying Lemma 3.1 together with (H1) and (H3) with . By the general rationality conditions in Section 2, this implies that is rational as well, and hence the result for follows from (4). ∎

Remark 4.8.

Proposition 4.7 is false if we drop the assumption of the hypotheses. In fact, if (H2) and (H3) do not hold, then may even have a natural boundary along its circle of convergence (see Example 5.9 below).

Lemma 4.9.

[(H2)] If is not coseparable, then .

Proof.

If , then for all , contradicting the assumption that is not coseparable. ∎

Lemma 4.10.

[(H2)] Suppose that and are such that . Then and commute.

Proof.

Since and , we have . Let as in Lemma 4.11, so that . It follows that

We conclude that , and hence . ∎

We will now prove the announced result on the existence of the numbers defined in Notation 3.2.

Lemma 4.11.

[(H2)]

  1. If is coseparable, then none of the numbers exist for .

  2. If is not coseparable, then all of the numbers exist.

Proof.

1 If is coseparable, then by Lemma 4.6 all the maps for and are separable isogenies, and hence . Hence do not exist for .

2 Since is not coseparable, there is some such that , and hence by Proposition 4.3.6 the values of can be arbitrarily large. This proves the existence of