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“fm˙08˙01” — 2008/6/24 — 14:42 — page 1 — #1 Forum Mathematicum Forum Math. (2008), 1–19 DOI 10.1515/FORUM.2008. © de Gruyter 2008 Minimal number of periodic points for compact simply-connected manifolds C1 self-maps of Grzegorz Graff, Jerzy Jezierski (Communicated by Michael Brin) Abstract. Let f be a self-map of a smooth compact connected and simply-connected manifold of dimension m ≥ 3, r a fixed natural number. In this paper we define a topological invariant Drm[f ] which is the best lower bound for the number of r-periodic points for all C 1 maps homotopic to f . In case m = 3 we give the formula for Dr3 [f ] and calculate it for self-maps of S 2 × I. 2000 Mathematics Subject Classification: 37C25, 55M20; 37C05. 1 Introduction In 1983 Albrecht Dold [D] formulated the necessary conditions that must be fulfilled by a sequence of indices of iterations. He considered a map f : U → X , where U ⊂ X is an open subset of an ENR, satisfying the following fixed-compactness condition. We denote inductively U0 = U , Un+1 = f −1 (Un ) i.e. Un = {x ∈ U : x, f (x), ..., f n (x) ∈ U }. We assume that the fixed point set Fix(f n ) = {x ∈ Un : f n (x) = x} is compact for each n ∈ N. In such situation the fixed point index ind(f n ) = ind(f n , Un ) is well-defined. Dold proved that the sequence of fixed point indices {ind(f n )}∞ n=1 must satisfy the following congruences, (called Dold relations): ∑k |n μ (n/k)ind(f k ) ≡ 0 (mod n) for each n ∈ N, where μ denotes the Möbius function. The classical issue in periodic point theory is the question of what is the minimal number of periodic points in the homotopy class of a given f , a self-map of a compact manifold. It appears that this problem is strictly related to the question what are possible forms of local indices (at an isolated fixed point) of iterations of maps homotopic to f . Namely, let f : U → R m be a continuous map defined on a neighborhood of the point 0 ∈ R m . We define the iterations f n as above, and assume that 0 is the only fixed point of f n for each n ∈ N in some small neighborhood Vn ⊂ Un of 0 and consider ind(f n , 0) = ind(f n , 0, Vn). Which n ∞ integer sequences {cn }∞ n=1 can be obtained as {ind(f , 0)}n=1? It is shown in [BaBo] and [GNP2] that once the Dold congruences are satisfied, there is a continuous map f realizing an arbitrary sequence {cn }∞ n=1 , for m ≥ 2, as a sequence of indices at a single fixed point. This fact has a strong impact on the possibility of cancelling periodic points in the given homotopy class. If f : M → M is a self-map of a compact connected and simplyResearch supported by KBN grant No 1 P03A 03929. Manuscript nr.: 08/1-B30 “fm˙08˙01” — 2008/6/24 — 14:42 — page 2 — #2 2 G, Graff, J. Jezierski connected manifold of dimension ≥ 3, then it is homotopic to a continuous map g with Fix(g r ) consisting of a single point, where r is a prescribed natural number ([Je2] Thm. 5.1). In the smooth category however, the situation is much different. (We say that f is smooth in the meaning that it is C 1 .) In 1974 Shub and Sullivan [SS] proved that fixed point indices of iterations of a smooth map, at an isolated fixed point, are periodic. This is a very strong restriction making the theory quite different than in the continuous case. Later, in 1981, Chow, Mallet-Paret and Yorke [CMPY] showed that in the smooth case a sequence of indices may be represented as the sums of some elementary sequences whose coefficients depend on the derivative of a map at the fixed point. This leads to a system of conditions 1 that any sequence {ind(f n , 0)}∞ n=1, for f being C map, must satisfy. It was conjectured in [CMPY] that these conditions are also sufficient to realize given sequence as the sequence of indices of iterations of some C 1 map. The conjecture was answered positively by Babenko and Bogatyi [BaBo] in dimension 2 and by Graff and Nowak-Przygodzki in dimension 3 [GNP1]. Basing on the results mentioned above and methods of Nielsen fixed point theory developed in [Je2] (cf. also [HK1], [HK2]), we are able to deal with the smooth case. First of all we show in this paper that, unlike in the continuous case, Fix(f r ) cannot be reduced to a point by a smooth map g, homotopic to the given f , where f : M → M , M satisfies the same assumptions as in the continuous case, r is a fixed natural number. Next, we express in terms of local indices of iterations the least cardinality of Fix(g r ) which can be obtained by a smooth map g in the homotopy class of f (cf. Theorem 3.8). We reduce this problem to the decomposition of the sequence of Lefschetz numbers L(f n ), where n|r, into the sum of sequences each of which can be locally realized as a sequence of indices at an isolated periodic orbit for some C 1 map. As a result we get the topological invariant Drm [f ] which is the best lower bound for the number of r-periodic points for all C 1 maps homotopic to f . Let us notice that Drm [f ] may be interpreted purely in the terms of smooth category, namely we may assume that f is smooth and approximate the homotopy which joins f with g by a smooth one. Then Drm [f ] gives the minimal number of periodic points in the smooth homotopy class of the given f . In the second part of the paper, we use the classification of sequences of indices of iterations of smooth maps in dimension 3, described in [GNP1], to give the explicit formulae for Dr3 [f ]. We also calculate Dr3 [f ] for a 3-manifold with simple homology groups, namely S2 × I . Let us remark that in the non simply-connected case the problem of finding minimal number of periodic points in smooth category is much more complicated. In the possible version of the non simply-connected counterpart of Theorem 3.8, which we intend to prove in the forthcoming paper, the restrictions for local indices for a smooth map must be combined with topological properties of a space expressed in terms of orbits of Reidemeister classes of the iterated map. “fm˙08˙01” — 2008/6/24 — 14:42 — page 3 — #3 Minimal number of periodic points for C 1 maps 3 2 Theorem of Chow, Mallet-Paret and Yorke and DD m sequences Definition 2.1. Let f : U → R m , where U is an open subset of R m , be a map such that 0 is an isolated fixed point for each iteration of f . For each n we define integers (called local Dold coefficients) in (f , 0) by the following equality: in (f , 0) = ∑ μ (n/k)ind(f k , 0), k |n where μ is the Möbius function, i.e. μ : N → Z is defined by three properties: μ (1) = 1, μ (k) = (−1)r if k is a product of r different primes, μ (k) = 0 otherwise. 2 We may write down Dold relations, mentioned in the introduction, as: for each natural n (1) in (f , 0) ≡ 0 (mod n). Dold relations make it possible to decompose a sequence of indices of iterations into a sum of elementary sequences. This decomposition is called a periodic expansion. Definition 2.2. For a given k we define the basic sequence (basic expansion): k if k|n, regk (n) = 0 if k |n. 2 Notice that each basic sequence regk is a periodic sequence of the form: (0, . . ., 0, k, 0, . . ., 0, k, . . ., . . .), where the non-zero entries appear only for indices of the sequence which are multiples of k. Theorem 2.3 (cf. [JM]). Any sequence ψ : N → C can be written uniquely in the following form of a periodic expansion: ∞ ψ (n) = ∑ ak (ψ )regk (n), k =1 where an (ψ ) = in (ψ )/n = 1n ∑k |n μ (k)ψ (n/k). Moreover, ψ takes integer values and satisfies Dold relations iff an (ψ ) ∈ Z for every n ∈ N. Definition 2.4. Assume that a periodic expansion for a sequence {ψ (n)}∞ n=1 is given. Let B(ψ ) = {n ∈ N : an = 0}. The set B(ψ ) will be called the basic set for a periodic expansion of ψ . 2 Now consider f : U → R m , where U is an open subset of R m containing 0, a C 1 map with 0 an isolated fixed point for each iterations. Let us denote by Δ the set of degrees of all primitive roots of unity which are contained in σ (Df (0)), the spectrum of derivative at 0. 1 Chow, Mallet-Paret and Yorke showed that the basic set B({ind(f n , 0)}∞ n=1), for a C map f , is finite and depends on the set Δ. We denote by σ+ the number of real eigenvalues of Df (0) greater than 1, σ− the number of real eigenvalues of Df (0) less than −1, in both cases counting with multiplicity. Let O = {LCM(K ) : K ⊂ Δ}, “fm˙08˙01” — 2008/6/24 — 14:42 — page 4 — #4 4 G, Graff, J. Jezierski where by LCM(K ) we mean the least common multiple of all elements in K with the convention that LCM(0) / = 1. Let Oodd = O ∩ {n ∈ N : n = 2k − 1, k = 1, 2, . . .}. We present below the result of Chow, Mallet-Paret and Yorke ([CMPY]) expressed in the language of periodic expansions (cf. [GNP1], [MP], [JM]). Theorem 2.5. Let U ⊂ R m be an open neighborhood of 0, f : U → R m be a C 1 map having 0 as an isolated fixed point for each iteration. Then: (∗) ind(f n , 0) = ∑ ak regk (n), where k ∈O O= O if σ− is even, O ∪ 2Oodd if σ− is odd. (∗∗) If σ− is odd and k ∈ 2Oodd \ O, then ak = −ak /2 . (∗ ∗ ∗) The bounds for the coefficients a1 and a2 are the following: (1) a1 = (−1)σ+ if 1 ∈ σ (Df (0)). (2) a1 ∈ {−1, 0, 1} if 1 is an eigenvalue of Df (0) with multiplicity 1. (3) a2 ∈ {0, (−1)σ++1 } if 1 ∈ σ (Df (0)) and −1 is the eigenvalue of Df (0) with multiplicity 1. 2 Remark 2.6. Let us notice that there are three different types of restrictions described in the theorem. Namely: bounds for the set O (which contains the basic set B), relations between coefficients ak and ak /2 and conditions for a1 and a2 . Theorem 2.5 gives the description of indices for a map f for which 0 is an isolated fixed point for all iterations. In the further considerations we take some fixed r and consider more general situation, in which 0 is an isolated fixed point only for n|r. Nevertheless, for a finite sequence {ind(f n , 0)}n|r the formula and conditions for indices of iterations given in Theorem 2.5 remain true. Remark 2.7. Let r be a fixed natural number, U ⊂ R m be an open neighborhood of 0, f : U → R m be a C 1 map having 0 as an isolated fixed point for f n , where n|r. Then, for n|r, ind(f n , 0) = ∑k |r,k ∈O ak regk (n) and the coefficients ak (if they appear, so if k|r) satisfy the conditions (∗) and (∗∗) of Theorem 2.5. Notice that by the assumption {ind(f n , 0)}n|r is well-defined. Remark 2.7 may be relatively easily obtained by analyzing the original proof of Theorem 2.5. Namely, it results from two facts. First, so-called Virtual Period Proposition (Prop. 3.2 [CMPY]) remains true without the assumption that 0 is an isolated fixed point. Second, if we choose as in [CMPY] the approximation of f by a map with simple fixed points, the contribution to the sequence {ind(f n , 0)}n|r comes only from such n-orbits, for which n|r (by an n-orbit we understand an orbit whose elements are periodic points with the minimal period equal to n). Definition 2.8. In the rest of the paper we will use the following notation (cf. Definition 2.4): B(f , x0 ) = B({ind(f n , x0 )}∞ n=1 ), “fm˙08˙01” — 2008/6/24 — 14:42 — page 5 — #5 Minimal number of periodic points for C 1 maps 5 Br (f , x0 ) = B({ind(f n , x0 )}n|r ), B(f ) = B({L(f n )}∞ n=1 ), Br (f ) = B({L(f n )}n|r . m Definition 2.9. A sequence of integers {cn }∞ n=1 is called DD (p) sequence if there is an m 1 isolated p-orbit P, a neighborhood U ⊂ R of P and a C map φ : U → R m such that cn = ind(φ n , P) (notice that cn = 0 if n is not a multiple of p). The finite sequence {cn }n|r will be called DDm (p|r) sequence if this equality holds for n|r, where r is fixed. In other words, DDm (p) sequence is a sequence that can be realized as a sequence of indices of iterations on an isolated p-orbit for some smooth map φ . The following lemma gives the description of DDm (p) sequences. m Lemma 2.10. A sequence {cn }∞ n=1 is a DD (1) sequence if and only if 0 for p |n, c̃n = p cn/p for p|n m is a DDm(p) sequence. We will say that a DDm (p) sequence {c̃n }∞ n=1 comes from a DD (1) ∞ sequence {cn }n=1 . Proof. “⇒” Let φ : R m → R m be a smooth map with the isolated fixed point x1 , such that ind(φ k , x1 ) = ck . We fix different points x1, ..., xp ∈ R m . Let us choose small enough mutually disjoint neighborhoods Ui xi (for i = 1, ..., p), and diffeomorphisms φi : Ui → Ui+1 (for i = 1, ..., p − 1). At last we define φp : Up → U1 by the formula: (2) −1 φp (x) = φ (φ1−1(φ2−1(· · · φp−1 (x)))) for x near xp . We define Φ : (3) p i=1 Ui → R m putting Φ(x) = φi (x) for x ∈ Ui . Then (4) Φp (x) = φp (φp−1 (· · · φ1 (x))) = φ (x) in a neighborhood of x1 . What is more, ind(Φkp, x1 ) = ind(φ k , x1 ) = ck . Since the indices in all points of the orbit are the same, ind(Φkp, {x1 , ..., xp}) = p ck . Finally, for each natural n ind(Φn , {x1 , ..., xp}) is equal either to p · cn/p if p|n, or to zero otherwise, which proves that c̃n is a DDm (p) sequence. p 1 “⇐” If φ is a map from Definition 2.9 which realizes {c̃n }∞ n=1 , then φ is the C map m ∞ which gives the needed DD (1) sequence {cn }n=1 . 2 “fm˙08˙01” — 2008/6/24 — 14:42 — page 6 — #6 6 G, Graff, J. Jezierski Remark 2.11. Let us notice that the same construction, as in the proof of “⇒”, can be done for an arbitrary orbit of points in a manifold. Remark 2.12. Lemma 2.10 gives an easy procedure which enables one to obtain all forms of DDm (p) sequences once we know the forms of DDm (1) sequences. Namely by the relation between respective basic sets: B(c̃n ) = pB(cn ) = {pk : k ∈ B(cn )}, we find that in order to get any DDm (p) sequence it is enough to replace all basic sequences ak regk by ak regpk in the periodic expansion of some DDm (1) sequence. 3 Minimal number of periodic points Let f : X → X be a self-map of a topological space and r ∈ N a fixed number. What is the minimal number of periodic points #Fix(g r ), where g is a map homotopic to f ? The Nielsen periodic point theory gives an invariant NFr (f ), introduced by Boju Jiang in [Ji], which is a homotopy invariant and the lower bound for the number of periodic points: NFr (f ) ≤ #Fix(g r ) for each g homotopic to f . In [Je1] it is shown that, for X – a compact PL-manifold of dimension not less than 3, NFr (f ) is the best such lower bound, i.e. for each f there is a map g homotopic to f such that #Fix(g r ) = NFr (f ). In this paper we confine ourselves to the simply-connected case. Then Nielsen theory is trivial and NFr (f ) is equal to 0 or 1. In such a case Theorem 5.1 in [Je2] gives: Theorem 3.1. Any continuous self-map f of a compact connected simply-connected PLmanifold of dimension ≥ 3 is homotopic to a map g such that 0/ if L(f n ) = 0 for all n|r, r Fix(g ) = {∗} otherwise, where {∗} - Fix(g) denotes the set which consists of one point. 2 Now we study a similar question in the smooth category. We are given a manifold M . We will assume through the rest of the paper that M is a smooth compact connected and simplyconnected manifold. Let f be a continuous self-map of M and r ∈ N a given number. If the boundary ∂ M = 0, / then we assume that all considered maps have no periodic points on the boundary. The question we ask is: what is the minimal number of periodic points #Fix(g r ), where g is a smooth map homotopic to f ? The results presented in the previous section show that here the situation is drastically different. It is easy to notice that a self-map of a simply-connected manifold is not homotopic, in general, to a smooth map g with Fix(g r ) = {∗}, a point. In fact: Remark 3.2. If a self-map f of M is homotopic to a smooth map g with Fix(g r ) = {∗}, then the sequence of Lefschetz numbers {(L(f n )}n|r is a DDm(1|r) sequence. Proof. For every n|r we have: L(f n ) = L(g n ) = ind(g n ) = ind(g n , ∗). 2 “fm˙08˙01” — 2008/6/24 — 14:42 — page 7 — #7 Minimal number of periodic points for C 1 maps 7 In the next section we will see that the set of periodic points can be reduced to a single point in rather exceptional cases. On the other hand the following lemma holds: Lemma 3.3. If f , a self-map of M , is homotopic to a smooth map g with #Fix(g r ) = N finite, then the sequence of Lefschetz numbers of f decomposes as L(f n ) = c1 (n) + . . .+ cs (n), for n|r, where ci is a DDm (li |r) sequence and l1 + . . .+ ls = N . Proof. If #Fix(g r ) = N , then Fix(g r ) = a1 ∪ . . .∪ as, where ai is an orbit of the length li . We get for n|r: s s i=1 i=1 L(f n ) = L(g n ) = ind(g n ) = ∑ ind (g n , ai ) = ∑ ci (n), where ci (n) = ind (g n , ai ) is a DDm (li |r) sequence, since g is smooth. It remains to notice that l1 + . . . + ls = #a1 + . . .+ #as = #Fix(g r ) = N . 2 The above lemma suggests the following definition. Definition 3.4. Let {ξn }n|r be a sequence of integers satisfying Dold relations. Assume that we are able to decompose it into the sum: ξ (n) = c1 (n) + . . .+ cs (n), where ci is a DDm (li |r) sequence for i = 1, . . ., s. Each such decomposition determines the number l = l1 +. . .+ls . We define the number Drm [ξ ] as the smallest l which can be obtained in this way. 2 We will denote briefly Drm [f ] = Drm [{L(f n )}n|r ]. It is easy to see that Drm [f ] is a homotopy invariant. On the other hand, by Kupka-Smale Theorem (cf. for example [R]) every f is homotopic to some smooth map g which has a finite number of r-periodic points (r is fixed). Thus, Lemma 3.3 shows that Drm[f ] is a lower bound of the number of r-periodic points for g: #Fix(g r ) ≥ Drm [f ]. The aim of this section is to prove Theorem 3.8 which states that Drm [f ] is the best such lower bound, provided the dimension of the manifold is not less than 3. The main idea of the proof may be described as follows. We fix a decomposition L(f n ) = c1 (n) + · · · + cs (n) realizing Drm [f ] i.e. the sum l = l1 + · · ·+ ls in Definition 3.4 is the smallest. Then we deform f to a map f¯ adding an isolated orbit of points Oi satisfying ind(f¯ n ; Oi ) = ci (n) for each i = 1, ..., s (Creating Procedure, Theorem 3.5). It turns out that the remaining periodic points Fix(f¯ r ) \ (O1 ∪ · · · ∪ Os ) can be removed (Canceling Lemma, Lemma 3.7). “fm˙08˙01” — 2008/6/24 — 14:42 — page 8 — #8 8 G, Graff, J. Jezierski The map f1 , obtained after this cancelation, satisfies #Fix(f1r ) = #(O1 ∪ · · ·∪ Os ) = #O1 + · · · + #Os = l1 + · · · + ls = Drm[f ], hence f1 realizes the lower bound of the number of periodic points Drm [f ]. The following two results i.e. Creating Procedure and Cancelling Lemma, proved in [Je2] (valid not only for smooth but also PL-manifold M ) will be crucial in the proof of Theorem 3.8. The Creating Procedure enables one to create an additional orbit in the homotopy class of f , by a use of a homotopy ft which is constant near periodic points of f (up to the given period r) and such that f1r near the created orbit may be given by an arbitrarily prescribed formula. Theorem 3.5. (Creating Procedure, [Je2] Theorem 3.3) Given numbers p, r ∈ N, p|r and a map f : M → M , where dimM ≥ 3, such that Fix(f r ) is finite and a point x0 ∈ Fix(f r ). Then there is a homotopy {ft }0≤t≤1 satisfying: 1. f0 = f . 2. {ft } is constant in a neighborhood of Fix(f r ). p 3. f1 (x0 ) = x0 and f1i (x0 ) = x0 for i = 1, . . ., p − 1. p 4. The map f1 is given near x0 by an arbitrarily prescribed formula φ with the property r φ p (x) = x ⇐⇒ x = x0 . p−1 2 5. The orbit x0 , f1(x0 ), . . ., f1 (x0 ) is isolated in Fix(f1r ). Proposition 3.6. The map f1 in Theorem 3.5 can realize an arbitrarily prescribed DDm (p|r) sequence c̃n on the created p-orbit O, i.e f1 can be smooth in a neighborhood of O and c̃n = ind(f1n , O). Proof. The statement may be obtained by a slight modification of the proof of Creating Procedure from [Je2]. Namely, it results from the fact that f1 may be chosen in such a way that f1 (x) = Φ(x) near O, where Φ has the form from the formula (3) in Lemma 2.10, i.e. φi are some fixed diffeomorphisms and φ is an arbitrary smooth map that realizes any DDm(1) sequence (for more details cf. [Je2]). Now we take as φ a map that realizes DDm (1| pr ) sequence cn = 1p c̃np and by Lemma 2.10 obtain that: ind(f1n , O) = ind(Φn , O) = c̃n . 2 The next lemma makes possible to cancel subsets of periodic points which have indices of iterations equal to zero. Lemma 3.7. (Cancelling Lemma, [Je2] Lemma 5.2) Let f be a continuous self-map of M . Suppose that S ⊂ Fix(f r ) satisfies: 1. S is finite and f -invariant i.e. f (S) = S. 2. Fix(f r ) \ S is compact. 3. ind(f n , Fix(f n ) \ S) = 0 for all n|r. Then there is a homotopy ft , starting from f0 = f , constant near S and such that Fix(f1r ) = S. 2 “fm˙08˙01” — 2008/6/24 — 14:42 — page 9 — #9 Minimal number of periodic points for C 1 maps 9 Now we are in a position to prove the main result of this section. Theorem 3.8. (Main Theorem). Let M be a smooth compact connected and simply-connected manifold of dimension m ≥ 3 and r ∈ N a fixed number, then min{#Fix(g r ) : g is C1 and is homotopic to f } = Drm [f ]. In other words Drm [f ] is the best lower bound for #Fix(g r ) for smooth g in the homotopy class of f . 2 Proof. The fact that the minimum is greater or equal to Drm [f ] follows from Lemma 3.3. Let us decompose s (5) L(f n ) = ∑ ci (n), i=1 where n|r, ci is a DDm (li |r) sequence and the sum ∑si=1 li is minimal i.e. Drm [f ] = ∑si=1 li . We have to show that Drm [f ] is reachable by some map in the homotopy class of f i.e. find a smooth map f1 homotopic to f with Fix(f1r ) = ∑si=1 li . First we use Creating Procedure and Proposition 3.6, deforming f to f¯ in such a way that we subordinate to every i ∈ {1, . . ., s} an isolated orbit Oi of the length li for which ind(f¯ n , Oi ) = ci (n). Each ci is a DDm(li |r) sequence, so by Proposition 3.6 f¯ is smooth near the created orbits. If the boundary of M is nonempty we take Oi inside M . We denote S = si=1 Oi . We will show that there is a deformation of f¯ , rel. a neighborhood of S, to a smooth map f1 satisfying Fix(f1r ) = S. Then f1 is the desired map, since s #Fix(f1r ) = #S = ∑ li = Drm [f ]. i=1 It remains to perform the above deformation of f¯ . We will apply Lemma 3.7. We check that the assumptions of this lemma are satisfied. S, as the sum of orbits, is f¯ -invariant. Since S is isolated, Fix(f¯ r ) \ S is compact. Now we notice that for each n|r: ind(f¯ n ) = ind (f¯ n , Fix(f¯ n ) \ S) + ind(f¯ n , S) s = ind (f¯ n , Fix(f¯ n ) \ S) + ∑ ind (f¯ n , Oi ) i=1 s = ind (f¯ n , Fix(f¯ n ) \ S) + ∑ ci (n) = ind (f¯ n , Fix(f¯ n ) \ S) + ind(f n ), i=1 where the last equality results from the equation (5) and from the fact that Lefschetz number and global index coincide. Thus ind (f¯ n ) = ind (f¯ n , Fix(f¯ n ) \ S) + ind(f n ). Since ind(f¯ n ) = ind(f n ) = L(f n ), ind (f¯ n , Fix(f¯ n ) \ S) = 0, so the third assumption is also satisfied. The map f1 is smooth in some neighborhood W of S and f1r has no fixed points outside W . Thus, if f1 is not smooth as the global map, we may easily approximate it by a smooth map constantly equal to f1 on W without adding any new r-periodic points in the compact set M \ W . 2 “fm˙08˙01” — 2008/6/24 — 14:42 — page 10 — #10 10 G, Graff, J. Jezierski Remark 3.9. Let us consider the set MPr (f ) = min{#Pr (g) : g is homotopic to f }, where Pr (g) denotes the set of periodic points of g with the minimal period r. In the simplyconnected case we have, by Remark 3.1, that MPr (f ) = 0 for r > 1. If we define its smooth counterpart: MPsr (f ) = min{#Pr (g) : g is smooth and homotopic to f }, then it follows from the proof of Theorem 3.8 that the same statement holds: MPsr (f ) = 0 for r > 1. Namely, we may find smooth g homotopic to f such that all its r-periodic points are fixed points (which however, does not imply that Fix(g r ) = Drm [f ]). The problem of determining the least number of Fix(f r ), i.e. Drm [f ], reduces to the decomposition of the sequence of Lefschetz numbers {L(f n )}n|r into the sum of some elementary sequences i.e. DDm (p|r) sequences. This becomes a combinatorial task, once we know what they are. In the next section we will use the complete description of DD3 (1) sequences from [GNP1] to give explicit formulae for the least number of periodic points in dimension 3. 4 3-dimensional case First of all we remind the reader of the result from [GNP1] (cf. Theorem 4.1 below) which describes all DD3 (1) sequences (and thus, by Lemma 2.10, all DD3 (p) sequences). Theorem 4.1. There are seven kinds of DD3 (1) sequences: (A) cA (n) = a1 reg1 (n) + a2 reg2 (n), (B) cB (n) = reg1 (n) + ad regd (n), (C) cC (n) = −reg1 (n) + ad regd (n), (D) cD (n) = ad regd (n), (E) cE (n) = reg1 (n) − reg2 (n) + ad regd (n), (F) cF (n) = reg1 (n) + ad regd (n) + a2d reg2d (n), where d is odd, (G) cG (n) = reg1 (n) − reg2 (n) + ad regd (n) + a2d reg2d (n), where d is odd. In all cases d ≥ 3 and ai ∈ Z. Proposition 4.2. Each DD3 (p|r) sequence is also a DDm (p|r) sequence (for m ≥ 3), as a consequence Dr3 [ξ ] ≥ Drm [ξ ]. Proof. Let {cn }n|r be a DD3 (p|r) sequence, and let f be a map whose existence is guaranteed by Definition 2.9. Then {cn }n|r may be realized as the sequence of indices of the map f × g, where g : R m−3 → R m−3 is a sink type map for which ind(g n , 0) = 1, for example g(x) = 12 x. 2 Lemma 4.3. ([BaBo] Corollary 3.6) Let f be a C 1 self-map of a smooth m-dimensional manifold and let x0 be a fixed point of f . Then the basic set B(f , x0 ) for the periodic expansion m+1 of indices of iterations at x0 satisfies: #B(f , x0 ) ≤ 2[ 2 ] . 2 The same inequality holds for each orbit of periodic points. “fm˙08˙01” — 2008/6/24 — 14:42 — page 11 — #11 Minimal number of periodic points for C 1 maps 11 Corollary 4.4. Let O(p) = {x1 , . . ., xp } be a p-orbit of a smooth map f : M → M (dim M = m). By B(f , O(p) ) we denote B({ind(f n , O(p) )}∞ n=1 ) (cf. Definitions 2.4 and 2.8). Let us consider the map f p . Notice that ind((f p )n , x1 ) = ∑ ak regk (n) k ∈B(f p ,x1 ) implies ind(f n , O(p) ) = ∑ as/p regs (n) s∈B(f ,O(p) ) and B(f , O(p) ) = pB(f p , x1 ) = {pk : k ∈ B(f p , x1 )} (cf. [BaBo]).Thus #B(f , O(p) ) = #B(f p , x1 ). As a consequence by Lemma 4.3 applied to f p we get: #B(f , O(p) ) ≤ 2[ m+1 2 ]. Theorem 4.5. Let f be a self-map of a smooth compact simply-connected m-dimensional manifold. Then: Drm [f ] ≥ #Br (f ) 2[ . m+1 ] 2 Proof. Each basic sequence regk which appears with a non-zero coefficient in the periodic expansion of Lefschetz numbers of f must also appear in expansions of some periodic orbits of f , thus: Br (f ) ⊂ Br (f , O(p) ). {O(p) :p|r} As a consequence, by Corollary 4.4: m+1 #Br (f ) ≤ ∑ #Orbp (f ) · 2[ 2 ] , (6) p|r where Orbp (f ) denotes the set of all p-orbits of f . Assume now that g is a smooth map homotopic to f for which #Fix(g r ) is minimal. We get: Br (f ) = Br (g), next we apply the inequality (6) for g and obtain: #Br (f ) m+1 2[ 2 ] = #Br (g) 2[ m+1 ] 2 ) ≤ ∑ #Orbp (g) ≤ ∑ #Orbp (g) · p = Drm [f ]. p|r p|r This ends the proof. 2 Corollary 4.4 leads to the following observation: Remark 4.6. In the definition of Drm [f ] it is enough to consider only DDm (p|r) sequences m+1 such that p < 2[ 2 ] . In order to justify this fact, consider {cn }n|r a DDm (p|r) sequence with p ≥ 2[ m+1 ] 2 , which gives the contribution p in Definition 3.4. By Theorem 4.1 and Proposition “fm˙08˙01” — 2008/6/24 — 14:42 — page 12 — #12 12 G, Graff, J. Jezierski 4.2 it can be replaced by #B(f , O(p) ) expressions of the type (D), so DDm (1|r) sequences, each of which gives the contribution equal to 1. Finally, as by Corollary 4.4 these sequences m+1 in sum give the contribution equal at most 2[ 2 ] , we can always replace {cn }n|r by them during the calculation of Drm[f ]. Notice that, in particular for m = 3, we have p < 22 , hence p ≤ 3. 2 Definition 4.7. Let us list three DD3 (2) sequences which come from DD3 (1) sequences of the form (E), (F) and (G): (E’) cE (n) = reg2 (n) − reg4 (n) + a2d reg2d (n), where d ≥ 3, (F’) cF (n) = reg2 (n) + a2d reg2d (n) + a4d reg4d (n), where d ≥ 3 is odd, (G’) cG (n) = reg2 (n) − reg4 (n) + a2d reg2d (n) + a4d reg4d (n), where d ≥ 3 is odd. 2 Let us emphasize that the contribution to Dr3 [f ] of each sequence (A)-(G) is 1, while of each (E’)-(G’) is 2. Lemma 4.8. Let f : M → M be a C 1 map, dim M = 3, then in the definition of Dr3 [f ] it is enough to consider only DD3 (1|r) sequences i.e. sequences which, for n|r, are of the forms (A)–(G); and DD3 (2|r) sequences of the forms (E’), (F’) and (G’). Proof. By Remark 4.6 it is enough to use only DD3 (p|r) sequences with p ≤ 3. Let us now consider a DD3 (3|r) sequence. If it comes from one of DD3 (1|r) sequences of the type (A) - (F), then the basic set Br (f , O) for the sequence of indices of its realization on some 3-orbit has no more than 3 elements, thus we can replace it by at most three DD3 (1|r) sequences of the types (D) or (A). If the DD3 (3|r) sequence comes from (G), then it has the expansion: g(n) = reg3 (n) − reg6 (n) + ad reg3d (n) + a2d reg6d (n), where d is odd. It can be replaced by three DD3 (1|r) sequences, namely: c1F (n) = reg1 (n) + ad reg3d (n) + a2d reg6d (n), c2F (n) = reg1 (n) + reg3 (n) − reg6 (n), c3A (n) = −2reg1 (n). Finally, DD3 (2|r) sequence which comes from one of the DD3 (1|r) sequences of the types (A) - (D) may be replaced by two DD3 (1|r) sequences of the types (A) or (D). 2 4.1 Formula for D3r [f] In this section we add the assumption that M is 3-dimensional, thus we consider f , a selfmap of M , where M is a smooth compact connected simply-connected 3-manifold. We will give a formula for the least number of periodic points, up to r periodic, for a smooth map homotopic to f . By Theorem 3.8 the number is equal to Dr3 [f ] = Dr3 [{L(f n )}n|r ]. We will apply Lemma 4.8 which allows us, during the calculation of Dr3 [f ], to use only sequences (A)–(F), (E’), (F’) and (G’). Let r be a fixed natural number. The sequence of Lefschetz numbers {L(f n )}n|r satisfies Dold relations, hence it can be written in the form of a periodic expansion, for n|r: (7) L(f n ) = b1 reg1 (n) + b2 reg2 (n) + b4reg4 (n) + ∑ bk regk (n), k ∈G where b1 , b2 , b4 are arbitrary integers, G = {k ∈ N : k|r and k ∈ {1, 2, 4} and bk = 0} = Br (f ) \ {1, 2, 4}. “fm˙08˙01” — 2008/6/24 — 14:42 — page 13 — #13 Minimal number of periodic points for C 1 maps 13 Let us define H , the subset of G: H = {k ∈ G : k is odd and bk = 0, b2k = 0}. Definition 4.9. We will say that a sequence ψ of one of the types (A)-(G), (E’)-(G’) reduces a basic sequence bk regk in the periodic expansion of Lefschetz numbers (7) if k ∈ Br (ψ ) and ak = bk , i.e. bk regk appears in the periodic expansion of ψ . Theorem 4.10. If r is odd, then: #G if |L(f )| ≤ #G, (∗) Dr3 [f ] = #G + 1 otherwise. If r is even and r > 4, then: (∗∗) Dr3 [f ] ∈ [#G − #H , #G − #H + 2]. Proof. Let r be odd so b2 and b4 are not needed and the sequences of Definition 4.7 are not relevant. Then, seeking for the exact value of Dr3 [f ], it is enough to use only the sequences ci (n) of the types (A)-(D). Each of the sequences (B), (C), (D) reduces exactly one basic sequence bk regk in the periodic expansion of {L(f n )}n|r for k ≥ 3. Thus we need at least #G basic expansions to get the equality (7): (8) L(f n ) = c1 (n) + ... + c#G (n) for n|r, n > 1. On the other hand notice that to get the above equality for n > 1 we can use tB , tC , tD sequences of the types (B), (C), (D) respectively, where tB , tC , tD are prescribed non-negative integers satisfying tB + tC + tD = #G. Since the contribution of each of these sequences to b1 = L(f ) is +1, −1, 0 respectively, we may choose them to get the equality (8) also for n = 1 if and only if −#G ≤ L(f ) ≤ #G. Otherwise, i.e. for |L(f )| > #G, we have to use one additional sequence of the type (A) with a2 = 0 to get the equality for n = 1. This ends the proof of part (∗). Now we assume that r is even. We need at least #G − #H basic expansions to get the equality (7) for r = 2, 4. In fact, only sequences (F) and (G) reduce two regk ’s, k = 1, 2, 4, and we may use them in such a way at most #H times ((F’) and (G’) also reduces two regk ’s but each of them is counted twice). To reduce the remaining regk ’s we will need #G − 2#H of them. Thus Dr3 (f ) ≥ #H + (#G − 2#H ) = #G − #H . On the other hand notice that we may always realize {L(f n )}n|r by #H sequences (F), #G − 2#H sequences (D), one sequence (A) with the coefficients a1 = b1 − #H , a2 = b2 and one sequence (D) with d = 4 and a4 = b4 . Thus Dr3 [f ] ≤ #G − #H + 2. 2 Now we consider some particular cases, which are not included in Theorem 4.10 Proposition 4.11. 1. If {L(f n )}n|r is constantly equal to 0, then Dr3 [f ] = 0. Assume that {L(f n )}n|r is not constantly equal to 0, then “fm˙08˙01” — 2008/6/24 — 14:42 — page 14 — #14 14 2. D23 (f ) = 1. 3. D43 (f ) = G, Graff, J. Jezierski 1 if (a) or (b) or (c) is satisfied, 2 otherwise, where (a) L(f ) = 1 and L(f 2 ) = −1, (b) L(f 4 ) = L(f 2 ), (c) L(f 2 ) = L(f ) and L(f ) ∈ {−1, 0, 1}. Proof. 1. L(f n ) ≡ 0 iff the sequence {L(f n )}n|r is the sum of zero basic sequences iff Dr3 [f ] = 0. 2. Each nonzero sequence of two elements b1 , b2 is of the type (A). 2 ) 3. Let us find when the sequence of three elements b1 = L(f ), b2 = L(f )−L(f , b4 = 2 L(f 4 )−L(f 2 ) 4 is one of the types (A)-(G): (A) ⇐⇒ b4 = 0, which gives the condition (b). (B), (C) or (D) ⇐⇒ (|b1| ≤ 1 and b2 = 0) ⇐⇒ (|L(f )| ≤ 1 and L(f ) = L(f 2 )) which gives the condition (c). (E) ⇐⇒ (b1 = 1 and b2 = −1) ⇐⇒ (L(f ) = 1 and L(f 2 ) = −1, which leads to condition (a). (F) and (G) are reduced to (A). It remains to notice that each sequence b1 , b2, b4 is the sum of two sequences of the types (A), (D) which implies D43 (f ) = 2 in the remaining cases. 2 Corollary 4.12. Let r = 1. As D13 [f ] ≤ 1, each map f is homotopic to a smooth map with (at most) one fixed point, in other words MPs1 (f ) ≤ 1 (compare Remarks 3.9 and 3.2 for r = 1). Let us remind the reader that by ζ (r) we denote the number of all divisors of r. Corollary 4.13. Let r ∈ N. Each map f is homotopic to a smooth map g with #Fix(g r ) ≤ ζ (r). #Br (f ) m+1 2[ 2 ] ≤ Proof. We begin with the proof of the second inequality. By definition, G = Br (f ) \ {1, 2, 4}, on the other hand Br (f ) is the number of basic sequences bk regk in the expansion of Lefschetz numbers and thus #Br (f ) ≤ ζ (r). By Proposition 4.2 and Theorem 4.10 Drm [f ] ≤ Dr3 [f ] ≤ ζ (r). Finally we apply Theorem 3.8 and get the desired inequality. The first inequality in Corollary results from Theorem 4.5 and Theorem 3.8. 2 Corollary 4.14. Let r = q be a prime number. Then the map f is homotopic to a smooth map g with #Fix(g r ) ≤ 1 if and only if at least one of the conditions holds: 1. q = 2, 2. |L(f )| ≤ 1, 3. L(f q ) = L(f ). Proof. For q = 2 the equivalence results from part 2 of Proposition 4.11. Let us consider now prime q > 2. By Corollary 4.13 Dr3 [f ] ≤ 2. We must find all conditions in Theorem 4.10 part (∗) under which Dr3 [f ] ≤ 1. We get that Dr3 [f ] ≤ 1 is equivalent to the following “fm˙08˙01” — 2008/6/24 — 14:42 — page 15 — #15 Minimal number of periodic points for C 1 maps 15 alternative: either #G ≤ 1 and L(f ) ∈ [−#G, #G], or #G = 0. The first possibility leads to q ) |L(f )| ≤ 1 and the second, by the definition of G, forces bq = L(f )−L(f = 0. 2 2 4.2 Calculation of D3r [f ] for self maps of S 2 × I As in the previous sections we assume that M is a smooth compact connected and simplyconnected 3-dimensional manifold. Now, assume that additionally M has non-empty boundary. We remind the reader that for a manifold with boundary we consider self-maps which have no periodic points on the boundary. We take homology with the coefficients in the field of rational numbers Q. Notice that connectivity of M implies that H0 (M ) = Q, furthermore H1 (M ) = 0 as M is simply-connected. Because M is a manifold with boundary, H3 (M ) = 0. Thus the homology spaces of M have the following form: H0 (M ) = Q, H1 (M ) = 0, H2 (M ) = Q i , H3 (M ) = 0. Equivalently, the boundary of M consists of i + 1 pairwise disjoined 2-spheres (cf. [Gr]). In this section we will give the complete description of the least number of periodic points of smooth maps, up to homotopy, in case i = 1, i.e. for M = S 2 × I , the only connected and simply-connected 3-manifold with boundary and H2 (M ) = Q. We define global Dold coefficients in (f ) by the formula: in (f ) = ∑ μ (n/k)L(f k ). k |n Recall that in (f )/n = bn (f ), where bn (f ) = bn is the nth coefficient of the periodic expansion of the sequence {L(f n )}∞ n=1 in the formula (7) (cf. also Theorem 2.3). Let f : M → M . Then the homomorphism f∗2 : H2 (M ; Q) → H2 (M ; Q) is a multiplication by a number β ∈ Z (called the degree of f ). Assume r ∈ N is fixed. Then, finding the least cardinality of Fix(g r ), where g is a smooth map homotopic to f , is equivalent to determining the number Dr3 [f ] = Dr3 [{L(f n )}n|r ], where L(f n ) = 1 + β n . We will need the exact form of the basic set Br (f ) (cf. Definitions 2.4 and 2.8). This knowledge is provided by the following result proved in [LPR]. Lemma 4.15. ([LPR] Theorem 1.2) (a) b1 (f ) = 1 + β . (b) b2 (f ) = 0 if and only if β ∈ {0, 1}. (c) If n > 2, then bn (f ) = 0 if and only if β ∈ {−1, 0, 1}. 2 Corollary 4.16. If |β | > 1, then B(f ) = N, so Br (f ) = {n ∈ N : n|r}, or equivalently #Br (f ) = ζ (r), where ζ (r) is the number of all divisors of r. 2 Under the assumption that |β | > 1 Corollary 4.16 implies: Corollary 4.17. For r odd G = Br (f ) \ {1}, thus #G = ζ (r) − 1. 2 Corollary 4.18. For r even there is: if 4|r, then G = Br (f ) \ {1, 2, 4}, thus #G = ζ (r) − 3, if 4 |r, then G = Br (f ) \ {1, 2}, thus #G = ζ (r) − 2. 2 “fm˙08˙01” — 2008/6/24 — 14:42 — page 16 — #16 16 G, Graff, J. Jezierski Notice that #H is the number of pairs k, 2k, where k|r, k > 1 is odd, in G. By Lemma 4.15 each n, such that n|r, belongs to G, thus we obtain for even r: Corollary 4.19. #H= η (r) − 1, where η (r) is the number of all odd divisors of r. Taking into account that L(f ) = 1 + β we get, by Theorem 4.10 and Corollary 4.17, the following description of Dr3 [f ] for odd r and all f with the absolute value of the degree greater than 1: Theorem 4.20. If r is odd, |β | > 1 then: ζ (r) − 1 if − ζ (r) ≤ β ≤ ζ (r) − 2, 3 Dr [f ] = ζ (r) otherwise. Example 4.21. Let f : M → M have the degree β ∈ {−1, 0, 1} and r = 15. The Lefschetz numbers L(f n ) for n|15 are: 1 + β , 1 + β 3 , 1 + β 5 , 1 + β 15 . We represent this sequence (for n = 1, 3, 5, 15) in the form of a periodic expansion. (9) L(f n ) = (1 + β )reg1 (n) + ( +( β3 −β β5−β )reg3 (n) + ( )reg5 (n) 3 5 β 15 − β 5 − β 3 + β )reg15 (n). 15 Now we apply Theorem 4.20: ζ (15) = 4, r = 15 is odd, hence: 3 for β ∈ {−4, −3, −2, 2}, 3 D15 [f ] = 4 for β < −4 or β > 2. 2 Proposition 4.2 allows us to derive some additional information in the case of higher dimensional manifolds. Example 4.22. Let f be a self-map of a smooth compact connected and simply-connected m-dimensional manifold M̃ , such that {L(f n )}n|15 is the same as in Example 4.21, i.e. L(f n ) are expressed by the right hand-side of the formula (9). By Proposition 4.2 there is a C 1 map g homotopic to f such that Fix(g 15 ) has at most 3 points if β ∈ {−4, −3, −2, 2} and at most 4 points if β < −4 or β > 2. Moreover, if m = 3 then Fix(g 15 ) cannot have less points than this number. 2 Now we analyze the case of r even. Theorem 4.23. Assume that r is even and |β | > 1. Then Dr3 [f ] = ζ (r) − η (r). Proof. For r = 2, 4 the thesis follows from Proposition 4.11. Assume r > 4. First we consider the case 4|r. By Corollaries 4.18 and 4.19, we may replace in the thesis ζ (r) − η (r) by #G − #H + 2. On the other hand, by Theorem 4.10, Dr3 [f ] ∈ [#G − #H , #G − #H + 2]. “fm˙08˙01” — 2008/6/24 — 14:42 — page 17 — #17 Minimal number of periodic points for C 1 maps 17 Let (10) L(f n ) = b1 reg1 (n) + b2reg2 (n) + b4 reg4 (n) + ∑ bk regk (n) = ∑ci (n), k ∈G i where ci i |r) sequences which give minimal decomposition of Lefschetz numbers of iterations, i.e. ∑i pi = Dr3 [f ]. We remind the reader that by Lemma 4.8 ci have one of the forms (A)-(G), (E’)-(G’) (thus pi ≤ 2). Suppose that: are DD3 (p (11) Dr3 [f ] = #G − #H . We show that this cannot happen. The equation (11) forces that the coefficients b1 , b2 , b4 , of the periodic expansion of {L(f n )}n|r , must be obtained as a sum of coefficients a1 , a2 , a4 , respectively; of #G − #H sequences (B)-(G) and (E’)-(G’), where the last are counted twice and where each involve ad regd with d ∈ G. This is impossible because any sum of such sequences gives always non-positive contribution to b4 ((E’) and (G’) allow to produce 4 2 −reg4 ), while b4 = β −4 β > 0. As a result we must use an additional sequence, say cα , of the type (B), (C), (D), or (E), with the coefficient a4 = 0, which makes possible to reduce b4 reg4 . Thus we get: (12) Dr3 [f ] ∈ [#G − #H + 1, #G − #H + 2]. Suppose now that Dr3 [f ] = #G − #H + 1. As we require minimal realization, (E’) is not valid, because it reduces only one basic sequence bk regk , k ≥ 6, but is counted twice. By the same reason (F’), (G’) could be used only under the condition that each of them reduces two basic sequences. Now we show that (F’), (G’) may always be replaced in the minimal realization by sequences of the types (A)-(G). For odd d (d > 1, 4d|r) we consider a triple: (13) bd regd (n) + b2d reg2d (n) + b4d reg4d (n), which appears in the periodic expansion of Lefschetz numbers of iterations. By Lemma 4.15 each coefficient near the basic sequences in the formula (13) is nonzero. Notice that (F’), (G’) may be used only if each of them reduces a part of such triple (two last terms). Then we have to use one sequence (B)-(E) to reduce bd regd , which gives the contribution 2 + 1 = 3 to Dr3 [f ]. On the other hand, we may use one (F) or (G) to reduce two first terms and one sequence (B)-(E) to reduce the last and one sequence (A) with a1 = b1 and a2 = b2 , getting the same contribution 1 + 1 + 1 = 3, independently of the values of b1 and b2 . Thus in the minimal realization, we may consider only sequences (A)-(G). We need at least #G − #H sequences (B)-(G) to reduce bk regk for k = 1, 2, 4 (cf. also proof of Theorem 4.10), and one cα of the type (B)-(E) to reduce b4 reg4 . Finally, we need 2 also one sequence of the type (A) to reduce b2 reg2 with b2 = β 2−β > 0, because (B)-(G) have coefficients a2 non-positive. As a result Dr3 [f ] = #G − #H + 2. Now consider the case 4 r. By Corollary 4.17 and the second part of Corollary 4.18 we know that ζ (r) − η (r) = #G − #H + 1. Thus, to finish the proof, it is enough to state that Dr3 [f ] = #G − #H + 1. “fm˙08˙01” — 2008/6/24 — 14:42 — page 18 — #18 18 G, Graff, J. Jezierski Because b4 = 0, we see that Dr3 [f ] ∈ [#G − #H , #G − #H + 1] (compare the proof of Theorem 4.10). On the other hand, if Dr3 [f ] = #G − #H , then b2 > 0 must be a sum of a2 -coefficients of #G − #H sequences of minimal realization. This is impossible, because the only sequences which may be used to realize b2 in such a way, are of the types (E’), (F’) or (G’), but in the case 4 r each of them reduces only one sequence (for k = 1, 2) in the expansion of {L(f n )}n|r , but is counted twice. This ends the proof. 2 Proposition 4.24. In case β ∈ {−1, 0, 1} Dr3 [f ] = 1 as we can use one DD3 (1|r) sequence of the type (A). Namely: 3 β {L(f n )}∞ n=1 Dr [f ] 0 reg1 (n) 1 1 2reg1 (n) 1 −1 reg2 (n) 1 Theorems 4.20 and 4.23 together with Proposition 4.24 give the complete description of the least number of periodic points of smooth self-maps, up to homotopy, for S 2 × I . References [BaBo] Babenko I. K., Bogatyi S. 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Benjamin, Menlo Park, CA 1971 [HK1] Heath P., Keppelmann E.: Fibre techniques in Nielsen periodic point theory on nil and solvmanifolds. I. Topology Appl. 76 (1997), 217–247 [HK2] Heath P., Keppelmann E.: Fibre techniques in Nielsen periodic point theory on nil and solvmanifolds. II. Topology Appl. 106 (2000), 149–167 [Je1] Jezierski J.: Weak Wecken’s Theorem for periodic points in dimension 3. Fund. Math. 180 (2003), 223–239 [Je2] Jezierski J.: Wecken’s theorem for periodic points in dimension at least 3. Topology Appl. 153, (2006), 1825–1837 [Ji] Jiang B. J: Lectures on the Nielsen Fixed Point Theory. Contemp. Math. 14. Amer. Math. Soc., Providence 1983 [JM] Jezierski J., Marzantowicz W.: Homotopy methods in topological fixed and periodic points theory. Topological Fixed Point Theory and Its Applications 3. Springer, Dordrecht 2005 “fm˙08˙01” — 2008/6/24 — 14:42 — page 19 — #19 Minimal number of periodic points for C 1 maps [LPR] [MP] [R] [SS] 19 Llibre J., Paranõs J., Rodriguez J. A.: Periods for transversal maps on compact manifolds with a given homology. Houston J. Math. 24 (1998), 397–407 Marzantowicz W., Przygodzki P.: Finding periodic points of a map by use of a k-adic expansion. Discrete Contin. Dynam. Systems 5 (1999), 495–514 Robinson C.: Dynamical systems. Stability, symbolic dynamics, and chaos. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL 1999 Shub M., Sullivan P.: A remark on the Lefschetz fixed point formula for differentiable maps. Topology 13 (1974), 189–191 Received July 20, 2007; revised December 4, 2007; in final form December 10, 2007 Grzegorz Graff, Faculty of Applied Physics and Mathematics, Gdansk University of Technology, ul. Narutowicza 11/12, 80-952 Gdansk, Poland [email protected] Jerzy Jezierski, Institute of Applications of Mathematics, Agricultural University of Warsaw (SGGW), 00-665 Warsaw, Poland [email protected] “fm˙08˙01” — 2008/6/24 — 14:42 — page 20 — #20