Nielsen type numbers and homotopy minimal periods for maps on 3 solvmanifolds

Transcription

1 Algebraic & Geometric Toology 8 (8) Nielsen tye numbers and homotoy minimal eriods for mas on solvmanifolds JONG BUM LEE XUEZHI ZHAO For all continuous mas on solvmanifolds, we give exlicit formulas for a comlete comutation of the Nielsen tye numbers NP n.f / and N ˆn.f /. The most general cases were exlored by Heath and Keelmann [] and the comlementary art is studied in this aer. While studying the homotoy minimal eriods of all mas on solvmanifolds, we give a comlete descrition of the sets of homotoy minimal eriods of all such mas, including a correction to Jezierski, Kȩdra and Marzantowicz s results in []. M; S 1 Introduction In Nielsen fixed oint theory, there are well known invariants which are the Lefschetz number L.f / and the Nielsen number N.f /. It is known that the Nielsen number is much more owerful than the Lefschetz number but comuting it is very hard. For eriodic oints, two Nielsen tye numbers NP n.f / and N ˆn.f / were introduced by Jiang, which are lower bounds for the number of eriodic oints of least eriod exactly n and the set of eriodic oints of eriod n, resectively. It is obvious that these Nielsen numbers are much more owerful than the Lefschetz number in describing the eriodic oint sets of self-mas, but the comutation of those homotoy invariants is difficult in general. However using fiber techniques on nilmanifolds and some solvmanifolds, Heath and Keelmann [] (see also Keelmann and McCord [8]) succeeded in showing that the Nielsen numbers and the two Nielsen tye numbers are related to each other under certain conditions. One of the natural roblems in dynamical systems is the study of the existence of eriodic oints of least eriod exactly n. Homotoically, a new concet, namely homotoy minimal eriods, HPer.f / D \ fm j g m.x/ D x; g q.x/ x; q < m for some xg g'f Published: 1 May 8 DOI: 1.14/agt.8.8.6

2 64 Jong Bum Lee and Xuezhi Zhao was introduced by Alsedà, Baldwin, Llibre, Swanson and Szlenk in [1]. Since the homotoy minimal eriod is reserved under a small erturbation of a self-ma f on a manifold X, we can say that the set HPer.f / of homotoy minimal eriods of f describes the rigid art of the dynamics of f. A comlete descrition of the set of homotoy minimal eriods of all self-mas was obtained on nilmanifolds by Jezierski and Marzantowicz [6], and Lee and Zhao [1] and on solvmanifolds by Jezierski, Kȩdra and Marzantowicz []. There are six simly connected dimensional unimodular Lie grous: the abelian Lie grou R, the nilotent Lie grou Nil, the solvable Lie grou Sol, the universal covering grou ze./ of the connected comonent of the Euclidean grou, the universal covering grou e PSL.; R/ of the secial linear grou and the secial unitary grou SU./. There are also infinitely many simly connected dimensional non-unimodular solvable Lie grous, see Ha and Lee []. It is easy to see that the solvable grou ze./ is not of tye.nr/ (see the next section for the definition). Since a solvable Lie grou which admits a lattice is unimodular, the non-unimodular solvable Lie grous do not concern us. In this aer, we are concerned with closed manifold nsol which are quotient saces of Sol by its lattices, and continuous mas between them. Those manifolds will be called solvmanifolds with Sol-geometry or in short solvmanifolds. Jezierski and Marzantowicz in [6], and the authors in [1] studied homotoy minimal eriods for mas on manifolds with Nil-geometry. In [] (see also Kim, Lee and Yoo [9]), Jezierski, Kȩdra and Marzantowicz carried out a further study of homotoy minimal eriods for mas on solvmanifolds. One of motivations of the resent work is to correct Proosition 4. of [], in which the entries of the matrices are claimed to be integers. However this is not true; see Examle.6. Every continuous ma on nsol which is induced from a homomorhism on Sol must reserve the lattice. This will rovide the necessary conditions for those entries; see Remark.. The lattices of Sol The Lie grou Sol is one of the eight geometries that one considers in the study of manifolds, see Scott [11]. One can describe Sol as a semi-direct roduct R Ì ' R where t R acts on R via the ma: e t '.t/ D e t Algebraic & Geometric Toology, Volume 8 (8)

3 Nielsen tye numbers and homotoy minimal eriods 6 Its Lie algebra sol is given as sol D R Ì R where: s.s/ D s The Lie grou Sol can be faithfully reresented into Aff./ by e t x 6 e t y 4 1 t 1 where x, y and t are real numbers, and hence its Lie algebra sol is isomorhic to the algebra of matrices: t a 6 t b 4 t Now we will discuss the existence of a lattice in Sol (cf Thurston [1, Theorem 4..1] or Scott [11, age 4]). Lemma.1 of the form A grou is a lattice of Sol D R Ì ' R if and only if it is an extension 1! Z!! Z! 1 where the action of Z on Z is generated by an element A of SL.; Z/ with trace greater than. Proof Suose is a lattice. Since R is the nil-radical of R Ì ' R, \ R is a lattice in R. Thus is of the form Z Ì Z, where Z D ht i R. We obtain that kx 1 C lx ˆ< D 6 1 ky 1 C ly >= 4 1 ˇ k; l Z Ì ' ht i ˆ: >; 1 8 e ˆ< mt 9 kx 1 C lx D 6 e mt ky 1 C ly >= 4 1 mt ˆ: ˇ k; l; m Z >; 1 Algebraic & Geometric Toology, Volume 8 (8)

4 66 Jong Bum Lee and Xuezhi Zhao where.x 1 ; y 1 / and.x ; y / are linear indeendent, and t. Since is a subgrou of Sol, the elements e t 1 x i e t e t x i 6 e t 6 1 y i 4 1 t 4 1 D 6 e t e t y i 4 1 t must lie in. Hence, there are integers `11 ; `1 ; `1 ; ` such that e t x i e t y i x1 x D `1i C `i ; i D 1; : y 1 y.i D 1; / This imlies that: e t e t x1 x y 1 y D x1 x y 1 y `11 `1 `1 ` By a comutation, we have D Z Ì Z, where the action of Zon Z is generated by: `11 `1 e t A D Since A is conjugate to e t, the determinate of A is 1, ie, `1 ` A SL.; Z/.The trace of A is `11 C ` D e t C e t and hence is greater than. `11 `1 Let us consider the converse. Given a matrix A D SL.; Z/ with trace `11 C ` >, A has two distinct irrational eigenvalues `11C` D, where D D.`11 C ` / 4. With two corresonding eigenvectors, we form a real invertible " # `11 ` CD `11 ` D matrix P D `1 `1. Then we have: 1 1 The semi-roduct P 1 AP D 8 e mt k `1 D ˆ< C l ` 6 e mt k `1 D C l ` mt ˆ: 1 `1 " `11 C` CD `11 C` `11 CD D ` CD D ` D # 9 >= ˇ k; l; m Z D Z Ì A Z >; Algebraic & Geometric Toology, Volume 8 (8)

5 Nielsen tye numbers and homotoy minimal eriods 6 of the linear san of the column vectors " `1 D `1 D # and " ` `11 CD D `11 ` CD D # of P 1 and Z D ht i, where t D ln `11C` C.`11 C` / 4, forms a lattice of the Sol grou R Ì ' R. Immediately, we have: Corollary. Any lattice of Sol D R Ì ' R has a resentation where A D ha; b; s j Œa; b D 1; sas 1 D a`11 b`1 ; sbs 1 D a`1 b` i; `1 SL.; Z/ and `11 `1 ` 1 x 1 1 x e t a D 6 1 y b D 6 1 y 4 1 s D 6 e t 4 1 t satisfying 1 x1 x e t `11 y 1 y e t D `1 `1 ` x1 x y 1 y 1 1 ie, e t and e t x1 x are two irrational eigenvalues of A and consists of two y 1 y eigenvectors of A. For an element A D `11 `1 `1 ` SL.; Z/ generating the action of Z on Z, since its trace is greater than, A must have different real eigenvalues: one is greater than 1, the other is less than 1. Such a matrix is said to be a hyerbolic element of SL.; Z/. We shall write A D Z Ì A Z or D Z Ì Z for such a which fits in the extension 1! Z!! Z! 1. Furthermore, neither `1 nor `1 vanishes. In fact, if `1`1 D, the determinant of A would be `11`. Since A SL.Z/, `11` D 1. The trace of A would be `11 C ` D. This contradicts Lemma.1. Now we will discuss the homomorhisms on the lattices of Sol. Algebraic & Geometric Toology, Volume 8 (8)

6 68 Jong Bum Lee and Xuezhi Zhao Theorem. Let D ha; b; s j Œa; b D 1; sas 1 D a`11 b`1 ; sbs 1 D a`1 b` i be a lattice of Sol. Then ƒ D ha; bi D Z is a fully invariant subgrou of, ie, every homomorhism on restricts to a homomorhism on ƒ. Proof Consider the commutator subgrou Œ ; of. Since every element of is of the form a m b n s`, we have: Œ ; D hœa m b n s`; a m b n s` j m; m ; n; n ; `; ` Zi D ha m m b n n `.a m b n /`.a m b n / j m; m ; n; n ; `; ` Zi where the homomorhism W! is given by.a/ D a`11b`1 and.b/ D a`1b`. Thus Œ ; ƒ, and =Œ ; is generated by xa D aœ ; ; b x D bœ ; ; xs D sœ ;. Since det.i A/, the rank of Œ ; is. Thus hxa; bi x generates the torsion subgrou of =Œ ; D hxa; bi x hxsi, and note also that the inverse image of hxa; bi x under the natural homomorhism W! =Œ ; is the subgrou of generated by a and b. Thus ƒ D 1.hxa; bi/. x Since Œ ; is fully invariant in, and since hxa; bi x is fully invariant in =Œ ;, we see that ƒ is fully invariant in. Theorem.4 Let D ha; b; s j Œa; b D 1; sas 1 D a`11 b`1 ; sbs 1 D a`1 b` i be a lattice of Sol. Then any homomorhism on is one of the following: Tye (I).a/ D a u b `1 `1 v,.b/ D a v b uc ` `11 `1 v,.s/ D a b q s Tye (II).a/ D a u b v `11 ` u,.b/ D a `1 v `1 `1 b u,.s/ D a b q s 1 Tye (III).a/ D 1,.b/ D 1,.s/ D a b q s m with m 1 Proof Let W! be a homomorhism on the lattices of Sol. Theorem. imlies that.a/ D a m 11 b m 1 ;.b/ D a m 1 b m ;.s/ D a b q s m where m 11 ; m 1 ; m 1 ; m ; ; q; m Z. Let W! be the homomorhism given by.a/ D a`11b`1 and.b/ D a`1b` so that is the conjugation by s. Note that must reserve the relations sas 1 D.a/ and sbs 1 D.b/. Since.s/.a/.s/ 1 D.a b q s m /.a m 11 b m 1 /.a b q s m / 1 D m.a m 11 b m 1 / Algebraic & Geometric Toology, Volume 8 (8)

7 Nielsen tye numbers and homotoy minimal eriods 69.s/.b/.s/ 1 D.a b q s m /.a m 1 b m /.a b q s m / 1 D m.a m 1 b m / we must have m.a m 11b m 1/ D..a//, m.a m 1b m / D..b//. These equations are equivalent to: m `11 `1 m11 m 1 m11 m D 1 `11 `1 `1 ` m 1 m m 1 m `1 ` By Lemma.1, there is a real invertible matrix `11 ` C.`11 C` / 4 `11 ` `1.`11 C` / 4 P D 4 `1 1 1 e such that P 1 t AP D e t, where t D ln `11C` C.`11 C` / 4 is a ositive real number. Thus, we have: e t m P e t P 1 m11 m 1 m11 m D 1 e t P m 1 m m 1 m e t P 1 Let Q denote P 1 m11 m 1 P. It follows that: m 1 m e mt e t e mt Q D Q e t There are three ossibilities: (1) m D 1, Q D ; ˇ () m D 1, Q D ; ı () m 1, Q D. They yield our three tyes. From this Theorem, there is a one-to-one corresondence from the set of homomorhisms on the lattice Z Ì A Z to the subset of Z : D f.u; v; ; q; 1/ Z j `1 v; ` `11 v Zg `1 `1 S f.u; v; ; q; 1/ Z j `11 ` `1 u v Zg `1 `1 S f.; ; ; q; m/ Z j m 1g Algebraic & Geometric Toology, Volume 8 (8)

8 Jong Bum Lee and Xuezhi Zhao We shall write.u;v;;q;m/ for the homomorhism on Z Ì A Z corresonding to the integer vector.u; v; ; q; m/, which is indicated in above theorem. By a direct comutation, we have: Proosition. Two homomorhisms.u;v;;q;m/ and.u ;v ; ;q ;m / on Z Ì A Z are conjugate if and only if one of the following holds for some integers k; k ; k :.1/ m D m D 1, Œu v D Œu v A k k, q D A k C.I A/ q k k u./ m D m D 1, v D A k u v./ m D m 1, u D u D v D v D, Continuous mas on nsol, q D A k q D A k q q C.I A 1 / C.I A m / k k Let f W nsol! nsol be a continuous ma. U to a homotoy, we may assume that f reserves the base oint of nsol, where is the unit of Sol. Then f induces a homomorhism f W 1. nsol; /! 1. nsol; /. Note that 1. nsol; / is identified with in a classical way. We can extend this homomorhism on in an obvious way to a homomorhism on Sol D R Ì ' R. Since the Lie grou homomorhism W Sol! Sol sends into, it induces a ma on the -solvmanifold nsol. Note that a homomorhism on induced by the ma is the same as the homomorhism induced by f W nsol! nsol. Since a solvmanifold nsol is asherical, we see that f is homotoic to. In all, we have a classification theorem of self mas on nsol. Theorem.1 Let D Z Ì A Z be a lattice of Sol, where A D `11 `1 `1 ` k. Then any continuous ma on nsol is homotoic to a ma W nsol! nsol induced from a homomorhism W Sol! Sol with the restriction.u;v;;q;m/ W! in Theorem.4. Moreover, two homomorhisms on Sol induce homotoic mas on nsol if and only if their restrictions on satisfy one of relations in Proosition.. Since the invariants that we are going to deal with are all homotoy invariants, we will assume in what follows that every continuous ma on nsol is induced by a homomorhism on Sol. Algebraic & Geometric Toology, Volume 8 (8)

9 Nielsen tye numbers and homotoy minimal eriods 1 Definition. (Keelmann and McCord [8]) Let f W nsol! nsol be a continuous ma inducing a homomorhism W!. Then there is a commutative diagram: 1! ƒ!! =ƒ! 1? yy??y??y x 1! ƒ!! =ƒ! 1 The induced homomorhisms " # W x Z! Z and W y Z! Z is called a linearization of y f, denoted F D x. By Theorem.4, we obtain: Proosition. Let f W nsol! nsol be a continuous ma, where D Z Ì A Z `11 `1 with A D. Then any linearization matrix is an integer matrix of one of the following `1 ` u v u `11 ` `1 u v 6 `1 4 v u C ` `11 6 `1 `1 v ; 4 v u ; 4.m 1/ `1 `1 1 1 m which are resectively conjugate to:.i/.ii/.iii/ u u C `11 `11 `.`11 C` / 4 ` C 4.m 1/ m v `1 `11 ` C.`11 C` / u 4 v ; `1 1 u C `11 `.`11 C` / 4 v `1.`11 C` / 4 v ; `1 1 Algebraic & Geometric Toology, Volume 8 (8)

10 Jong Bum Lee and Xuezhi Zhao Proof The three integer matrices are induced from Theorem.4. After a conjugation T 1./T by the following matrix: 6 T D 4 `11 we arrive at our conclusion. ` C.`11 C` / 4 `11 `.`11 C` / 4 `1 ` Note that homomorhisms of distinct tyes are not conjugate to each other. We shall say a ma on nsol is of tye (I), (II) or (III) according to its homomorhism on. Corollary.4 Each linearization matrix of a continuous ma on nsol is conjugate to one of the following ().I/ 4 ˇ ;.II/ 4ı ;.III/ 4. 1/; 1 1 where C ˇ; ˇ; ı; Z. Moreover, D if and only if ˇ D, and D if and only if ı D. `11 Proof Since the trace `11 C `1 of `1.`11 C ` / 4 must be an irrational number. In tye (I), if D u `11 `.`11 C` / 4 `1 `1 ` is an integer greater than, v D, then v must be zero and hence u D because u; v; `11 ; `1 ; ` D are all integers. It follows that ˇ D D. The converse is the same. The roof of the relation between and ı in tye (II) is similar. In fact, nsol is a torus bundle over the circle. It is known that any ma on nsol is homotoic to a fibre ma with resect to the above bundle structure. Then x is the degree of the induced ma on the base sace S 1. By Proosition. and Theorem.1, we obtain: Proosition. Let f and f be two continuous mas on nsol induced by homomorhisms and " # " # y y on Sol, with linearization matrices x and x, where D Z Ì A Z. If f and f are homotoic, then y D A k y and x D x. Algebraic & Geometric Toology, Volume 8 (8)

11 Nielsen tye numbers and homotoy minimal eriods By this result, det and x are both homotoy invariants. A ma is of tye (I), (II) or (III) according to x is 1, 1 or the others. It should be noticed that the y is not a homotoy invariant. The following is an exlicit examle of mas on nsol and their linearizations. 1 Examle.6 Let D Z Ì A Z be a lattice of Sol, where A D is an element 1 1 " 1C 1 # of SL.; Z/ with trace and eigenvalues. We take P to be. 1 1 Then: # P 1 AP D " C 1 1 D 4 C eln C ln e Three generators of, as elements in Sol, are: a D 6 4 ; b D e s D 6 4 C ln 1 C1 1 1 e C ln 1 ln C 1 The linearization matrix F of any ma on nsol induced by a homomorhism W Sol! Sol is one of the following u v u v C u 4v u v ; 4 v u ; 4.m 1/, 1 1 m where u; v; m are integers. The corresonding matrices P are re- 1 sectively: u P F 1 v u C 1 v 1C 6 u v ; 4u C 1C v ; m Algebraic & Geometric Toology, Volume 8 (8)

12 4 Jong Bum Lee and Xuezhi Zhao Remark. The entries ; ˇ; ; ı in the linearization F of tye (I), (II) (see the list ()) need not be integers. The above examle shows in articular that the only conditions for which we can imose are: C ˇ; ˇ; ı; Z Therefore the conditions for the entries in [, Proosition 4.] should be changed as above. Notice that a linearization F in the form of () with all entries integers is ossible only when v D if F is of tye (I) or (II). The above examle also shows that all homomorhisms on Sol need not reserve the lattice A D Z Ì A Z and hence need not induce continuous mas on A nsol. 4 Lefschetz numbers and Nielsen tye numbers A connected solvable Lie grou G is called of tye.nr/ (for no roots ) if the eigenvalues of Ad.x/W G! G are always either equal to 1 or else they are not roots of unity. Solvable Lie grous of tye.nr/ were considered first in Keelmann and McCord [8]. Since our solvmanifold nsol is of tye.nr/, the following is a direct consequence of the main result, Theorem.1, of [8]. Proosition 4.1 Let f W nsol! nsol be any continuous ma on the solvmanifold " # y nsol with linearization matrix F D x. Then for all ositive integers n: and N.f n / D jl.f n /j. 8 ˆ< if f is of tye.i/ L.f n / D.1. 1/ ˆ: n /.1 C.det / y n / if f is of tye.ii/ 1 xn if f is of tye.iii/ Proof First, we consider the case n D 1. Since nsol is of tye.r/ and hence of tye.nr/, by [8, Theorem.1], we have L.f / D det.i F/, and N.f / D jl.f /j. It is sufficient to consider the Lefschetz number. If f is of tye (I), then x D 1. It follows that N.f / D L.f / D. If f is of tye (II), then x D 1. By Proosition., there is an invertible matrix P such that P 1 P y D. Hence, ı " #! y L.f / D det I 1 Algebraic & Geometric Toology, Volume 8 (8)

13 Nielsen tye numbers and homotoy minimal eriods D det I D P 1 " y # P! ı A 1 D.1 ı/ D.1 C det /: y If f is of tye (III), we have y D, and hence: 1 L.f / D 4 A D 1 x " # y For general n, f n has a linearization matrix n xn. The arguments are the same. It should be ointed out that f n is of tye (I) if f is of tye (II) and n is even. Recall from Proosition. that the numbers det y and x are indeendent of the choice of linearizations of f. Theorem 4. Let f W nsol! nsol be a continuous ma with a linearization matrix " # y F D x. We have.1/ If f is of tye.i/, then: NP n.f / D N ˆn.f / D./ If f is of tye.ii/, then: ( P NP n.f / D mjn.m/j1 C.det / y m j n is odd, n is even, N ˆn.f / D j1 C.det y / n j; n D r n ; n is odd:./ If f is of tye.iii/, then: NP n.f / D X mjn.m/j1 x m j; N ˆn.f / D j1 x n j x Algebraic & Geometric Toology, Volume 8 (8)

14 6 Jong Bum Lee and Xuezhi Zhao Proof In case (1), the Nielsen number N.f m / is for all ositive integer m. The ma f has no essential eriodic orbit classes of any eriod. It follows that NP n.f / D N ˆn.f / D. Since nsol is a solvmanifold of tye.nr/, by Heath and Keelmann [, Theorems 1.], we have N ˆm.f / D N.f m / and NP m.f / D P qjm.q/n.f m q / for all mjn rovided N.f n /. This roves our case () for odd n and case (), by Proosition 4.1. In case () for even n, by Proosition 4.1, we have N.f n / D. It follows that NP n.f / D. Let n D r n for odd n. By Proosition 4.1 again, f q has no essential fixed oint class for every even factor q of n. Thus, the set of essential fixed oint classes of f q with q j n is the same as the set of essential fixed oint classes of f q with q j n. Thus, N ˆn.f / D N ˆn.f /, which is just N.f n / D j1c.det y / n j. Homotoy minimal eriods HPer.f / In this section, we shall resent the homotoy minimal eriods for all mas on solvmanifolds. Theorem.1 Let f W nsol! nsol be a continuous ma with a linearization matrix " # y F D x. Then: 8 f1g if x D if x D 1 ˆ< if x D 1; det y D 1 HPer.f / D f1g if x D 1; det y D ; 1 N N if x D 1; j det j y > 1 N fg if ˆ: x D N otherwise Proof Since NP n.f / is a lower bound for the number of eriodic oints with least eriod n, by definition, we have HPer.f / fn j NP n.f / g. By Jezierski [4], we obtain HPer.f / D fn j NP n.f / g. Note that N.f n / D j det.i F n /j for any n. Using the same argument as in Jiang and Llibre [, Theorem.4], we have: fn j NP n.f / g D fn j N.f n / ; N.f n / N.f n q / for all rime q j n; q ng Algebraic & Geometric Toology, Volume 8 (8)

15 Nielsen tye numbers and homotoy minimal eriods Table 1 HPer.f / f1g N fg N N N Linearization matrix F is conjugate to 4 ˇ. C ˇ; ˇ Z/ or 4. / / or 4 or ; 1; ; 1/ 4ı. ı 1; ; 1; ı Z/ 1 Hence: () HPer.f / D fn j N.f n / ; N.f n / N.f n q / for all rime q j n; q ng (1) If x D, then N.f n / D 1 for all n by Proosition 4.1. Hence HPer.f / D f1g. () If x D 1, then f is of tye (I). We have N.f n / D for all n. This imlies that HPer.f / D. () If x D 1, then f is of tye (II). By Proosition 4.1, N.f n / D for even n. It follows that HPer.f / does not contain any even number, ie HPer.f / N N. Let us consider its subcases: (.1) If det y D 1, by Proosition 4.1, N.f n / D for all n. Thus, HPer.f / D. (.) If det y D, by Proosition 4.1, N.f n / D for all odd n. Since N.f /, we have 1 HPer.f /. By (), we have n 6 HPer.f / for all odd n with n > 1. Thus, HPer.f / D f1g. (.) If det y D 1, by Proosition 4.1, N.f n / D 4 for all odd n. Since N.f /, we have 1 HPer.f /. By (), we have n 6 HPer.f / for all odd n with n > 1. Thus, HPer.f / D f1g. Algebraic & Geometric Toology, Volume 8 (8)

16 8 Jong Bum Lee and Xuezhi Zhao Table Tye./ Linearization matrix HPer./ e t x (I) 6 e t y t (II) (II) (II) (III) (III) (III) e t 6 e t C 4 1 t 1 y x e t 1 y 6 e t 1C x 4 1 t 1 e t.1 /y 6 e t.1 C /x 4 1 t e t 6 e t 4 1 t 1 e t 6 e t 4 1 t f1g 1 4 N N 1 4 f1g 4 N fg 4 N (4) If x D, by Proosition 4.1, we have N.f n / D j1 x n j. Esecially, N.f / D N.f / D. By (), 1 HPer.f / but 6 HPer.f /. Note that N.f nc1 / > N.f n / for all n >. By (), HPer.f / D N fg. () In the final case, x ; 1; ; 1. We still have N.f n / D j1 x n j. Notice that N.f nc1 / > N.f n / for all n. By (), HPer.f / D N. Algebraic & Geometric Toology, Volume 8 (8)

17 Nielsen tye numbers and homotoy minimal eriods 9 In Table 1 we tabulate this result according to the Linearization matrix. In fact, for each subset S N aearing as HPer.f / and each form of linearization F listed above, there exists a self-ma f W nsol! nsol such that HPer.f / D S. Examles are given next. 1 Examle. Fix the lattice D Z Ì A Z of Sol, where A D. We list the 1 1 continuous mas on nsol by the exlicit lifting homomorhism W Sol! Sol, which e t 1 x are determined by./ D B6 e t y 4 1 t A 1 Hence for each subset S N aearing as HPer.f / and each form of linearization F listed in Table can be realized. Acknowledgments The authors would like to thank the referee for thorough reading and valuable comments in their original version. The first author was suorted in art by the Sogang University (KOREA) Research Grant in 6 and in art by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD) (H1, KRF-6-11-C16), and the second author was suorted by natural science foundation of China (1114) and a BMEC grant (KZ 11). References [1] L Alsedà, S Baldwin, J Llibre, R Swanson, W Szlenk, Minimal sets of eriods for torus mas via Nielsen numbers, Pacific J. Math. 169 (199) 1 MR1464 [] Y Ha, J Lee, Left invariant metrics and curvatures on simly connected threedimensional Lie grous, to aear in Mathematische Nachrichten [] P R Heath, E Keelmann, Fibre techniques in Nielsen eriodic oint theory on nil and solvmanifolds. I, Toology Al. 6 (199) 1 4 MR14414 [4] J Jezierski, Wecken s theorem for eriodic oints in dimension at least, Toology Al. 1 (6) MR9 [] J Jezierski, J Kȩdra, W Marzantowicz, Homotoy minimal eriods for NRsolvmanifolds mas, Toology Al. 144 (4) 9 49 MR91 [6] J Jezierski, W Marzantowicz, Homotoy minimal eriods for mas of threedimensional nilmanifolds, Pacific J. Math. 9 () 8 11 MR199 [] B Jiang, J Llibre, Minimal sets of eriods for torus mas, Discrete Contin. Dynam. Systems 4 (1998) 1 MR1616 Algebraic & Geometric Toology, Volume 8 (8)

18 8 Jong Bum Lee and Xuezhi Zhao [8] E C Keelmann, C K McCord, The Anosov theorem for exonential solvmanifolds, Pacific J. Math. 1 (199) MR199 [9] H J Kim, J B Lee, W S Yoo, Comutation of the Nielsen tye numbers for mas on the Klein bottle, to aear in J. Korean Math. Soc. [1] J B Lee, X Zhao, Nielsen tye numbers and homotoy minimal eriods for all continuous mas on the -nilmanifolds, Sci. China Ser. A 1 (8) 1 6 [11] P Scott, The geometries of -manifolds, Bull. London Math. Soc. 1 (198) MR [1] W P Thurston, Three-dimensional geometry and toology. Vol. 1, Edited by Silvio Levy, Princeton Mathematical Series, Princeton University Press, Princeton, NJ (199) MR149 Deartment of Mathematics, Sogang University, Seoul 11 4, KOREA Deartment of Mathematics, Institute of Mathematics and Interdiscilinary Science Caital Normal University, Beijing 1, CHINA jlee@sogang.ac.kr, Received: 6 February zhaoxve@mail.cnu.edu.cn Algebraic & Geometric Toology, Volume 8 (8)