minimal lefschetz sets of periods for morse–smale diffeomorphisms

8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms

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Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms

on the n-dimensional torus

Juan Luis Garca Guiraoa and Jaume Llibreb*

aDepartamento de Matematica Aplicada y Estadstica, Universidad Politecnica de Cartagena,Antiguo Hospital de Marina, 30203 Cartagena (Region de Murcia), Spain; bDepartament de

Matematiques, Universitat Autonoma de Barcelona, Bellaterra, 08193 Barcelona (Catalonia), Spain

(Received 16 July 2009; final version received 23 July 2009)

Dedicated to Robert Devaney on the occasion of his 60th birthday

We present a complete description of the minimal Lefschetz set of periods for everyhomological class of MorseSmale diffeomorphisms defined on the n-dimensionaltorus forn 1; 2; 3; 4, by using the Lefschetz zeta function. The techniques applied forobtaining these results also work for n . 4.

Keywords: periodic point; minimal Lefschetz set of periods; Morse Smalediffeomorphism

AMS Subject Classification: 58F20

1. Introduction and statement of the main results

We consider discrete dynamical systems given by a self-diffeomorphism fdefined on agiven compact manifold M. In this setting, usually the periodic orbits play an important

role. In dynamical systems, often the topological information can be used to study

qualitative and quantitative properties of the system. Perhaps, the best known example in

this direction is the results contained in the paper entitledPeriod three implies chaos for

continuous self-maps on the interval [17].

For continuous self-maps on compact manifolds, one of the most useful tools for

proving the existence of fixed points, or more generally of periodic points, is the Lefschetz

fixed point theorem and its improvements, see for instance [2,3,7 10,12,18,20].

The Lefschetz zeta function Zftsimplifies the study of the periodic points off. This is a

generating function for the Lefschetz numbers of all iterates off.In this work, we put our attention in the class of discrete smooth dynamical systems

defined by theMorseSmale diffeomorphisms on the tori.

We denote by Diff(M) the space ofC1 diffeomorphisms on a compact manifold M.

This space is a topological space endowed with the topology of the supremum with respect

to fand its differential Df. In this paper, all the diffeomorphisms will be C1.

We say that two diffeomorphisms f; g [ Diff(M) are topologically equivalent if andonly if there exists a homeomorphismh : M!M such thath +f g + h. A diffeomorphism

fisstructurally stableif there exists a neighborhood Uoffin Diff(M) such that eachg [ U

ISSN 1023-6198 print/ISSN 1563-5120 online

q 2010 Taylor & Francis

DOI: 10.1080/10236190903203887

http://www.informaworld.com

*Corresponding author. Email: [email protected]

Journal of Difference Equations and Applications

Vol. 16, Nos. 56, MayJune 2010, 689703


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is topologically equivalent to f. The class of MorseSmale diffeomorphisms is structurally

stable inside the class of all diffeomorphisms [23 25]. Therefore, it is interesting to

understand the dynamics of this class of diffeomorphims.

Many authors have published papers analysing the relationships between the dynamics

of the MorseSmale diffeomorphisms and the topology of the manifold where they aredefined, see for instance [4 6,10,11,13,14,21,24,26 28]. In the class of all diffeomorph-

isms, the MorseSmale have a relative simple dynamics. In particular, the sets of all their

periodic orbits are finite, and their structure is preserved under small perturbations.

Our main objective is to describe the minimal Lefschetz sets of periods of the

MorseSmale diffeomorphisms on the n-dimensional torus for n 1; 2; 3; 4, and showthat the algorithm used for obtaining these results extends to higher dimensions. In fact,

the results for the n-dimensional torus for n 1; 2 are well known, see [14] for thetwo-dimensional torus and [2] for the circle.

In Section 2, we provide the definitions of Lefschetz number and Lefschetz zeta

function. Additionally, we recall a fundamental result on this last function which works for

the MorseSmale diffeomorphisms (Theorem 2.1) and which will be one of the main tools

of this paper.

We define the minimal set of periods of a continuous map in Section 3. We describe

the known results about these sets for the continuous maps on the n-dimensional torus

for n 1; 2; 3. These results will be later on compared, when this is possible, with the minimalset of periods and the minimal Lefschetz set of periods of a Morse Smale diffeomorphism.

In Section 4, first we give the definition of a MorseSmale diffeomorphism and of its

minimal set of periods. For the circle and the two-dimensional torus are characterized not

only the minimal set of periods but also their set of periods.

The definition of minimal Lefschetz set of periods for MorseSmale diffeomorphisms

is given in Section 5, there we also provide the references for finding the description ofthe minimal Lefschetz set of periods for MorseSmale diffeomorphisms on T1 and T2,

respectively.

As we will see, the characteristic polynomials of the homomorphisms induced in

the homological groups of Tn by the MorseSmale diffeomorphisms are products of

cyclotomic polynomials. We define them and describe their properties in Section 6. These

polynomials will play a main role in the proof of our results.

In Section 7, we describe how to compute the action induced by a continuous map on

Tn on its homological groups knowing their action on the first homological group. These

calculations are based on the exterior algebra structure of the homological groups of Tn

.

This knowledge is another key point in this work and allows to compute the Lefschetz zetafunctions of the MorseSmale diffeomorphisms on Tn.

The main results of this paper are Theorems 8.2 and 9.2, where we describe the

minimal Lefschetz set of periods of the MorseSmale diffeomorphims on T3 and T4,

respectively. These theorems are the main results of Sections 8 and 9, respectively.

Finally, in Section 10, we provide a result of the MorseSmale diffeomorphisms onTn

for n . 4.

2. Lefschetz zeta function

Probably, the main goal of Lefschetzs work in 1920s was to link the homology class of a

given map with an earlier work on the indices of Brouwer on the continuous self-maps on

compact manifolds. These two notions provide equivalent definitions for the Lefschetz

numbers, and from their comparison can be obtained information about the existence of

fixed points.

J.L.G. Guirao and J. Llibre690


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Let M be ann-dimensional manifold. We denote byHkM;Qfor k 0; 1; . . .; nthehomological groups with coefficients in Q. Each of these groups is a finite linear space

over Q.

Given a continuous map f : M!M, there exist n1 induced linear maps f*k :

HkM;Q!

HkM;Q by f. Every linear map f*k is given by an nk

nk matrix withinteger entries, where nkis the dimension ofHkM;Q for k 0; 1; . . .; n.

Given a continuous map f : M!M on a compact n-dimensional manifold M,

itsLefschetz number L(f) is defined as

Lf Xnk0

21k trace f*k:

One of the main results connecting the algebraic topology with the fixed point theory is the

Lefschetz fixed point theorem, see for instance [7].

Our aim is to obtain information on the set of periods off. For this purpose, it is usefulto have information on the whole sequence {Lfm}1m0 of the Lefschetz numbers of all

iterates off. Thus, we define the Lefschetz zeta function offas

Zft expX1m1

Lfm

mtm

!:

This function generates the whole sequence of Lefschetz numbers, and it may be

independently computed through

Zft Ynk0

detInk2 tf*k21k1 ; 1

where n dimM, nk dimHkM;Q, Ink is the nk nk identity matrix, and we takedetInk2 tf*k 1 ifnk0. Note that the expression (1) is a rational function int. So the

information on the infinite sequence of integers {Lfm }1m0 is contained in two

polynomials with integer coefficients, for more details see [10].

Letfbe a diffeomorphism on a compact manifold M having finitely many hyperbolic

periodic orbits. Ifg is a hyperbolic periodic orbit of period p, then for each x [ glet Eux

denote the subspace ofTxM

generated by the eigenvectors ofDfp

xcorresponding to theeigenvalues whose moduli are greater than one. Let Esx be the subspace ofTxM generated

by the remaining eigenvectors. We define the orientation type D of g to be 1 if

Dfpx: Eux ! Eux preserves orientation, and 21 if it reverses orientation. The index uofg

is the dimension ofEux for some x [ g. We note that the definitions ofD and u do not

depend on the point x, but only on the periodic orbit g. Finally, we associated the triple

p; u;D to the periodic orbit g.For f the periodic data is defined as the collection composed by all triples p; u;D,

where a same triple can occur more than once provided it corresponds to different periodic

orbits. Franks [10] proved the following result, which will play a key role for establishing

our main results.

Theorem2.1.Let f be a C1 map on a compact manifold having finitely many hyperbolic

periodic orbits, and letS be the period data of f. Then the Lefschetz zeta function of f

Journal of Difference Equations and Applications 691


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satisfies

Zft

Yp;u;D[S1 2 Dtp21

u1

: 2

Clearly, the MorseSmale diffeomorphisms satisfy the hypotheses of Theorem 2.1.

3. Minimal sets of periods for continuous torus maps

Letf : M!M be a continuous map. Assume thaty [M. Iffpy y, thenyis aperiodic

point of fof period p if f jy y for all 0 # j , p. We denote by Perf {m [ N :

fhas a periodic point of period m} the set of periods of f. We define the minimal set of

periods offas

MPerf \h,f

Perh; 3

whereh: M!M runs over all the continuous maps which are homotopic to f.

The minimal sets of periods have been studied for different classes of maps. For

instance, it is known that for a continuous map on the interval its minimal set of periods is

{1}. For continuous self-maps defined on the circle, the set MPer(f) is described in [2].

In [1] and [15] are studied the minimal sets of periods for the continuous self-maps on the

n-dimensional torus obtaining, in particular, the complete characterization of these sets in

dimensions 2 and 3, respectively.

4. Minimal set of periods of Morse Smale diffeomorphisms

Letfm be themth iterate off[ Diff(M). A pointx [M is anonwandering pointoffif

for any neighbourhood U of x, there is a positive integer m such that fm U> U Y.

We denote by V(f) the set of nonwandering points off.

Assume that x [M. If fx x and the derivative of f at x, Df(x), has all its

eigenvalues disjoint from the unit circle, then x is called a hyperbolic fixed point.

Let y be a periodic point of period p. Then y is a hyperbolic periodic point ify is a

hyperbolic fixed point offp. We call the set {y;fy; . . .;fp21y} theperiodic orbitof theperiodic point y.

Suppose that dis the metric onM

induced by the norm of the supremum, and p isa hyperbolic fixed point of f. Then the stable manifold of x is Wsx {y [M :

dx;fmy! 0 as m!1}, and the unstable one is Wux {y [M :dx;f2my!0 as m!1}. We extend these notions to a hyperbolic periodic point x of period p as

follows. The stable and unstable manifolds are defined as the stable and unstable

manifolds ofx underfp.

A diffeomorphismf : M!M isMorseSmale if

(1) V(f) is finite;

(2) all periodic points are hyperbolic; and

(3) for each x;y [ Vf, Wsx, and Wuy have a transversal intersection.

Condition (1) implies that V(f) is the set of all periodic points off.

The following result on the MorseSmale diffeomorphism on the circle is due to

the minimal set of periods of a MorseSmale diffeomorphism f defined on a compact



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manifold M is

MPermsf \h,f

Perh; 4

whereh runs over all the Morse Smale diffeomorphisms ofM which are homotopic to f.

Clearly MPermsf $ MPerf.

The minimal sets MPermsffor Tk withk 1; 2 are known. Thus, for k 1 we have

that MPermsf is the empty set if f is orientation preserving, and the set {1} if f is

orientation reversing. In [14] are described completely the sets of periods of the Morse

Smale diffeomorphisms on T2 for every homotopy class associated to the orientation type.

These results are possible to obtain by the existence of a deep knowledge on the

Morse Smale dynamics on T2 due to Batterson [4,5]. For MorseSmale diffeomorphisms

defined in a dimension higher than 2, we do not have such a knowledge on the dynamics of

the MorseSmale diffeomorphisms. In fact, in a dimension larger than 2, we do not know

how to compute the minimal set of periods MPermsfof a Morse Smale diffeomorphisms

fon Tn, but we can characterize the so calledminimal Lefschetz set of periods, see the nextsection for definition.

5. Minimal Lefschetz set of periods of Morse Smale diffeomorphisms

The statement of Theorem 2.1 allows us to define the minimal Lefschetz set of periods

for a C1 map on a compact manifold having finitely many hyperbolic periodic points.

Such a map has a Lefschetz zeta function of the form (2).

Note that in general the expression of one of these Lefschetz zeta functions is not

unique as product of the elements of the form 1^ tp^1. For instance the following

Lefschetz zeta function for a Morse Smale diffeomorphism on T4 can be written in four

different ways in the form given by (2):

Zft 12 t321t3

12 t61t3

1 2 t312 t6

12 t61t3

1 2 t312 t6

12 t312 t23

12 t321t3

12 t312 t23:

According to Theorem 2.1, the first expression will provide the periods {1,3} for f, the

second the periods {1,6}, the third the periods {1,2,6}, and finally the fourth the periods

{1,2,3}. Then for this Lefschetz zeta function Zft, we will define its minimal Lefschetz

set of periods as

MPerLf {1; 3}> {1; 3; 6}> {1; 2; 3; 6}> {1; 2; 3} {1; 3}:

In general, for the Lefschetz zeta function Zftof aC1 mapfon a compact manifold

having finitely many hyperbolic periodic points, we define its minimal Lefschetz set of

periods as the intersection of all sets of periods forced by the finitely many different

representations ofZftas products of the form 1^ tp^1.

For MorseSmale diffeomorphisms and from the definition of minimal Lefschetz set

of periods it follows always that

MPerLf # MPermsf:

For Morse Smale diffeomorphims on the circle T1, the above # is in fact

an equality. In general, it is unknown when MPerLf MPermsf. However, for thetwo-dimensional torus, we have that MPerLf can be a proper subset of MPermsf,

for more information see [14,19].



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Our aim is to study MPerLf for the MorseSmale diffeomorphisms on Tn, n . 2.

Thus, we provide an algorithm for computing the Lefschetz zeta funtions in orden to apply

Theorem 2.1. This algorithm is based on the special properties of the homological groups

of torus maps presented in Section 7. Using this algorithm, we describe completely

MPerLf for the MorseSmale diffeomorphisms on T

3

and T

4

.Let f : Tn ! Tn be a MorseSmale diffeomorphism. It is known (cf. [26]) that if a

homotopy class admits a Morse Smale diffeomorphism, then the linear maps corresponding

to its homology must be quasi-unipotent, i.e. all their eigenvalues are roots of unity.

Using Theorem III.13 of [22], it follows that any two quasi-unipotent integer matrices

having the same irreducible characteristic polynomial overQ are similar over the integers.

Let A be a matrix, which is equivalent over the integers to f*1. If the characteristic

polynomial ofA is reducible over Q, then A is similar over the integers to a block upper

triangular matrix

A

~A11 ~A12 . . . ~A1r

0 ~A22 . . . ~A2r

.

.

....

.

.

.

0 0 . . . ~Arr

0BBBBBB@1CCCCCCA

;

where the characteristic polynomial of ~Ajj is irreducible over Q for j 1; . . .; r(cf. Theorem III.12 of [22]). So the characteristic polynomial ofA is the product of the

characteristic polynomials of the ~Ajjs.

6. Cyclotomic polynomials

By definition, thenth cyclotomic polynomial is given by

cnt Y

k

wk2 t;

where wk e2pik=n and k runs over all the relative primes smaller than or equal to n.

For more details about these polynomials, see [16]. An alternative way to express cntis

cnt 1 2 tnQ

djn;d,n

cdt: 5

Let w(n) be the degree ofcnt. Thenn P

djnwd. So w(n) is the Euler function, whichsatisfies wn n

Qpjn;p prime1 2 1=p. Therefore, if the prime decomposition of n is

pa11 . . .pak

k , then wn Qk

j1paj21

i pj 2 1:

From formula (5), we have

cnt Ydjn

1 2 tdmn=d; 6

where mis the Mobius function, i.e.

mm

1 if m 1;

0 if k2

jm for some k[ N;21r if m p1 . . .pr distinct primes factors

8>>>:



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Proposition 6.1.For0 , k, n, wkisarootofcnt ifand only if w21k isalsoarootofcnt.

Proof. If wk e2pik=n then w21k e

22pik=n e2pin2k=n with n 2 k and n coprime

if and only if k and n are coprime. Then, the proposition follows directly from the

definition ofcnt. A

We note that if a homotopy class admits a MorseSmale diffeomorphism, then the

linear maps corresponding to its homology must be quasi-unipotent, the characteristic

polynomials associated to the matrix associated to f*1 must be the product of cyclotomic

polynomials (Table 1).

7. Action on the homological groups of the tori

We recall now the homological groups of the n-dimensional torus Tn with coefficients in

Q. More precisely

HkTn

;Q Q%. . .nk

%Q;

wherenkn

k

!fork 0; 1; . . .; n.

Given a continuous map f : Tn ! Tn, there exist n1 induced linear maps f*k :HkT

n;Q!HkTn;Q by f. Every linear map f*k is given by an nk nk matrix with

integer entries. It is well known that f*0 is the identity, i.e. f*0 1.

Now we can recall how to compute f*k in terms off*1 fork 2; . . .; n.

7.1 The three-dimensional torus

Now we shall compute the actions on the homology off*2 andf*3 in function off*1 for

continuous maps f : T3 ! T3. Let

f*1

a d g

b e h

c f i

0BB@

1CCA:

Then we have f*3 det f*1 and

f*2

ae 2 bd ah 2 bg dh 2 eg

af2 cd ai 2 cg di 2fg

bf2 ce bi 2 ch ei2fh

0BB@

1CCA

a33 a32 a31

a23 a22 a21

a13 a12 a11

0BB@

1CCA; 7

Table 1. The first 30 cyclotomic polynomials.

c1t 1 2 t c2t 1 t c3t 1 2 t3=1 2 t

c4t 1 t2 c5t 1 2 t

5=1 2 t c6t 1 t3=1t

c7t 1 2 t7=1 2 t c8t 1 t

4 c9t 1 2 t9=1 2 t3

c10t 1 t5=1t c11t 1 2 t

11=1 2 t c12t 1 t6=1t2

c13t 1 2 t13=1 2 t c14t 1t

7=1t c15t 1 2 t151 2 t=12 t312 t5

c16t 1 t8 c17t 1 2 t

17=1 2 t c18t 1 t9=1t3

c19t 1 2 t19=1 2 t c20t 1 2 t

10=1 2 t2 c21t 1 2 t211 2 t=12 t312 t7

c22t 1 t11=1t c23t 1 2 t23=1 2 t c24t 1 t12=1t4c25t 1 2 t

25=1 2 t5 c26t 1t13=1t c27t 1 2 t

27=12 t9

c28t 1 t14=1t2 c29t 1 2 t

29=1 2 t c30t 1t151t=1t31t5



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where as usual aij is the determinant of the 2 2 matrix obtained by removing file i and

columnj off*1.

7.2 The four-dimensional torus

Similarly, we compute the actions on the homology off*2,f*3, andf*4 depending on f*1for continuous maps f : T4 ! T4. Let

f*1

a e i m

b f j n

c g k o

d h l p

0BBBBB@

1CCCCCA:

Then we have f*4 det f*1,

f*3

a44 a43 a42 a41

a34 a33 a32 a31

a24 a23 a22 a21

a14 a13 a12 a11

0BBBBB@

1CCCCCA; 8

where aij denotes the determinant of the 3 3 matrix obtained by removing file i and

columnj off*1. Finally

f*2

a34;34 a34;24 a34;23 a34;14 a34;13 a34;12

a24;34 a24;24 a24;23 a24;14 a24;13 a24;12

a23;34 a23;24 a23;23 a23;14 a23;13 a23;12

a14;34 a14;24 a14;23 a14;14 a14;13 a14;12

a13;34 a13;24 a13;23 a13;14 a13;13 a13;12

a12;34 a12;24 a12;23 a12;14 a12;13 a12;12

0BBBBBBBBBBB@

1CCCCCCCCCCCA

;

af2 be aj 2 bi an 2 bm ej 2fi en 2fm in 2jm

ag 2 ce ak 2 ci ao 2 cm ek 2 gi eo 2 gm io 2 km

ah 2 de al 2 di ap 2 dm el 2 hi ep 2 hm ip 2 lm

bg 2 cf bk 2 cj bo 2 cn fk 2 gj fo 2 gn jo 2 kn

bh 2 df bl 2 dj bp 2 dn fl2 hj fp 2 hn jp 2 ln

ch 2 dg cl 2 dk cp 2 do gl 2 hk gp 2 ho kp 2 lo

0BBBBBBBBBBB@

1CCCCCCCCCCCA

; 9

where theaij;kldenote the determinant of the 2 2 matrix obtained by removing filesiand

j and the columns kand l off*1.



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8. MPerL for MorseSmale diffeomorphisms on T3

From the information of the previous two paragraphs, any induced homology

homomorphism f*1 on the first rational homology group of a Morse Smale

diffeomorphismf : T3 ! T3 is similar to one of the 14 integer matrices Bis of Table 2,

where u; v; w [ {0; 1; 2; . . . }. We note that the orientation-preserving Morse Smalediffeomorphisms onT3 induces a homology homomorphism on the first rational homology

group, which are similar over the integers to the matrices Biwithi [ {1; . . .; 7}. MatricesBi, where i [ {7; . . .; 14}; are induced by the orientation-reversing ones.

We consider the seven matricesAi:

A1 1 r

0 1

!; A2

21 r

0 21

!; A3

1 0

0 21

!; A4

1 1

0 21

!;

A5

0 1

21 0 !

; A6

0 1

21 21 !

; A7

0 21

1 1 !

; 10

wherer[ {0; 1; 2; . . . }. Any induced homology homomorphism f*1 on the first rationalhomology group of a MorseSmale diffeomorphismfon T2 is similar to over the integers

to one of the previous seven matrices, see [4].

The 14 matrices Bi are obtained from the seven matrices Ai given in (10) adding a

block of size 1 1 formed by^1 taking into account the last two paragraphs of Section 5.

We must note that since there are no irreducible cyclotomic polynomials of degree 3

(Section 6), these 14 matrices obtained as we have explained are all the possible 3 3

matrices associated to f*1

, when fis MorseSmale diffeomorphisms on T3.

Table 2. f* 1 for the MorseSmale diffeomorphisms on T3.

B1

1 u v

0 1 w

0 0 1

0BB@

1CCA B2

21 u v

0 21 w

0 0 1

0BB@

1CCA B3

1 0 v

0 21 w

0 0 21

0BB@

1CCA

B4

1 1 v

0 21 w

0 0 21

0BB@

1CCA B5

0 1 v

21 0 w

0 0 1

0BB@

1CCA B6

0 1 v

21 21 w

0 0 1

0BB@

1CCA

B7

0 21 v

1 1 w

0 0 1

0BB@

1CCA B8

1 u v

0 1 w

0 0 21

0BB@

1CCA B9

21 u v

0 21 w

0 0 21

0BB@

1CCA

B10

1 0 v

0 21 w

0 0 1

0BB@

1CCA B11

1 1 v

0 21 w

0 0 1

0BB@

1CCA B12

0 1 v

21 0 w

0 0 21

0BB@

1CCA

B13

0 1 v

21 21 w

0 0 21

0BB@ 1CCA B140 21 v

1 1 w

0 0 21

0BB@ 1CCA



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Proposition8.1. Let f : T3 ! T3 be a MorseSmale diffeomorphism and let f*1 be the

induced homology homomorphism on its first rational homology group. Then the Lefschetz

zeta function Zft is

1 if f*1 < Bi; where i [ {1; 2; 3; 4; 5; 6; 7; 8; 10; 11};

1t

1 2 t

4if f*1 < B9;

1t

1 2 t

2if f*1 < B12;

1t31 2 t3

12 t31t3 if f*1 < B13;

1t1t3

12 t12 t3 if f*1 < B14:

Proof. By expression (1) for the case of a MorseSmale diffeomorphism on T3, the

Lefschetz zeta function is given by

Zft Y3k0

det Ink2 tf*k21k1

det I3 2 tf*1 det I1 2 tf*3

12 t det I2 2 tf*2 : 11

By the results of subsection 7.1, f*3 det f*1andf*2can be calculated in function off*1using expression (7). Therefore, using (11) we can compute the Lefschetz zeta function for

the 14 different homotopy classes of MorseSmale diffeomorphisms on T3. A

Theorem 8.2. Let f : T3 ! T3 be a MorseSmale diffeomorphism and let f*1, be the

induced homology homomorphism on its first rational homology group. Then

MPerLf

Y if f*1 < Bi for i [ {1; 2; 3; 4; 5; 6; 7; 8; 10; 11};

{1} if f*1 < Bi for i [ {9; 12};

{1; 3} if f*1 < Bi for i [ {13; 14}:

8>>>:Proof. From Proposition 8.1 and using Theorem 2.1 it is easy to check that the minimal

Lefschetz set of periods for the Morse Smale diffeomorphisms on T3 are the ones

described in the statement of the theorem. A

From Theorem 8.2, the next result immediately follows.

Corollary8.3. Let f : T3 ! T3, be a Morse Smale diffeomorphism. Then

MPerLf Y

if f is orientation preserving;Y; {1} or{1; 3} if f is orientation reversing:

(



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Then the Lefschetz zeta function Zft is

1 if f*1 < Ci for i [ K;

1t

1 2 t 8

if f*1

minimal lefschetz sets of periods for morse–smale diffeomorphisms

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