8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms
1/15
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
8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms
2/15
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
8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms
3/15
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
8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms
4/15
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
J.L.G. Guirao and J. Llibre692
8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms
5/15
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].
Journal of Difference Equations and Applications 693
8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms
6/15
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>>>:
J.L.G. Guirao and J. Llibre694
8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms
7/15
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
Journal of Difference Equations and Applications 695
8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms
8/15
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.
J.L.G. Guirao and J. Llibre696
8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms
9/15
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
Journal of Difference Equations and Applications 697
8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms
10/15
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:
(
J.L.G. Guirao and J. Llibre698
8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms
11/15
8/14/2019 Minimal Lefschetz sets of periods for MorseSmale diffeomorphisms
12/15
Then the Lefschetz zeta function Zft is
1 if f*1 < Ci for i [ K;
1t
1 2 t 8
if f*1