New zeta functions for dynamical systems and ...

NEW ZETA FUNCTIONS

FOR DYNAMICAL

SYSTEMS AND

NIELSEN FIXED POINT THEORY

A.L.Fel'shtyn

The Leningrad

Technology

In the paper we define new dynamical to study the Nielsen of the dynamical

Institute

zeta functions.

We continue

zeta function [I, 2] . The universal properties

zeta functions

are investigated.

§ I. INTRODUCTION

We assume everywhere

X

~" X namical

X

to be a connected compact polyhedron

to be a continuous

systems the following

zeta function

number of isolated

map. In the theory of discrete

zeta functions

~(~=~p(

~

are known:

)

fixed points of

and

the Artin-Mazur

, where

I~ ; the Lefschetz

dy-

r,, ~j, is the ~[

zeta function

~=0 H~[X~]is ~01

Z

Artin-Mazur

Mazur and Lefschetz group rings z m

the Lefschetz

and Lefschetz

The above ~eta functions function of algebraic zeta function

zeta functions

zeta functions,

orZz~of

~

[4]

; reduced

[ 5 ] ; twisted Artin-

which have coefficients

an abelian g r o u p H

in the

[6]

are directly analogous

manifolds

is rational:

number of

to the Wail zeta

over finite fields [ 7 ] . The Lefschetz

34

£{~

The Artin-Mazur

]l-i)L+{

smooth maps of a compact smooth manifold

Consider the case when

diffeomorphism. function

•

zeta function has a positive radius of convergence for

a dense set in the space of [3].

[4]

X

is a smooth manifold,

~

an Axiom

In this case the rationality of the Artin-Mazur

0 We define the minimal dynamical zeta functions M;(Z)_ and M~(~) as formal power series:

and

The minimal zeta functions

M ~ i~)

and

M~{~)

are the homotopy inva-

riants of We study zeta functions

KN~(~),

M ~ (~)

and ~ ( ~

in § 3 and

§ 4.

§ 2. NIELSEN ZETA FUNCTION

2.1. Preliminaries. The Nielsen zeta function gence

N~(~) has a positive radius of conver-

(see [I, 2 ] ) .

PROBLEM. For which spaces and maps Nielsen zeta function

N{(~)is

a rational function or a radical from a rational function ? Is

N~(~

an algebraic function ? When

N~(~)

is rational or a radical from rational function the

infinite sequence of the Nielsen numbers

{ N { I ~ )}~=~

by a finite set of complex numbers - zeroes and poles of Lemma I ). Let /~L(~) ~ £ ~ I N

be the Mobius function,

i.e.

is determined N~(z)

(see

37

4

d.=~

1

(_~)L d.=~, p,, p~. 0 , If I

is a periodic

Ntlz) =

if

p2, I C~

for some prime

map of least period ~I. , then [2]

0 J

primes

distinct

p

:

(,1-zd') - J.,l~

d, lm,

The Nielsen

zeta function T

is rational

automorphisms

of

for e x p a n d i n g

maps of the orientable

~ , for hyperbolic

nuous maps of the projective LEMMA

I. Zeta function

tion of degree ~ [ ~ plex numbers

for hyperbolic

4

endomorphisms

endomorphisms

of the nilmanifold,

compact manifolds

and for conti-

spaces [2] N{[~I

is a radical

from a rational

if and only if there exists a finite

oC~ and ~ {

N({~)=(7-Aj -

and

func-

set of com-

such that

Z~

)/n~

0

PROOF.

Suppose that the Nielsen

from a rational

zeta function

N~(~)

function of degree MI. ~ ~ . It is immediate

finition of the zeta function that if it is expanded about the origin,

where

P{Z)

is a radical

then the constant

and Q ( ~ l a r e

With this assumption

polynomials,

, N~(~)

term is I. If

from the de-

in a power series

N~(~ "%

we may assume that

P(~)

P(0I--~[0I-- 1 ,

can be factored as follows:

38 4

_ / R4, (4-~.+z)

Nf(z) where

(2)

~j[t-~j ~)

~ , ~ , # j ("

. Taking the logarithmic

derivative

of both sides

we have

!

Nl(z) N~(z)

Multiply

( >- -z~

both sides by

in a power

series.

We now compute

-

T- -#i

/- t

and then use the geometric

series to expand

One finds finally that

the left-hand

side in a different

way.

From the defi-

nition

N~(z) = e:~p ( ~

Differentiating sides by

~

rL

logarithimically

.z')

both sides and then multiply

both

, we find that

(4)

Comparing coefficients =(~ lation.

-- E % ~ % )/ ~ Q.E.D.

of - ~ in equations . The converse

(3) and

(4) we have

,i~{mj hl,0

is p r o v e d by a direct calcu-

39

COROLLARY. Suppose that

~

is an orientation - preserving homeo-

morphism of a compact orientable surface. Then there exists a finite set of complex numbers o6 i and ~j

and natural [[~~ J

such that

J

PROOF. For o r i e n t a t i o n - p r e s e r v i n g homeomorphisms

of a compact o r i -

entable surface the Nielsen zeta function is rational or a radical from rational function [I, 2 ] . 2.2. The Jiang subgroup and the Nielsen zeta function. Sometimes we may compute Nielsen numbers N(I ~) and prove rationality of

N~(Z)

. The trace subgroup of cyclic homotopies (Jiang sub-

group) U(I~ ~ 0 ) ~ ~ ( X ~ I (~0)) that Let

~(~0]) is defined by 7 ( { ~ 0 ] = [ ~ 6 ~z61IX,

there exists a cyclic homotopy

H= {~

(X,


i

which improves the above estimation of 3.2. The ~ 0 ~

the arbitrarily dimensional diffeomorphism of H . Then the

5 ~ p ~ ~0~ I ~ { ~ ( ~ ) A ~

measure on M

metric, implies the estimation

is the maximum

Geometrically

~I~

[25]

M induced by a given Riemannian I

~ .

M

K Nielsen zeta function of the periodic map.

PROBLEM. For which spaces and maps K N ~ ( ~ )

is a rational function

or a radical from a rational function ? Is ~ ~ ( ~ )

an algebraic func-

45 t ion ? Let

KN[{")

KN..

=

THEOREM 5. Suppose that {~(K/

and

~

K

is a periodic map of the least period

. Then

~1~ PROOF. S i n c e

show that

, we have

KN4=KN~if (.~,rtl,)=4

----~+~. Then = KN

= ~

(I~)~ =

I~ % =

KNj =KN~,j f o r

There are ~ , $ ~ Z +

K

~(~°~)-|~

/ o £. o K ',-1 u *%1 J ~"

) )=

Since

we have

from ~0to ~

~ ( ~ * ( { ° ~ ) - 4 ) --

~ ~ (~ °~) -4. (I o~),[{~o~) -4 = o6 w

It follows that 06*(~E°~) -|~ K is derived by the iteration

of this process. So class of {~

~--

~0 and ~4 of I" X --~ ~ belong to the

(I z° ~v]-4~ K and a product

= (~ o ~

. We

and KNII

same ~%0~K fixed point class. Then there exists a path ~ such as

j

such as ~

I"~+'~---(I"~)~'I----- ~

(I~ . Let two fixed points

every

~0

and ~

If two points ~

ed point classes

belong to the same ~ 0 & K f i x e d

point

and ~4 belong to the different ~ 0 ~ K f i x -

~ , then they belong to the different ~ 0 ~ K

fixed

point classes of {[ . So, every essential n%0& K fixed point class of I correspond to some an essentialS0& ~ fixed point class of ~ the different~0~Kfixed rent ~%0~ gous

K

point classes of ~

fixed point classes of ~ .

correspond to the diffe-

Then ~ N [ ~ ) ~ K N ( ~ )

KN(({)=KN(1)Hence ) KN(1)----KN(~).

it is proved that K N I

K N ( ~ if [{ i ~

)

and

. Analo-

The same way

, where ~l~b

• Using

this series of the equals numbers we receive by direct calculation:

K N t(z) = 6=p ( 4.=tEKN~.4.z~') = e=p( d.l,,. }-- >--4,=Ip(d.lcl. (.~d.)~.,[. ) =

46

I

= e:~p (~-

~,r~ ( t - zd')) = I--I ~v/k4- %d') -p(&)'

where the integer numbers P ( ~ ) P(~)=~--

as

~-~

KN~

= ~

used then:

P(~)

are calculated recursively via formula

. Moreover,

if the last formula is rewritten

P[~{) and the M6bius Inversion Theorem in number theory Js

P(~) = ~

~(~4)'K~I~

, where ~ ( ~ 4 ) is the M~bius

function. Q.E.D.

§ 4. MINIMAL DYN~4ICAL ZETA FUNCTIONS

4.1. A radius of convergence for the minimal zeta functions M~(~)

and

M{(~

THEOREM 6. Suppose pact polyhedron.

that { ~ X

~ X

be a continuous map of a com-

Then a minimal zeta function

~ ~ (~)

has a positive

radius of convergence. PROOF. We consider a smooth compact manifold M lar neighbourhood of X motopy type with

I

and a smooth map ~ ~ ~

. There is a smooth map ~ : M

such as every iterate points

of

~

M

of the same ho~ M

homotopic to

has only a finite number of fixed

(see [12] , p.62). According to Artin and Mazur [3] there exists

constant Then

~

, which is a regu-

~ = ~[~) ~ + o~

6m > F(~r~) ~ M ~ ( { I

, such as F ( ~ ) for every ~ 0

L ~~

for every ~ ~ 0

. Now the statement of the

theorem follows from the Cauchy-Adamar formula. Q.E.D.

47

Let ~

be a radius of convergence

R~.

if {

is homotope

for one of the zeta functions

to expansive homeomorp~ism ~: X

[14Zl , then from Conze inequality + ~ > ~ I ~ ) ~ m

- X

~p~t,0~F(~} ~) [~4] I%

from ~ e inequalities F ( ~ ) ~ M ~ [ { ] ~ formula it follows that Suppose

to an Axiom A la

~

"- M

diffeomorphism

M~

be a diffeomorphism of a smooth

. According to Smale and Shub [15]

+ o~ > ~ { ~ ) - - - ~

-22 M ~ [ { ]

anda ~auchy-Adamar

R ~ 9..,~p (- ~'~(g,l)> O.

now that ~ : M

compact manifold M

M~({)

~p

~

({)

~:M

~ M

~o~I~

~

is isotope

. Then from a Bowen's formu-

) [16] , the inequalities

and the Cauchy-Adamar

F( ~ ) ~

formula the next sta-

tement follows THEOREM 7. Suppose pact manifold.

be a diffeomorphism of a smooth com-

Then

R ~" ezp

RE~,LARK. Let nifold M

{[M --~M

I

(- [(~,))~ O. be a

~ I÷6

diffeomorphism of a Riemannian ma-

. Then the Przytycki's

inequality implies

the estimation

R~ M

Computation of the nu

ers

A periodic point class of period with a fixed point class of isreducible to period ~ L ~ of period

~

It/ ~

of

and

t:

X ~

X

is synonymous

. A periodic point class of period if it contains some periodic point class

; it is irreducible

if it is not reducible to any lower

period. The set of periodic point classes decomposes

into

{ -orbits.

If a periodic point class is reducible to period ~I, , so is its ~-image. Thus, the reducibility DEFINITION

is a property of an

~ -orbit.

[12] . The Nielsen type number for the

~-th

iterate,

48 the minimal height of

denoted

such

~-invariant

sets of

periodic point classes, that each essential class of any period ~ I ~ contains at least one class in the set. A procedure for finding of

~

: take the

N F~(~)

from the periodic-point-class

~ -invariant set 5

data

of all the essential classes, of

any period ~ I ~

, which do not contain any essential classes of lower

period. To each

~ -orbit in

5

, find the lowest period which it can

be reduced too. The sum of these nunJ3ers is

NF~(~)

•

B.Halpern has proved the following results

M[({)If

THEOREM 8. [12] . For all [~ connected differentiable

N ( I ~) is feasible

[12] ), we have a fair chance to compute N F ~ [ I I M~,({) and

M~[I

] .

For which spaces and maps

M~(~)~

functions or radicals from rational functions gebraic functions

(see

and consequently by

4.3. Two theorems about the minimal zeta function PROBLEM.

is compact

manifold of dimension ~ 5 , then for all

In the cases where, the computation of

the above theorem

X

M ~ (~)

M~ (~ are rational

? are M ~ ( ~ ) , M ¢ ( ~ ) a l -

?

THEOREM 9. Suppose that ~

is a periodic map of the least period

. Then

Me (z) =

PROOF. Let ( ~ where ~I ~

(I-z &) &,m

Fl

£I~ ?

~)=

t

°

M~I~,

Every essential class of the period

, is reduced to the period I. There is I - I correspondence

between the sets of the essential classes of the period I and the period

~

. Hence

NF4(~)=N~

(~). In the same way it is proved that

49

NF (f)

I]

8 it follows that

la for

HI(Z)

,if M

(~,-~)=~

, where C1.1111..

From Theorem

({]=M~

follows from the calculation as in Theorem 5. Q.S.D.

Proof of the next theorem is based on the Thurston theory of homeomorphisms of surfaces [ 17 ] . THEOREM 10. Suppose that {" M~---~-M Z is a orientation-preserving homeomorphism of an orientable compact surface M Z . Then M ~ ( ~

is a

rational function or a radical from rational function. PROOF. The case of the surface with ~ ( M ~ ) ~ 0 dared in 4.4. In the general case % ( M Z ) x 0 [17] , [26]

~

is

(S Z ? TZ)is consi-

, according to Thurston

isotopic to the homeomorphism ~'MZ---~ M Z such

that either: I) ~

is periodic; 2) ~

is pseudo-Anosov; 3) ~

is re-

ducible. The case I) has already been discussed in Theorem 9. If ~is an orientation-preserving pseudo-Anosov homeomorphism of an orientable com-

F(~m):M~[~]--M~(~)--~(~[17].

pact surface, then for e v e r y ~ > 0 Hence M~( ~)~- M ~ ( ~ ) : M ; ( ~ ) : N ~ ( ~ ) :

~(~).Since Shub and Fathi [27]

constructed the Markov's partions for the pseudo-Anosov homeomorphism, then the rationality of ~ ( Z ) M~(Z)

is rational also. If the homeomorphism

there exists a system F such that aM Z

i) F{

; ii) F~

has an open

. Each component 5~

smallest positive iterate

rate ~

of

~

have

is invariant

~-invariant tubular neighbourhood

cf M Z \ % ( F ) i s ~J

; iii) F

of ~

mapped to itself by some

, and each f ~ / S j

satisfies I)

is mapped to itself by sime smallest positive ite, and each

ralized twist.'Since , then

is reducible, then

of disjoint simple closed curves F4 ~...7F m

is not isotopic to F~ ~ =~ ]

or 2). Each ~(F{]

~

is not isotopic to either a point or a component of

by ~ . The system F %(F)

is proved as in [8] by Manning. Then

% (Z]

%[S{]

"%~/ % [ F%]

is a (possibly trivial) gene-

is homotopically equivalent to the circle

is a rational function (see (4.4)). On (~)

, on

%[F~I

we have

(~)='~

Sj we "

~"~%)

50

The number

M~(~]

is

• Then M ~ ( ~ ] = M ~ ( ~ ]

is the product (by the property of the

exponent) of the minimal zeta-function M ~ (~)

M~I(~/Sj)and M~(C~/I],[~")/

the sum of numbers

M W~,~j

and M ~%1%([~) " Hence

is a radical from a rational function• Q.E.D.

4.4. Examples. Let X = a degree ~

, ~,>

~ . Then

Now let

0

aria

I"

b

M ~ I~] ~ ]~-_ ~

X ~- $Z~and I

{ ~ 5 z~

= b

~S a continuous map of

, if I ~ I

and ~ I ~ ) = l

~ ~2~has a degree i

, if ~ - - I

and

if ~----I

• Then M ~ ( ~ ) =

M~(~)~_M~I~)~--N~(%)_~_~z

if 4 = - 4 In the next example X = 7

~and f " T ~

~< T

is a hyperbolic endo-

morphism or automorphism. Then M ~ [ { ] : M~({] ~ F({~):N({~)=IL({~)I = =l~6t(E--~)l [12, 18] , where ~ ' ~ ~--~ R ~ is a linear lifting of { Thus M { ( Z ] = ~ { • ( Z ] = < ~ ( Z ) : N{(Z)~--(L~(~. Z ]]tl)~ are the rational functions (here ~----(-I) ~ , where 5~[0{Z(~) such that . {