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In mathematics , the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems . Given a continuous map
f
:
X
→
X
{\displaystyle f\colon X\to X}
, the zeta-function is defined as the formal series
ζ
f
(
t
)
=
exp
(
∑
n
=
1
∞
L
(
f
n
)
t
n
n
)
,
{\displaystyle \zeta _{f}(t )=\exp \left(\sum _{n=1}^{\infty }L(f^{n}){\frac {t^{n}}{n}}\right),}
where
L
(
f
n
)
{\displaystyle L(f^{n})}
is the Lefschetz number of the
n
{\displaystyle n}
-th iterate of
f
{\displaystyle f}
. This zeta-function is of note in topological periodic point theory because it is a single invariant containing information about all iterates of
f
{\displaystyle f}
.
Examples
[ edit ]
The identity map on
X
{\displaystyle X}
has Lefschetz zeta function
1
(
1
−
t
)
χ
(
X
)
,
{\displaystyle {\frac {1}{(1-t)^{\chi (X )}}},}
where
χ
(
X
)
{\displaystyle \chi (X )}
is the Euler characteristic of
X
{\displaystyle X}
, i.e., the Lefschetz number of the identity map.
For a less trivial example, let
X
=
S
1
{\displaystyle X=S^{1}}
be the unit circle , and let
f
:
S
1
→
S
1
{\displaystyle f\colon S^{1}\to S^{1}}
be reflection in the x -axis, that is,
f
(
θ
)
=
−
θ
{\displaystyle f(\theta )=-\theta }
. Then
f
{\displaystyle f}
has Lefschetz number 2, while
f
2
{\displaystyle f^{2}}
is the identity map, which has Lefschetz number 0. Likewise, all odd iterates have Lefschetz number 2, while all even iterates have Lefschetz number 0. Therefore, the zeta function of
f
{\displaystyle f}
is
ζ
f
(
t
)
=
exp
(
∑
n
=
1
∞
2
t
2
n
+
1
2
n
+
1
)
=
exp
(
{
2
∑
n
=
1
∞
t
n
n
}
−
{
2
∑
n
=
1
∞
t
2
n
2
n
}
)
=
exp
(
−
2
log
(
1
−
t
)
+
log
(
1
−
t
2
)
)
=
1
−
t
2
(
1
−
t
)
2
=
1
+
t
1
−
t
{\displaystyle {\begin{aligned}\zeta _{f}(t )&=\exp \left(\sum _{n=1}^{\infty }{\frac {2t^{2n+1}}{2n+1}}\right)\\&=\exp \left(\left\{2\sum _{n=1}^{\infty }{\frac {t^{n}}{n}}\right\}-\left\{2\sum _{n=1}^{\infty }{\frac {t^{2n}}{2n}}\right\}\right)\\&=\exp \left(-2\log(1-t)+\log(1-t^{2})\right)\\&={\frac {1-t^{2}}{(1-t)^{2}}}\\&={\frac {1+t}{1-t}}\end{aligned}}}
[ edit ]
If f is a continuous map on a compact manifold X of dimension n (or more generally any compact polyhedron), the zeta function is given by the formula
ζ
f
(
t
)
=
∏
i
=
0
n
det
(
1
−
t
f
∗
|
H
i
(
X
,
Q
)
)
(
−
1
)
i
+
1
.
{\displaystyle \zeta _{f}(t )=\prod _{i=0}^{n}\det(1-tf_{\ast }|H_{i}(X,\mathbf {Q} ))^{(-1)^{i+1}}.}
Thus it is a rational function. The polynomials occurring in the numerator and denominator are essentially the characteristic polynomials of the map induced by f on the various homology spaces.
Connections
[ edit ]
This generating function is essentially an algebraic form of the Artin–Mazur zeta function , which gives geometric information about the fixed and periodic points of f .
See also
[ edit ]
References
[ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Lefschetz_zeta_function&oldid=1151924651 "
C a t e g o r i e s :
● Z e t a a n d L - f u n c t i o n s
● D y n a m i c a l s y s t e m s
● F i x e d p o i n t s ( m a t h e m a t i c s )
● T h i s p a g e w a s l a s t e d i t e d o n 2 7 A p r i l 2 0 2 3 , a t 0 2 : 2 9 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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