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Lefschetz zeta function
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<h2 id="examples">Examples</h2> <p>The identity map on X {\displaystyle X} has Lefschetz zeta function </p> 1 ( 1 − t ) χ ( X ) , {\displaystyle {\frac {1}{(1-t)^{\chi (X)}}},} <p>where χ ( X ) {\displaystyle \chi (X)} is the <a href="/facts/Euler_characteristic/JCjzWUr2">Euler characteristic</a> of X {\displaystyle X} , i.e., the Lefschetz number of the identity map. </p><p>For a less trivial example, let X = S 1 {\displaystyle X=S^{1}} be the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a>, and let f : S 1 → S 1 {\displaystyle f\colon S^{1}\to S^{1}} be reflection in the <i>x</i>-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 </p> 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}}} <h2 id="formula">Formula</h2> <p>If <i>f</i> is a continuous map on a compact manifold <i>X</i> of dimension <i>n</i> (or more generally any compact polyhedron), the zeta function is given by the formula </p> 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}}.} <p>Thus it is a rational function. The polynomials occurring in the numerator and denominator are essentially the characteristic polynomials of the map induced by <i>f</i> on the various homology spaces. </p> <h2 id="connections">Connections</h2> <p>This generating function is essentially an <a href="/facts/Algebra/nMdqczjo">algebraic</a> form of the <a href="/facts/Artin%E2%80%93Mazur_zeta_function/vaR7IcvR">Artin–Mazur zeta function</a>, which gives <a href="/facts/Geometry/wTQf5ffQ">geometric</a> information about the fixed and periodic points of <i>f</i>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Lefschetz_fixed-point_theorem/TEjN2DQ8">Lefschetz fixed-point theorem</a></li> <li><a href="/facts/Artin%E2%80%93Mazur_zeta_function/vaR7IcvR">Artin–Mazur zeta function</a></li> <li><a href="/facts/Ruelle_zeta_function/prssMear">Ruelle zeta function</a></li></ul> <ul><li>Fel'shtyn, Alexander (2000), "Dynamical zeta functions, Nielsen theory and Reidemeister torsion", <i><a href="/facts/Memoirs_of_the_American_Mathematical_Society/L6BHqgsE">Memoirs of the American Mathematical Society</a></i>, 147 (699), <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/chao-dyn/9603017">chao-dyn/9603017</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697460">1697460</a></li></ul>