Lefschetz zeta function

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 fcolon X o X} , the zeta-function is defined as the formal series 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 fcolon X o X} , the zeta-function is defined as the formal series 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} . The identity map on X {displaystyle X} has Lefschetz zeta function 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 fcolon S^{1} o S^{1}} be reflection in the x-axis, that is, f ( θ ) = − θ {displaystyle f( heta )=- heta } . 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 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 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. 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.