Algebraic properties of the Lefschetz zeta function, periodic points and topological entropy (Q1802368)

The authors study the dynamical properties of the Lefschetz zeta function \(Z_ f\) associated to a continuous self-map \(f:M\to M\) of a compact manifold \(M\). These properties are consequences of some simple algebraic properties of \(Z_ f\) which is a rational function \(Z_ f(t)=P(t)/Q(t)\) where \(P(t)\), \(Q(t)\) are polynomials. For \(R(t)\) a polynomial, the authors define \(R^*(t)\) by \(R(t)=\) \((1-t)^ \alpha(1+t)^ \beta R^*(t)\) where \(\alpha,\beta\) are nonnegative integers such that \(1-t\), \(1+t\) do not divide \(R^*(t)\). Theorem 3.1. Let \(f:M\to\text{Int}(M)\) be a \(C^ 1\)-map on the compact manifold \(M\) and let \(P(t)/Q(t)=Z_ f(t)\) be its Lefschetz zeta function. If \(P^*(t)\) or \(Q^*(t)\) has odd degree then \(f\) has infinitely many periodic points. Theorem 3.2. Let \(f:M\to M\) be continuous on the compact connected surface \(M\). If \(P^*(t)\) or \(Q^*(t)\) has odd degree then \(f\) has positive topological entropy. Theorem 3.3. Let \(T^ n\) be the \(n\)-dimensional torus and let \(f:T^ n\to T^ n\) be a continuous map. If \(P^*(t)\) or \(Q^*(t)\) has odd degree then \(f\) has infinitely many (least) periods.