Estimation of the number of periodic orbits
The author is concerned with estimating the asymptotic growth rate of the Nielsen numbers \(N(f^n)\) for a map \(f : X \to X\) of a compact connected polyhedron. Pick a point \(v \in X\) and a path \(w\) from \(v\) to \(f(v)\). Denote \(G : = \pi_1 (X,w)\) and let \(f_G : = w_* f_* : G \to G\). Two elements \(g,g' \in G\) are said to be \(f_G\)-conjugate if there is an \(h \in G\) with \(g' = f_G (h) gh^{-1}\). Let \(f(x) = x\) and pick a path \(c\) from \(v\) to \(x\). The \(G\)-coordinate of \(x\) is defined to be the \(f_G\)-conjugacy class of \(w (f \circ c) c^{-1}\). The \(G\)-coordinate is constant on each fixed point class \({\mathbf F}\), so one defines \(\text{cd}_G ({\mathbf F}, f)\) to be this common value. The generalized Lefschetz number (or Reidemeister trace) is then defined to be \(L_G (f) = \sum \text{ind} ({\mathbf F}, f) \text{cd}_G ({\mathbf F}, f)\) where the summation extends over all fixed point classes \({\mathbf F}\) of \(f\). Let \(T_f\) be the mapping torus of \(f\), let \(\varphi\) be the semiflow on \(T_f\) generated by \(f\) and \(\Gamma : = \pi_1 (T_f,v)\) and denote by \(\Gamma_c\) the set of conjugacy classes in \(\Gamma\). If \(f^n x = x\) the \(\Gamma\)-coordinate of \((x,n)\), \(\text{cd}_\Gamma (x,n)\), is defined to be the free homotopy class of the loop \([0,n] \to X\) defined by \(t \mapsto \varphi_tx\). Points \(x\) and \(x'\) with \(f^n x = x\) and \(f^n x' = x'\) are said to be in the same \(n\)-orbit class \({\mathbf O}^n\) if \(\text{cd}_\Gamma (x,n) = \text{cd}_\Gamma (x',n)\). This common value is again denoted by \(\text{cd}_\Gamma ({\mathbf O}^n)\). The generalized Lefschetz number of \(f^n\) is then defined to be \(L_\Gamma (f^n) = \sum \text{ind} ({\mathbf O}^n, f^n) \text{cd}_\Gamma ({\mathbf O}^n) \in \mathbb{Z} \Gamma_c\) the summation being over all \(n\)-orbit classes. Consider a commutative ring \(R\) with unity and a representation \(\rho : G \to \text{GL}_\ell (R)\). This extends to a representation \(\rho : \mathbb{Z} \Gamma \to {\mathcal M}_\ell (R)\) in the ring of \((\ell, \ell)\)-matrices over \(R\). The \(\rho\)-twisted Lefschetz number of \(f^n\) then is defined as \(L_\rho (f^n) = \sum \text{ind} ({\mathbf O}^n, f^n) \text{tr(cd}_\Gamma ({\mathbf O}^n))^\rho \in R\) (summation being over all \(n\)-orbit classes), and the \(\rho\)-twisted Lefschetz zeta function is \(\zeta_\rho(f) = \exp \sum^\infty_{n = 0} L_\rho (f^n) t^n/n! \in R t\). The author estimates the growth rate of \(|L_\Gamma (f^n) |\) and of \(N (f^n)\) from above and he shows that each zero or pole of \(\zeta_\rho (f)\) (in case \(R = \mathbb{C})\) gives rise to a lower estimate for the growth rate of \(|L(f^n) |\). There is a wealth of examples and much more material in this article including an elegant proof of \textit{N. V. Ivanov}'s theorem [Sov. Math., Dokl. 26, 63-66 (1982); translation from Dokl. Akad. Nauk SSSR 265, 284-287 (1982; Zbl 0515.54016)] that the logarithm of the growth rate of \(N(f^n)\) is majorized by the topological entropy of \(f\). The same question for the growth rate of Lefschetz numbers is still open (even for smooth maps on compact manifolds). This article is clearly written and a pleasure to read though not always easy. For more information, see the author's contribution in [Contemp. Math. 152, 183-202 (1993; Zbl 0798.55001)].
scientific article; zbMATH DE number 910251