Periodic structure of the transversal maps on surfaces (Q2065130)

The authors consider the set of periods of transversal maps on orientable and non-orientable compact surfaces without boundary. A transversal map is defined as a \(C^1\) map \(f: X \rightarrow X\) on a compact differentiable manifold \(X\) such that \(f(X) \subset \mathrm{Int}(X)\) and for all positive integers \(m\) at each point \(x\) fixed by \(f^m\), the value \(1\) is not an eigenvalue of \(Df^m(x)\). The authors' main results are two theorems, one for orientable and one for non-orientable closed surfaces of genus \(g\). Their results are comparable to the results in [the first author and \textit{R. Swanson}, Houston J. Math. 19, No. 3, 395--403 (1993; Zbl 0787.55002)]. At the end of the paper they provide a detailed description of the differences between their results and those in [loc. cit.]. The main findings identify sufficient conditions so that the map \(f\) has infinitely many periods, and -- in particular -- odd periods. The sufficient conditions are described in terms of the spectra of the induced maps on the homology. The assumptions for the two theorems include a variety of technical conditions on the eigenvalues of \(f\). The authors use weaker assumptions than those in [loc. cit.], and their proofs are different. In particular, the current paper relies on the use of Lefschetz numbers of period \(m\).