\(C^1\) self-maps on closed manifolds with finitely many periodic points all of them hyperbolic. (English) Zbl 1413.37014

Llibre, Jaume; Sirvent, Víctor F.

Math. Bohem. 141, No. 1, 83-90 (2016).

Summary: Let \(X\) be a connected closed manifold and \(f\) a self-map on \(X\). We say that \(f\) is almost quasi-unipotent if every eigenvalue \(\lambda\) of the map \(f_{*k}\) (the induced map on the \(k\)-th homology group of \(X\)) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of \(\lambda\) as eigenvalue of all the maps \(f_{*k}\) with \(k\) odd is equal to the sum of the multiplicities of \(\lambda\) as eigenvalue of all the maps \(f_{*k}\) with \(k\) even. We prove that if \(f\) is \(C^1\) having finitely many periodic points all of them hyperbolic, then \(f\) is almost quasi-unipotent.

MSC:

37C30	Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37C05	Dynamical systems involving smooth mappings and diffeomorphisms
37C25	Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics

Keywords:

hyperbolic periodic point; differentiable map; Lefschetz number; Lefschetz zeta function; quasi-unipotent map