$C^1$ self-maps on closed manifolds with finitely many periodic points all of them hyperbolic

Llibre, Jaume ; Sirvent, Víctor F.

Mathematica Bohemica, Tome 141 (2016) no. 1, pp. 83-90.

Résumé

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 \mbox {$k$-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. \endgraf We prove that if $f$ is $C^1$ having finitely many periodic points all of them hyperbolic, then $f$ is almost quasi-unipotent.

Détail

DOI : 10.21136/MB.2016.6

Classification : 37C05, 37C25, 37C30
Keywords: hyperbolic periodic point; differentiable map; Lefschetz number; Lefschetz zeta function; quasi-unipotent map; almost quasi-unipotent map