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

Abstract

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.

Keywords

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