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Keywords:
hyperbolic periodic point; differentiable map; Lefschetz number; Lefschetz zeta function; quasi-unipotent map; almost quasi-unipotent map
hyperbolic periodic point; differentiable map; Lefschetz number; Lefschetz zeta function; quasi-unipotent map; almost quasi-unipotent map
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 \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.
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