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Keywords:
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.
References:
[1] Alsedà, L., Baldwin, S., Llibre, J., Swanson, R., Szlenk, W.: Minimal sets of periods for torus maps via Nielsen numbers. Pac. J. Math. 169 (1995), 1-32. DOI 10.2140/pjm.1995.169.1 | MR 1346243 | Zbl 0843.55004
[2] Berrizbeitia, P., Sirvent, V. F.: On the Lefschetz zeta function for quasi-unipotent maps on the $n$-dimensional torus. J. Difference Equ. Appl. 20 (2014), 961-972. DOI 10.1080/10236198.2013.872637 | MR 3210324 | Zbl 1305.37021
[3] Brown, R. F.: The Lefschetz Fixed Point Theorem. Scott, Foresman London (1971). MR 0283793 | Zbl 0216.19601
[4] Santos, N. M. dos, Urzúa-Luz, R.: Minimal homeomorphisms on low-dimensional tori. Ergodic Theory Dyn. Syst. 29 (2009), 1515-1528. MR 2545015
[5] Franks, J. M.: Homology and Dynamical Systems. CBMS Regional Conference Series in Mathematics 49 American Mathematical Society, Providence (1982). MR 0669378 | Zbl 0497.58018
[6] Franks, J. M.: Some smooth maps with infinitely many hyperbolic periodic points. Trans. Am. Math. Soc. 226 (1977), 175-179. MR 0436221 | Zbl 0346.58011
[7] Guirao, J. L. García, Llibre, J.: Minimal Lefschetz sets of periods for Morse-Smale diffeomorphisms on the $n$-dimensional torus. J. Difference Equ. Appl. 16 (2010), 689-703. DOI 10.1080/10236190903203887 | MR 2675600
[8] Llibre, J., Sirvent, V. F.: $C^1$ self-maps on closed manifolds with all their points hyperbolic. Houston J. Math 41 (2015), 1119-1127. MR 3455349
[9] Llibre, J., Sirvent, V. F.: Minimal sets of periods for Morse-Smale diffeomorphisms on non-orientable compact surfaces without boundary. J. Difference Equ. Appl. 19 (2013), 402-417. DOI 10.1080/10236198.2011.647006 | MR 3037282
[10] Llibre, J., Sirvent, V. F.: Minimal sets of periods for Morse-Smale diffeomorphisms on orientable compact surfaces. Houston J. Math. 35 (2009), 835-855 erratum ibid. 36 335-336 (2010). MR 2534284 | Zbl 1214.37027
[11] Shub, M., Sullivan, D.: Homology theory and dynamical systems. Topology 14 (1975), 109-132. DOI 10.1016/0040-9383(75)90022-1 | MR 0400306 | Zbl 0408.58023
[12] Smale, S.: Differentiable dynamical systems. With an appendix to the first part of the paper: Anosov diffeomorphisms'' by J. Mather Bull. Am. Math. Soc. 73 (1967), 747-817. MR 0228014 | Zbl 0202.55202
[13] Vick, J. W.: Homology Theory. An Introduction to Algebraic Topology. Graduate Texts in Mathematics 145 Springer, New York (1994). DOI 10.1007/978-1-4612-0881-5 | MR 1254439 | Zbl 0789.55004

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