Abstract
Let X be a topological manifold and \(f:X\rightarrow X\) a continuous map. We say the map f is partially periodic point free up to period n if f does not have periodic points of periods smaller than \(n+1\). A weaker notion is Lefschetz partially periodic point free up to period n, i.e. the Lefschetz numbers of the iterates of f up to n are all zero. Similarly f is Lefschetz periodic point free if the Lefschetz numbers of all iterates of f are zero. In the present article we consider continuous self-maps on products of any number spheres of different dimensions. We give necessary and sufficient conditions for such maps to be Lefschetz periodic point free and Lefschetz partially periodic point free. We apply these results to continuous self-maps on Lie groups.
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Alsedà, Ll, Llibre, J., Misiurewicz, M.: Combinatorial Dynamics and Entropy in Dimension One, Advanced Series in Nonlinear Dynamics, vol. 5, 2nd edn. World Scientific, Singapore (2000)
Barge, M.M., Walker, R.B.: Periodic point free maps of tori which have rotation sets with interior. Nonlinearity 6, 481–489 (1993)
Berrizbeitia, P., González, M.J., Sirvent, V.F.: On the Lefschetz zeta function and the minimal sets of Lefschetz periods for Morse-Smale diffeomorphisms on products of \(\ell \)-spheres. Topology Appl. 235, 428–444 (2018)
Brown, R.F.: The Lefschetz Fixed Point Theorem. Scott, Foresman and Company, Glenview, IL (1971)
Conner, P.E., Floyd, E.E.: On the construction of periodic maps without fixed points. Proc. Am. Math. Soc. 10, 354–360 (1959)
Duan, H.: The Lefschetz numbers of iterated maps. Topol. Appl. 67(1), 71–79 (1995)
Franks, J., Llibre, J.: Periods of surface homeomorphisms. Contemp. Math. 117, 63–77 (1991)
Fuller, F.B.: The existence of periodic points. Ann. Math. 57, 229–230 (1953)
Gierzkiewicz, A., Wójcik, K.: Lefschetz sequences and detecting periodic points. Discrete Contin. Dyn. Syst. 32, 81–100 (2012)
Graff, G., Kaczkowska, A.P., Nowak-Przygodzki, Signerska J: Lefschetz periodic point free self-maps of compact manifolds. J. Topol. Appl. 159, 2728–2735 (2012)
Guirao, J.L.G., Llibre, J.: On the Lefschetz periodic point free continuous self-maps on connected compact manifolds. Topology Appl. 158, 2165–2169 (2011)
Hall, G.R., Turpin, M.: Robustness of periodic point free maps of the annulus. Topol. Appl. 69, 211–215 (1996)
Handel, M.: Periodic point free homeomorphism of\({\mathbb{T}}^{2}\). Proc. Am. Math. Soc. 107, 511–515 (1989)
Halpern, B.: Periodic points on tori. Pacific J. Math. 83(1), 117–133 (1979)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Jezierski, J., Marzantowicz, W.: Homotopy methods in topological fixed and periodic points theory, Topological Fixed Point Theory and Its Applications 3. Springer, Dordrecht (2006)
Jäger, T.: Periodic point free homeomorphisms of the open annulus: From skew products to non-fibred maps. Proc. Am. Math. Soc. 138, 1751–1764 (2010)
Kumpel, P.G.: Lie groups and product of spheres. Proc. Am. Math. Soc. 16, 1350–1356 (1965)
Kwapisz, J.: A priori degeneracy of one-dimensional rotation sets for periodic point free torus maps. Trans. Am. Math. Soc. 354, 2865–2895 (2002)
Lefschetz, S.: Intersections and transformations of complexes and manifolds. Trans. Am. Math. Soc. 28, 1–49 (1926)
Llibre, J.: A note on the set of periods of transversal homological sphere self-maps. J. Differ. Equ. Appl. 9, 417–422 (2003)
Llibre, J.: Periodic point free continuous self-maps on graphs and surfaces. Topol. Appl. 159, 2228–2231 (2012)
Llibre, J., Sirvent, V.F.: Partially periodic point free self-maps on surfaces, graphs, wedge sums and products of spheres. J. Differ. Equ. Appl. 19, 1654–1662 (2013)
Llibre, J., Sirvent, V.F.:On Lefschetz periodic point free self-maps. J. Fixed Point Theory Appl. 20 , no. 1, Art. 38, 9 pp (2018)
Llibre, J., Sirvent, V.F.: Periodic structure of transversal maps on sum-free products of spheres. J. Differ. Equ. Appl. 25, 619–629 (2019)
Llibre, J., Todd, M.: Periods, Lefschetz numbers and entropy for a class of maps on a bouquet of circles. J. Differ. Equ. Appl. 11, 1049–1069 (2005)
Samelson, H.: Topology of Lie groups. Bull. Am. Math. Soc. 58, 2–37 (1952)
Smith, P.A.: Fixed-point theorems for periodic transformations. Am. J. Math. 63, 1–8 (1941)
Vick, J.W.: Homology theory. An introduction to algebraic topology, Springer–Verlag, New York, 1994. Academic Press, New York, 1973
Wang, S.: Free degrees of homeomorphisms and periodic maps on closed surfaces. Topol. Appl. 46, 81–87 (1992)
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Sirvent, V.F. Partially Periodic Point Free Self-Maps on Product of Spheres and Lie Groups. Qual. Theory Dyn. Syst. 19, 84 (2020). https://doi.org/10.1007/s12346-020-00419-9
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DOI: https://doi.org/10.1007/s12346-020-00419-9