On minimal manifolds
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- by Jan P. Boroński and George Kozlowski PDF
- Proc. Amer. Math. Soc. 149 (2021), 2669-2672
Abstract:
Let $M$ be a compact manifold of dimension at least 2. If $M$ admits a minimal homeomorphism, then $M$ admits a minimal noninvertible map.References
- J. Auslander and Y. Katznelson, Continuous maps of the circle without periodic points, Israel J. Math. 32 (1979), no. 4, 375–381. MR 571091, DOI 10.1007/BF02760466
- R. H. Bing, Upper semicontinuous decompositions of $E^3$, Ann. of Math. (2) 65 (1957), 363–374. MR 92960, DOI 10.2307/1969968
- J. P. Boroński, Alex Clark, and P. Oprocha, A compact minimal space $Y$ such that its square $Y\times Y$ is not minimal, Adv. Math. 335 (2018), 261–275. MR 3836665, DOI 10.1016/j.aim.2018.07.011
- J. P. Boroński, J. Činč, and M. Foryś-Krawiec, On rigid minimal spaces, Journal of Dynamics and Differential Equations, DOI 10.1007/s10884-020-09845-4
- J. P. Boroński, J. Kennedy, X. Liu, and P. Oprocha, Minimal non-invertible maps on the pseudo-circle, J. Dynam. Differential Equations, https://doi.org/10.1007/s10884-020-09877-w.
- François Béguin, Sylvain Crovisier, and Frédéric Le Roux, Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 2, 251–308 (English, with English and French summaries). MR 2339286, DOI 10.1016/j.ansens.2007.01.001
- Robert J. Daverman, Decompositions of manifolds, Pure and Applied Mathematics, vol. 124, Academic Press, Inc., Orlando, FL, 1986. MR 872468
- Albert Fathi and Michael R. Herman, Existence de difféomorphismes minimaux, Dynamical systems, Vol. I—Warsaw, Astérisque, No. 49, Soc. Math. France, Paris, 1977, pp. 37–59 (French). MR 0482843
- A. B. Katok, Minimal diffeomorphisms on principal 1-bundles, Abstracts of Sixth All-Union Topological Conference, Metsniereba Tbilisi, 1972, p. 63.
- Sergiĭ Kolyada, L’ubomír Snoha, and Sergeĭ Trofimchuk, Noninvertible minimal maps, Fund. Math. 168 (2001), no. 2, 141–163. MR 1852739, DOI 10.4064/fm168-2-5
- R. L. Moore, Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Soc. 27 (1925), no. 4, 416–428. MR 1501320, DOI 10.1090/S0002-9947-1925-1501320-8
- William Parry, A note on cocycles in ergodic theory, Compositio Math. 28 (1974), 343–350. MR 352407
- J. H. Roberts and N. E. Steenrod, Monotone transformations of two-dimensional manifolds, Ann. of Math. (2) 39 (1938), no. 4, 851–862. MR 1503441, DOI 10.2307/1968468
- L’. Snoha and V. Špitalský, Minimal space with non-minimal square, arXiv:1803.06323, 2018.
- Jakub Šotola and Sergei Trofimchuk, Construction of minimal non-invertible skew-product maps on 2-manifolds, Proc. Amer. Math. Soc. 144 (2016), no. 2, 723–732. MR 3430848, DOI 10.1090/proc12749
Additional Information
- Jan P. Boroński
- Affiliation: AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland; and National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic
- ORCID: 0000-0002-1802-4006
- Email: boronski@agh.edu.pl
- George Kozlowski
- Affiliation: Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849
- MR Author ID: 105955
- Email: kozloga@auburn.edu
- Received by editor(s): December 20, 2018
- Received by editor(s) in revised form: July 13, 2019
- Published electronically: March 16, 2021
- Additional Notes: This work was supported by the National Science Centre, Poland (NCN), grant no. 2015/19/D/ST1/01184.
- Communicated by: Nimish Shah
- © Copyright 2021 the authors
- Journal: Proc. Amer. Math. Soc. 149 (2021), 2669-2672
- MSC (2010): Primary 54H20
- DOI: https://doi.org/10.1090/proc/15186
- MathSciNet review: 4246815