## Construction of minimal non-invertible skew-product maps on 2-manifolds

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- by Jakub Šotola and Sergei Trofimchuk PDF
- Proc. Amer. Math. Soc.
**144**(2016), 723-732 Request permission

## Abstract:

Applying the Hric-Jäger blow up technique, we give an affirmative answer to the question about the existence of non-invertible minimal circle-fibered self-maps of the Klein bottle. In addition, we present a simpler construction of a non-invertible minimal self-map of two-dimensional torus.## 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 - F. Béguin, S. Crovisier, Tobias Jäger, and F. Le Roux,
*Denjoy constructions for fibered homeomorphisms of the torus*, Trans. Amer. Math. Soc.**361**(2009), no. 11, 5851–5883. MR**2529917**, DOI 10.1090/S0002-9947-09-04914-9 - Alexander Blokh, Lex Oversteegen, and E. D. Tymchatyn,
*On minimal maps of 2-manifolds*, Ergodic Theory Dynam. Systems**25**(2005), no. 1, 41–57. MR**2122911**, DOI 10.1017/S0143385704000331 - Matúš Dirbák and Peter Maličký,
*On the construction of non-invertible minimal skew products*, J. Math. Anal. Appl.**375**(2011), no. 2, 436–442. MR**2735534**, DOI 10.1016/j.jmaa.2010.09.042 - Robert Ellis,
*The construction of minimal discrete flows*, Amer. J. Math.**87**(1965), 564–574. MR**185589**, DOI 10.2307/2373063 - S. Glasner and B. Weiss,
*On the construction of minimal skew products*, Israel J. Math.**34**(1979), no. 4, 321–336 (1980). MR**570889**, DOI 10.1007/BF02760611 - Roman Hric and Tobias Jäger,
*A construction of almost automorphic minimal sets*, Israel J. Math.**204**(2014), no. 1, 373–395. MR**3273462**, DOI 10.1007/s11856-014-1102-3 - Wen Huang and Yingfei Yi,
*Almost periodically forced circle flows*, J. Funct. Anal.**257**(2009), no. 3, 832–902. MR**2530846**, DOI 10.1016/j.jfa.2008.12.005 - T. Jäger, F. Kwakkel, and A. Passeggi,
*A classification of minimal sets of torus homeomorphisms*, Math. Z.**274**(2013), no. 1-2, 405–426. MR**3054336**, DOI 10.1007/s00209-012-1076-y - 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 - Sergiĭ Kolyada, Ľubomír Snoha, and Sergeĭ Trofimchuk,
*Proper minimal sets on compact connected 2-manifolds are nowhere dense*, Ergodic Theory Dynam. Systems**28**(2008), no. 3, 863–876. MR**2422019**, DOI 10.1017/S0143385707000740 - Sergii Kolyada, L’ubomír Snoha, and Sergei Trofimchuk,
*Minimal sets of fibre-preserving maps in graph bundles*, Math. Z.**278**(2014), no. 1-2, 575–614. MR**3267591**, DOI 10.1007/s00209-014-1327-1 - William Parry,
*A note on cocycles in ergodic theory*, Compositio Math.**28**(1974), 343–350. MR**352407** - M. Rees,
*A point distal transformation of the torus*, Israel J. Math.**32**(1979), no. 2-3, 201–208. MR**531263**, DOI 10.1007/BF02764916 - 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

## Additional Information

**Jakub Šotola**- Affiliation: Mathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01, Opava, Czech Republic
- Email: Jakub.Sotola@math.slu.cz
**Sergei Trofimchuk**- Affiliation: Instituto de Matemática y Fisica, Universidad de Talca, Casilla 747, Talca, Chile
- MR Author ID: 211398
- Email: trofimch@inst-mat.utalca.cl
- Received by editor(s): July 16, 2014
- Received by editor(s) in revised form: January 29, 2015
- Published electronically: August 11, 2015
- Communicated by: Yingfei Yi
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 723-732 - MSC (2010): Primary 37B05; Secondary 37E99, 54H20
- DOI: https://doi.org/10.1090/proc12749
- MathSciNet review: 3430848