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

Šotola, Jakub; Trofimchuk, Sergei

doi:10.1090/proc12749

Construction of minimal non-invertible skew-product maps on 2-manifolds
HTML articles powered by AMS MathViewer

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