대한수학회지 (Journal of the Korean Mathematical Society)

  • 제58권4호
  • /
  • Pages.977-1000
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  • 2021
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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A PERSISTENTLY SINGULAR MAP OF 𝕋n THAT IS C2 ROBUSTLY TRANSITIVE BUT IS NOT C1 ROBUSTLY TRANSITIVE

  • 투고 : 2020.07.10
  • 심사 : 2021.04.15
  • 발행 : 2021.07.01
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초록

Consider the high dimensional torus 𝕋n and the set 𝜺 of its endomorphisms. We construct a map in 𝜺 that is robustly transitive if 𝜺 is endowed with the C2 topology but is not robustly transitive if 𝜺 is endowed with the C1 topology.

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과제정보

The author would like to give thanks in the first place to Dr. Jorge Iglesias without whose help this article wouldn't have been possible. And in the second place to Dr. Jorge Groisman and Dr. Roberto Markarian whose thorough reading and insightful comments helped improving the quality of this work.

참고문헌

  1. P. Berger and A. Rovella, On the inverse limit stability of endomorphisms, Ann. Inst. H. Poincar'e Anal. Non Lin'eaire 30 (2013), no. 3, 463-475. https://doi.org/10.1016/ j.anihpc.2012.10.001
  2. J. M. Boardman, Singularities of differentiable maps, Inst. Hautes Etudes Sci. Publ. ' Math. No. 33 (1967), 21-57.
  3. C. Bonatti, L. J. D'iaz, and E. R. Pujals, A C1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2) 158 (2003), no. 2, 355-418. https://doi.org/10.4007/annals.2003.158.355
  4. M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14. Springer-Verlag, New York-Heidelberg, 1973.
  5. B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University Press, New York, 2003. https://doi.org/10.1017/CBO9780511998188
  6. J. Iglesias, C. Lizana, and A. Portela, Robust transitivity for endomorphisms admitting critical points, Proc. Amer. Math. Soc. 144 (2016), no. 3, 1235-1250. https://doi.org/10.1090/proc/12799
  7. J. Iglesias and A. Portela, An example of a map which is C2-robustly transitive but not C1-robustly transitive, Colloq. Math. 152 (2018), no. 2, 285-297. https://doi.org/10.4064/cm7131-5-2017
  8. C. Lizana and E. Pujals, Robust transitivity for endomorphisms, Ergodic Theory Dynam. Systems 33 (2013), no. 4, 1082-1114. https://doi.org/10.1017/S0143385712000247
  9. C. Lizana and W. Ranter, Topological obstructions for robustly transitive endomorphisms on surfaces, arXiv:1711.02218, 2017.
  10. C. Lizana and W. Ranter, New classes of C1 robustly transitive maps with persistent critical points, arXiv:1902.06781, 2019.
  11. R. Mane, An ergodic closing lemma, Ann. of Math. (2) 116 (1982), no. 3, 503-540. https://doi.org/10.2307/2007021
  12. R. Thom, Les singularit'es des applications diff'erentiables, (French) Ann. Inst. Fourier (Grenoble) 6 (1955/56), 43-87. https://doi.org/10.5802/aif.60