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A persistently singular map of $\mathbb{T}^n$ that is $C^2$ robustly transitive but is not $C^1$ robustly transitive

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Consider the high dimensional torus $\mathbb{T}^n$ and the set $\mathcal{E}$ of its endomorphisms. We construct a map in $\mathcal{E}$ that is robustly transitive if $\mathcal{E}$ is endowed with the $C^2$ topology but is not robustly transitive if $\mathcal{E}$ is endowed with the $C^1$ topology.

Abstract

Consider the high dimensional torus $\mathbb{T}^n$ and the set $\mathcal{E}$ of its endomorphisms. We construct a map in $\mathcal{E}$ that is robustly transitive if $\mathcal{E}$ is endowed with the $C^2$ topology but is not robustly transitive if $\mathcal{E}$ is endowed with the $C^1$ topology.

참고문헌 (12)

  1. Pierre Berger, "On the inverse limit stability of endomorphisms", Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 30(3), pp.463~475 (2013), European Mathematical Society - EMS - Publishing House GmbH[DOI]
  2. J. M. Boardman, "Singularities of differentiable maps", Inst. Hautes Etudes Sci. Publ. Math, (33), pp.21~57 (1967)
  3. Christian Bonatti, "A C1-generic dichotomy for diffeomorphisms:weak forms of hyperbolicity or infinitely many sinks or sources", Annals of Mathematics, 158(2), pp.355~418 (2003), Annals of Mathematics[DOI]
  4. M. Golubitsky, "Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14" (1973), Springer-Verlag
  5. B. Hasselblatt, "A First Course in Dynamics" (2003), Cambridge University Press
  6. J. Iglesias, "Robust transitivity for endomorphisms admitting critical points", Proc. Amer. Math. Soc, (3), pp.1235~1250 (2016)[DOI]
  7. Jorge Iglesias, "An example of a map which is C2-robustly transitive but not C1-robustly transitive", Colloquium Mathematicum, 152(2), pp.285~297 (2018), Institute of Mathematics, Polish Academy of Sciences[DOI]
  8. CRISTINA LIZANA, "Robust transitivity for endomorphisms", Ergodic Theory and Dynamical Systems, 33(4), pp.1082~1114 (2013), Cambridge University Press (CUP)[DOI]
  9. C. Lizana, "Topological obstructions for robustly transitive endomor-phisms on surfaces", arXiv:1711.02218, ~ (2017)
  10. C. Lizana, "New classes of C1 robustly transitive maps with persistent critical points", arXiv:1902.06781, ~ (2019)
  11. Ricardo Mane, "An Ergodic Closing Lemma", The Annals of Mathematics, 116(3), pp.503~ (1982), JSTOR[DOI]
  12. R. Thom, "Les singularites des applications differentiables, (French)", Ann. Inst. Fourier (Grenoble), 6, pp.43~87 (1955)

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