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Generalized hyperbolic sets on Banach spaces

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Abstract

We prove that every compact invariant set close to a generalized hyperbolic fixed point of a diffeomorphism of a Banach space is generalized hyperbolic. Therefore, infinite weakly transitive shifted-hyperbolic sets do exist in every neighborhood of a shifted-hyperbolic fixed point.

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References

  1. D. V. Anosov, Geodesic flows on Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math., vol. 90, American Mathematical Society (Providence, RI, 1967).

  2. N. C. Bernardes and A. Messaoudi, A generalized Grobman-Hartman theorem, Proc. Amer. Math. Soc., 148 (2020), 4351–4360.

    Article  MathSciNet  MATH  Google Scholar 

  3. N. C. Bernardes, P. R. Cirilo, U. B. Darji, A. Messaoudi and E. R. Pujals, Expansivity and shadowing in linear dynamics. J. Math. Anal. Appl., 461 (2018), 796–816.

    Article  MathSciNet  MATH  Google Scholar 

  4. S-N. Chow, X-B. Lin and K. J. Palmer, A shadowing lemma with applications to semilinear parabolic equations. SIAM J. Math. Anal., 20 (1989), 547–557.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Cirilo, B. Gollobit and E. R. Pujals, Generalized hyperbolicity for linear operators, arXiv:1907.01146v2 (2019).

  6. P. Cirilo, B. Gollobit and E. Pujals, Dynamics of generalized hyperbolic linear operators, Adv. Math., 387 (2021), Paper No. 107830, 37 pp.

  7. S. Crovisier, Periodic orbits and chain-transitive sets of \(C^1\)-diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 87–141.

  8. E. D'Aniello, U. B. Darji and M. Maiuriello, Generalized hyperbolicity and shadowing in \(L^p\) spaces, J. Differential Equations, 298 (2021), 68–94.

  9. D. M. Grobman, Topological classification of neighborhoods of a singularity in n-space, Mat. Sb. (N.S.), 56 (1962), 77–94.

  10. Ph. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc., 11 (1960), 610–620.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, vol. 583, Springer-Verlag (Berlin–New York, 1977).

  12. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, with a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press (Cambridge, 1995).

  13. B. Lani-Wayda, Hyperbolic Sets, Shadowing and Persistence for Noninvertible Mappings in Banach Spaces, Pitman Research Notes in Mathematics Series, vol. 334, Longman (Harlow); copublished in the US with John Wiley & Sons, Inc. (New York, 1995).

  14. J. Palis, On the local structure of hyperbolic points in Banach spaces, An. Acad. Brasil. Ci., 40 (1968), 263–266.

    MathSciNet  MATH  Google Scholar 

  15. M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101–120.

    Article  MathSciNet  MATH  Google Scholar 

  16. C. C. Pugh, On a theorem of P. Hartman, Amer. J. Math., 91 (1969), 363–367.

  17. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747–817.

    Article  MathSciNet  MATH  Google Scholar 

  18. H. Steinlein and H-O. Walther, Hyperbolic sets and shadowing for noninvertible maps, in: Advanced Topics in the Theory of Dynamical Systems (Trento, 1987), Notes Rep. Math. Sci. Engrg., vol. 6, Academic Press (Boston, MA, 1989), pp. 219–234.

  19. H. Steinlein and H-O. Walther, Hyperbolic sets, transversal homoclinic trajectories, and symbolic dynamics for \(C^1\)-maps in Banach spaces, J. Dynam. Differential Equations, 2 (1990), 325–365.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Chungnam National University, Daejeon, Republic of Korea

    K. Lee

  2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, 21945-970, Rio de Janeiro, Brazil

    A. Rojas

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  1. K. Lee
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  2. A. Rojas
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Correspondence to A. Rojas.

Additional information

Work supported by Basic Science Research Program through the NRF funded by the Ministry of Education (Grant Number 2022R1l1A3053628).

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Lee, K., Rojas, A. Generalized hyperbolic sets on Banach spaces. Acta Math. Hungar. 168, 63–77 (2022). https://doi.org/10.1007/s10474-022-01266-7

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  • DOI: https://doi.org/10.1007/s10474-022-01266-7

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