Non Hyperbolic Solenoidal Thick Bony Attractors

• Zaj, M ^[1] ; Ghane, F H ^[1]
  1. [1] Ferdowsi University of Mashhad

     Ferdowsi University of Mashhad

     Irán

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 18, Nº 1, 2019, págs. 35-55
• Idioma: inglés
• DOI: 10.1007/s12346-018-0274-3
• Enlaces
  • Texto completo
• Resumen

  • This paper investigates the geometric structures of non-uniformly hyperbolic attractors of a certain class of skew products. We construct an open set of skew products with the fiber an interval over a linear expanding circle map such that any skew product belonging to this set admits a non-uniformly hyperbolic solenoidal attractor for which the following dichotomy is ascertained. This attractor is either the image of a continuous invariant graph under a semi conjugacy with nonempty interior, the so-called massive attractor, or a thick generalized bony attractor. Here, a generalized bony attractor is, roughly speaking, the image of a bony graph attractor under a semi conjugacy. Also, an attractor is thick if it has positive but not full Lebesgue measure. Moreover, in both cases, the attractors are mixing. In our construction, the contraction in the fiber is non-uniform and hence it is specified in terms of fiber Lyapunov exponents. Furthermore, we provide some related results on the ergodic properties of attracting graphs and stability results for such graphs under deterministic perturbations. In particular, we show that there exists an invariant ergodic SRB measure whose support is contained in that attractor.

• Referencias bibliográficas
  • 1. Alves, J., Pinheiro, V.: Topological structure of partially hyperbolic sets with positive volume. Trans. Am. Math. Soc. 360(10), 5551–5569...
  • 2. Anzai, H.: Ergodic skew product transformations on the torus. Osaka Math. J. 3, 83–99 (1951)
  • 3. Arnold, L., Crauel, H.: Iterated function systems and multiplicative ergodic theory. In: Pinsky, M.A., Wihstutz, V. (eds.) Diffusion Processes...
  • 4. Avila, A., Gouezel, S., Tsujii, M.: Smoothness of solenoidal attractors. Discrete Contin. Dyn. Syst. 15(1), 21–35 (2006)
  • 5. Bielecki, A.: Iterated function systems analogues on compact metric spaces and their attractors. Univ. Iagel. Acta Math. 32, 187–192 (1995)
  • 6. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, vol. 470. Springer, Berlin...
  • 7. Bowen, R.: A horseshoe with positive measure. Invent. Math. 29, 203–204 (1975)
  • 8. Broomhead, D., Hadjiloucas, D., Nicol, M.: Random and deterministic perturbation of a class of skew-product systems. Dyn. Stab. Syst. 14,...
  • 9. Bugeaud, Y.: Distribution Modulo One and Diophantine Approximation. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge...
  • 10. Campbell, K.M.: Observational noise in skew product systems. Phys. D 107, 43–56 (1997)
  • 11. Campbell, K.M., Davies, M.E.: The existence of inertial functions in skew product systems. Nonlinearity 9, 801–817 (1996)
  • 12. Crauel, H.: Extremal exponents of random dynamical systems do not vanish. J. Dyn. Differ. Equ. 2, 245–291 (1990)
  • 13. Davies, M.E., Campbell, K.M.: Linear recursive filters and nonlinear dynamics. Nonlinearity 9, 487– 499 (1996)
  • 14. Diaz, L.J., Gelfert, K.: Porcupine-like horseshoes: transitivity, Lyapunov spectrum, and phase transitions. Fund. Math. 216(1), 55–100...
  • 15. Edalat, A.: Power domains and iterated function systems. Inf. Comput. 124, 182–197 (1996)
  • 16. Ehsani, A., Fakhari, A., Ghane, F.H., Zaj, M.: Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid...
  • 17. Elton, J.H.: An ergodic theorem for iterated maps. Ergod. Theory Dyn. Syst. 7, 481–488 (1987)
  • 18. Elton, J.H.: A multiplicative ergodic theorem for Lipschitz maps. Stoch. Process. Appl. 34, 39–47 (1990)
  • 19. Furstenberg, H., Kesten, H.: Products of random matrices. Ann. Math. Stat. 31, 457–469 (1960)
  • 20. Gorodetski, A., Ilyashenko, Y.: Certain new robust properties of invariant sets and attractors of dynamical systems. Funct. Anal. Appl....
  • 21. Hirsch, M., Pugh, C.: Stable manifolds and hyperbolic sets. Bull. Am. Math. Soc. 75, 149–152 (1969)
  • 22. Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds. Lecture Notes in Mathematics, vol. 583. Springer, Berlin (1977)
  • 23. Hunt, B.R., Ott, E., Yorke, J.A.: Fractal dimensions of chaotic saddles of dynamical systems. Phys. Rev. E 54, 4819–4823 (1996)
  • 24. Hunt, B.R., Ott, E., Yorke, J.A.: Differentiable generalized synchronization of chaos. Phys. Rev. E 55, 4029 (1997)
  • 25. Ilyashenko, Y.: Thick attractors of step skew products. Regul. Chaotic Dyn. 15(2–3), 328–334 (2010)
  • 26. Ilyashenko, Y.: Thick attractors of boundary preserving diffeomorphisms. Indag. Math. 22(3–4), 257– 314 (2011)
  • 27. Ilyashenko, Y., Kleptsyn, V., Saltykov, P.: Openness of the set of boundary preserving maps of an annulus with intermingled attracting...
  • 28. Jachymski, J.R.: An fixed point criterion for continuous self mappings on a complete metric space. Aequ. Math. 48, 163–170 (1994)
  • 29. Kakutani, S.: Random ergodic theorems and Markoff processes with a stable distribution. In: Proceedings of Second Berkeley Symposium on...
  • 30. Kan, I.: Open sets of diffeomorphisms having two attractors, each with everywhere dense basin. Bull. Am. Math. Soc. 31, 68–74 (1994)
  • 31. Kleptsyn, V., Nalskii, M.B.: Contraction of orbits in random dynamical systems on the circle. Funct. Anal. Appl. 38(4), 267–282 (2004)
  • 32. Kleptsyn, V., Volk, D.: Physical measures for random walks on interval. Mosc. Math. J. 14(2), 339–365 (2014)
  • 33. Krengel, U.: Ergodic Theorems. de Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter, Berlin (1985)
  • 34. Kudryashov, Y.G.: Bony attractors. Funkts. Anal. Prilozhen. 44(3), 73–76 (2010). (English transl. Functional. Anal. Appl. 44(3), 219–222...
  • 35. Kudryashov, Y.G.: Des orbites pe´riodiques et des attracteurs des syste´mes dynamiques. Ph.D. thesis. E´ cole Normale Supe´rieure de Lyon,...
  • 36. Mane, R.: Ergodic Theory and Differentiable Dynamics. Volume 8 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics...
  • 37. Nobili, F.: Minimality of one invariant foliation for partially hyperbolic attractors. Nonlinearity 28, 1897–1918 (2015)
  • 38. Nobili, F.: Partially hyperbolic sets with a dynamically minimal invariant lamination. arxiv.org/pdf/1703.07413 (2017)
  • 39. Okunev, A.V., Shilin, I.S.: On the attractors of step skew products over the Bernoulli shift. arXiv:1703.01763v1 (2017)
  • 40. Palis, J.: A global view of dynamics and a conjecture on the denseness of finitude of attractors. Asterisque 261(xiiixiv), 335–347 (2000)
  • 41. Palis, J.: A global perspective for non-conservative dynamics. Ann. Inst. H. Poincare Anal. Non Lineaire 22(4), 485–507 (2005)
  • 42. Pecora, L.M., Carroll, T.L.: Discontinuous and nondifferentiable functions and dimension increase induced by filtering chaotic data. Chaos...
  • 43. Rams, M.: Absolute continuity of the SBR measure for non-linear fat baker maps. Nonlinearity 16(5), 1649–1655 (2003)
  • 44. Stark, J.: Invariant graphs for forced systems. Phys. D 109, 163–179 (1997)
  • 45. Stark, J.: Regularity of invariant graphs for forced systems. Ergod. Theory Dyn. Syst. 9, 155–199 (1999)
  • 46. Stark, J., Davies, M.E.: Recursive filters driven by chaotic signals. In: IEE Colloquium on Exploiting Chaos in Signal Processing. IEE...
  • 47. Tsujii, M.: Fat solenoidal attractors. Nonlinearity 14(5), 1011–1027 (2001)
  • 48. Tsujii, M.: Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194, 37–132 (2005)
  • 49. Viana, M., Yang, J.: Physical measures and absolute continuity for one-dimensional center direction. Ann. de Inst. Henri Poincare (C)...
  • 50. Viana, M., Yang, J.: Measure-theoretical properties of center foliations. arXiv:1603.03609 (2016)
  • 51. Volk, D.: Persistent massive attractors of smooth maps. Ergod. Theory Dyn. Syst. 34(2), 693–704 (2012)
  • 52. Williams, R.F.: Expanding attractors. Inst. Ht. Etudes Sci. Publ. Math. 43, 169–203 (1974)