Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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INVARIANT GRAPH AND RANDOM BONY ATTRACTORS

  • Fateme Helen Ghane (Department of Mathematics Ferdowsi University of Mashhad) ;
  • Maryam Rabiee (Department of Mathematics Ferdowsi University of Mashhad) ;
  • Marzie Zaj (Department of Mathematics Ferdowsi University of Mashhad)
  • Received : 2021.04.24
  • Accepted : 2022.11.08
  • Published : 2023.03.01
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Abstract

In this paper, we deal with random attractors for dynamical systems forced by a deterministic noise. These kind of systems are modeled as skew products where the dynamics of the forcing process are described by the base transformation. Here, we consider skew products over the Bernoulli shift with the unit interval fiber. We study the geometric structure of maximal attractors, the orbit stability and stability of mixing of these skew products under random perturbations of the fiber maps. We show that there exists an open set U in the space of such skew products so that any skew product belonging to this set admits an attractor which is either a continuous invariant graph or a bony graph attractor. These skew products have negative fiber Lyapunov exponents and their fiber maps are non-uniformly contracting, hence the non-uniform contraction rates are measured by Lyapnnov exponents. Furthermore, each skew product of U admits an invariant ergodic measure whose support is contained in that attractor. Additionally, we show that the invariant measure for the perturbed system is continuous in the Hutchinson metric.

Keywords

Acknowledgement

We would like to thank anonymous reviewer whose remarks improved the presentation of the paper.

References

  1. Ya. I. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, in New results in operator theory and its applications, 7-22, Oper. Theory Adv. Appl., 98, Birkhauser, Basel, 1997.
  2. L. Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. https://doi.org/10.1007/978-3-662-12878-7
  3. L. Arnold and H. Crauel, Iterated function systems and multiplicative ergodic theory, in Diffusion processes and related problems in analysis, Vol. II (Charlotte, NC, 1990), 283-305, Progr. Probab., 27, Birkhauser Boston, Boston, MA, 1992.
  4. A. Bielecki, Iterated function system analogues on compact metric spaces and their attractors, Univ. Iagel. Acta Math. 32 (1995), 187-192.
  5. V. I. Bogachev, Measure Theory. Vol. I, II, Springer-Verlag, Berlin, 2007. https://doi.org/10.1007/978-3-540-34514-5
  6. D. Broomhead, D. Hadjiloucas, and M. Nicol, Random and deterministic perturbation of a class of skew-product systems, Dynam. Stability Systems 14 (1999), no. 2, 115-128. https://doi.org/10.1080/026811199282029
  7. H. Bruin, Notes on ergodic theory, Preprint, November 5, 2014.
  8. K. M. Campbell, Observational noise in skew product systems, Phys. D 107 (1997), no. 1, 43-56. https://doi.org/10.1016/S0167-2789(97)00056-0
  9. K. M. Campbell and M. E. Davies, The existence of inertial functions in skew product systems, Nonlinearity 9 (1996), no. 3, 801-817. https://doi.org/10.1088/0951-7715/9/3/010
  10. M. E. Davies and K. M. Campbell, Linear recursive filters and nonlinear dynamics, Nonlinearity 9 (1996), no. 2, 487-499. https://doi.org/10.1088/0951-7715/9/2/012
  11. L. J. D'iaz and K. Gelfert, Porcupine-like horseshoes: transitivity, Lyapunov spectrum, and phase transitions, Fund. Math. 216 (2012), no. 1, 55-100. https://doi.org/10.4064/fm216-1-2
  12. J. H. Elton, An ergodic theorem for iterated maps, Ergodic Theory Dynam. Systems 7 (1987), no. 4, 481-488. https://doi.org/10.1017/S0143385700004168
  13. J. H. Elton, A multiplicative ergodic theorem for Lipschitz maps, Stochastic Process. Appl. 34 (1990), no. 1, 39-47. https://doi.org/10.1016/0304-4149(90)90055-W
  14. S. Fadaei, G. Keller, and F. H. Ghane, Invariant graphs for chaotically driven maps, Nonlinearity 31 (2018), no. 11, 5329-5349. https://doi.org/10.1088/1361-6544/aae024
  15. H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist. 31 (1960), 457-469. https://doi.org/10.1214/aoms/1177705909
  16. M. W. Hirsch and C. C. Pugh, Stable manifolds for hyperbolic sets, Bull. Amer. Math. Soc. 75 (1969), 149-152. https://doi.org/10.1090/S0002-9904-1969-12184-1
  17. M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin, 1977.
  18. B. R. Hunt, E. Ott, and J. A. Yorke, Fractal dimensions of chaotic saddles of dynamical systems, Phys. Rev. E 54 (1996), 4819-4823. https://doi.org/10.1103/PhysRevE.54.4819
  19. B. R. Hunt, E. Ott, and J. A. Yorke, Differentiable generalized synchronization of chaos, Phys. Rev. E (3) 55 (1997), no. 4, 4029-4034. https://doi.org/10.1103/PhysRevE.55.4029
  20. J. Jachymski, An iff fixed point criterion for continuous self-mappings on a complete metric space, Aequationes Math. 48 (1994), no. 2-3, 163-170. https://doi.org/10.1007/BF01832983
  21. T. H. Jager, Quasiperiodically forced interval maps with negative Schwarzian derivative, Nonlinearity 16 (2003), no. 4, 1239-1255. https://doi.org/10.1088/0951-7715/16/4/303
  22. T. Jager and G. Keller, Random minimality and continuity of invariant graphs in random dynamical systems, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6643-6662. https://doi.org/10.1090/tran/6591
  23. V. Kleptsyn and D. Volk, Physical measures for nonlinear random walks on interval, Mosc. Math. J. 14 (2014), no. 2, 339-365, 428. https://doi.org/10.17323/1609-4514-2014-14-2-339-365
  24. A. S. Kravchenko, Completeness of the spaces of separable measures in the KantrovichRubinshtein metric, Siberian Math. J. 47 (2006), no. 1, 68-76; translated from Sibirsk. Mat. Zh. 47 (2006), no. 1, 85-96. https://doi.org/10.1007/s11202-006-0009-6
  25. Yu. G. Kudryashov, Bony attractors, Funct. Anal. Appl. 44 (2010), no. 3, 219-222; translated from Funktsional. Anal. i Prilozhen. 44 (2010), no. 3, 73-76. https://doi.org/10.1007/s10688-010-0028-8
  26. L. M. Pecora and T. L. Carroll, Discontinuous and nondifferentiable functions and dimension increase induced by filtering chaotic data, Chaos 6 (1996), no. 3, 432-439. https://doi.org/10.1063/1.166186
  27. M. Rabiee, F. H. Ghane, and M. Zaj, An open set of skew products with invariant multi-graphs and bony multi-graphs, Qual. Theory Dyn. Syst. 21 (2022), no. 4, Paper No. 147. https://doi.org/10.1007/s12346-022-00670-2
  28. J. Stark, Regularity of invariant graphs for forced systems, Ergodic Theory Dynam. Systems 19 (1999), no. 1, 155-199. https://doi.org/10.1017/S0143385799126555
  29. J. Stark and M. E. Davies, Recursive filters driven by chaotic signals, in IEEE Colloquium on Exploiting Chaos in Signal Processing 1 (1994), 431-516.
  30. O. Stenflo, A survey of average contractive iterated function systems, J. Difference Equ. Appl. 18 (2012), no. 8, 1355-1380. https://doi.org/10.1080/10236198.2011.610793
  31. R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. https://doi.org/10.1007/978-1-4684-0313-8
  32. M. Viana and K. Oliveira, Foundations of ergodic theory, Cambridge Studies in Advanced Mathematics, 151, Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781316422601
  33. P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982.
  34. M. Zaj, A. Fakhari, F. H. Ghane, and A. Ehsani, Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus, Discrete Contin. Dyn. Syst. 38 (2018), no. 4, 1777-1807. https://doi.org/10.3934/dcds.2018073
  35. M. Zaj and F. H. Ghane, Non hyperbolic solenoidal thick bony attractors, Qual. Theory Dyn. Syst. 18 (2019), no. 1, 35-55. https://doi.org/10.1007/s12346-018-0274-3