Skip to main content Skip to main navigation menu Skip to site footer
  • Login
  • Language
    • English
    • Język Polski
  • Menu
  • Home
  • Current
  • Online First
  • Archives
  • About
    • About the Journal
    • Submissions
    • Editorial Team
    • Privacy Statement
    • Contact
  • Login
  • Language:
  • English
  • Język Polski

Topological Methods in Nonlinear Analysis

On a class of cocycles having attractors which consist of singletons
  • Home
  • /
  • On a class of cocycles having attractors which consist of singletons
  1. Home /
  2. Archives /
  3. Vol 50, No 2 (December 2017) /
  4. Articles

On a class of cocycles having attractors which consist of singletons

Authors

  • Grzegorz Guzik

Keywords

Cocycle, skew-product semiflow, iterated function system, topological limit, pullback attractor

Abstract

We give a new simple sufficient condition for existence of the global pullback attractor which consists of singletons for general cocycle mappings on an arbitrary complete metric space. In particular, we need not have any structure on a parameter space, so the criterion can be applied in both cases: nonautonomous as well as random dynamical systems. Our considerations lead us also to new large class of iterated function systems with point-fibred attractors.

References

L. Alseda and M. Misiurewicz, Random interval homeomorphisms, Proceedings of New Trends in Dynamical Systems, Salou 2012, Publ. Math. (2014), 15–36.

L. Alseda and M. Misiurewicz, Skew product attractors and concavity, Proc. Amer. Math. Soc. 143 (2015), 703–716.

L. Arnold, Random Dynamical Systems, Springer, Berlin, 1998.

R. Atkins, M. Barnsley, A. Vince and D. Wilson, A characterization of hyperbolic affine iterated function systems, Topology Proc. 36 (2010), 189–211.

M.F. Barnsley and A. Vince, Development in fractal geometry, Bull. Math. Sci. 3 (2013), 299–348.

M.C. Bortolan, A.N. Carvalho and J.A. Langa, Structure of attractors for skew product semiflows, J. Differential Equations 257 (2014), 490–522.

F.E. Browder, On the convergence of successive approximations for nonlinear functional equation, Indag. Math. 30 (1968), 27–35.

R.S. Burachik and A.N. Iusem, Set-valued Mappings and Enlargements of Monotone Operators, Springer, New York, 2008.

I. Chueshov, Monotone Random Systems Theory and Applications, Springer, Berlin, 2002.

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relatated Fields 100 (1994), 365–393.

P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev. 4 (1999), 45–76.

C.M. Goldie and R.A. Maller, Stability of perpetuities, Ann. Probab. 28 (2000), 1195–1218.

G. Guzik, Asymptotic stability of discrete cocycles, J. Difference Equ. Appl. 21 (2015), no. 11, 1044–1057.

G. Guzik, Asymptotic properties of multifunctions, families of measures and Markov operators associated with cocycles, Nonlinear Anal. 130 (2016), 59–75.

G. Guzik, Semiattractors of set–valued semiflows, J. Math. Anal. Appl. 435 (2016), 1321–1334.

G. Gwóźdź-Lukawska and J. Jachymski, The Barnsley–Hutchinson theory for infinite iterated function systems, Bull. Austral. Math. Soc. 72 (2005), no. 3, 441–454.

Á. Jorba, J.C. Tatjer and C. Nunez, Old and new results on strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 17 (2007), no. 11, 3895–3928.

R. Kapica, Random iteration and Markov operators, J. Difference Equ. Appl. 22 (2016), no. 2, 295–305.

P.E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, AMS Mathematical Surveys and Monographs, vol. 176, American Mathematical Society, Providence, 2011.

K. Kuratowski, Topology, vol. I, Academic Press, New York, 1966.

A. Lasota and J. Myjak, Attractors of multifunctions, Bull. Pol. Acad. Sci. Math. 48 (2000), no. 3, 319–334.

J. Matkowski, Integrable solutions of functional equations, Dissertationes Math. 127 (1975).

R. Miculescu and A. Mihail, Alternative characterization of hyperbolic affine iterated function systems, J. Math. Anal. Appl. 407 (2013), 56–68.

J. Morawiec and R. Kapica, Refinement equations and Feller operators, Integr. Equ. Oper. Theory 70 (2011), 323–331.

Downloads

  • PREVIEW
  • FULL TEXT

Published

2017-10-28

How to Cite

1.
GUZIK, Grzegorz. On a class of cocycles having attractors which consist of singletons. Topological Methods in Nonlinear Analysis. Online. 28 October 2017. Vol. 50, no. 2, pp. 727 - 739. [Accessed 6 July 2025].
  • ISO 690
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
Download Citation
  • Endnote/Zotero/Mendeley (RIS)
  • BibTeX

Issue

Vol 50, No 2 (December 2017)

Section

Articles

Stats

Number of views and downloads: 0
Number of citations: 0

Search

Search

Browse

  • Browse Author Index
  • Issue archive

User

User

Current Issue

  • Atom logo
  • RSS2 logo
  • RSS1 logo

Newsletter

Subscribe Unsubscribe
Up

Akademicka Platforma Czasopism

Najlepsze czasopisma naukowe i akademickie w jednym miejscu

apcz.umk.pl

Partners

  • Akademia Ignatianum w Krakowie
  • Akademickie Towarzystwo Andragogiczne
  • Fundacja Copernicus na rzecz Rozwoju Badań Naukowych
  • Instytut Historii im. Tadeusza Manteuffla Polskiej Akademii Nauk
  • Instytut Kultur Śródziemnomorskich i Orientalnych PAN
  • Instytut Tomistyczny
  • Karmelitański Instytut Duchowości w Krakowie
  • Ministerstwo Kultury i Dziedzictwa Narodowego
  • Państwowa Akademia Nauk Stosowanych w Krośnie
  • Państwowa Akademia Nauk Stosowanych we Włocławku
  • Państwowa Wyższa Szkoła Zawodowa im. Stanisława Pigonia w Krośnie
  • Polska Fundacja Przemysłu Kosmicznego
  • Polskie Towarzystwo Ekonomiczne
  • Polskie Towarzystwo Ludoznawcze
  • Towarzystwo Miłośników Torunia
  • Towarzystwo Naukowe w Toruniu
  • Uniwersytet im. Adama Mickiewicza w Poznaniu
  • Uniwersytet Komisji Edukacji Narodowej w Krakowie
  • Uniwersytet Mikołaja Kopernika
  • Uniwersytet w Białymstoku
  • Uniwersytet Warszawski
  • Wojewódzka Biblioteka Publiczna - Książnica Kopernikańska
  • Wyższe Seminarium Duchowne w Pelplinie / Wydawnictwo Diecezjalne „Bernardinum" w Pelplinie

© 2021- Nicolaus Copernicus University Accessibility statement Shop