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Existence of invariant idempotent measures by contractivity of idempotent Markov operators

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Abstract

We prove that the idempotent Markov operator generated by contractive max plus normalized iterated function system (IFS) is also a contractive map w.r.t. natural metrics on the space of idempotent measures. This gives alternative proofs of the existence of invariant idempotent measures for such IFSs.

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References

  1. Akian, M.: Densities of idempotent measures and large deviations. Trans. Amer. Math. Soc. 351, 4515–4543 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barnsley, M.F.: Fractals Everywhere. Academic Press, Cambridge (1988)

    MATH  Google Scholar 

  3. Bazylevych, L., Repovš, D., Zarichnyi, M.: Spaces of idempotent measures of compact metric spaces. Topol. Appl. 157(1), 136–144 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. da Cunha, R.D., Oliveira, E.R., Strobin, F.: Fuzzy-set approach to invariant idempotent measures. Fuzzy Sets Syst. https://doi.org/10.1016/j.fss.2022.06.008 (2022)

  5. Del Moral, P., Doisy, M.: Maslov idempotent probability calculus. I. Theory Probab. Appl. 43(4), 562–576 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hutchinson, J.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jachymski, J., Jóźwik, I.: Nonlinear contractive conditions: a comparison and related problems. Banach Center Publ. 77, 123–146 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kolokoltsov, V.N., Maslov, V.P.: The general form of the endomorphisms in the space of continuous functions with values in a numerical semiring. Sov. Math. Dokl. 36, 55–59 (1988)

    Google Scholar 

  9. Kolokoltsov, V.N., Maslov, V.P.: Idempotent Analysis and its Applications. Kluwer Publishing House, Netherlands (1997)

    Book  MATH  Google Scholar 

  10. Litvinov, G.L., Maslov, V.P., Shpiz, G.B.: Idempotent (asymptotic) analysis and the representation theory. In: Asymptotic combinatorics with application to mathematical physics, pp. 267–278. Kluwer Academic Publications, Dordrecht (2002)

    Chapter  MATH  Google Scholar 

  11. Litvinov, G.. L.: Maslov dequantization, idempotent and tropical mathematics: A brief introduction. J. Math. Sci. 140(3), 426–444 (2007)

    Article  MathSciNet  Google Scholar 

  12. Mazurenko, N., Zarichnyi, M.: Idempotent ultrametric fractals. Visnyk Lviv Univ. Ser. Mech. Math 79, 111–118 (2014)

    Google Scholar 

  13. Mazurenko, N., Zarichnyi, M.: Invariant idempotent measures. Carpathian Math. Publ. 10(1), 172–178 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zaitov, A.A.: On a metric of the space of idempotent probability measures. Appl. Gen. Topol. 21(1), 35–51 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zarichnyi, M.M.: Spaces and maps of idempotent measures. Izv. Math. 74(3), 481–499 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank the referee for a careful reading of the paper, especially for pointing out several issues that were not sufficiently clear in a first version of the paper.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística - UFRGS, Av. Bento Gonçalves 9500, 91500-900, Porto Alegre - RS, Brazil

    Rudnei D. da Cunha & Elismar R. Oliveira

  2. Lodz University of Technology, Wólczańska 215, 90-924, Łódź, Poland

    Filip Strobin

Authors
  1. Rudnei D. da Cunha
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  2. Elismar R. Oliveira
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  3. Filip Strobin
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Correspondence to Elismar R. Oliveira.

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da Cunha, R.D., Oliveira, E.R. & Strobin, F. Existence of invariant idempotent measures by contractivity of idempotent Markov operators. J. Fixed Point Theory Appl. 25, 8 (2023). https://doi.org/10.1007/s11784-022-01004-0

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  • DOI: https://doi.org/10.1007/s11784-022-01004-0

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