Invariant measures for place-dependent idempotent iterated function systems. (English) Zbl 07875139
Summary: We study the set of invariant idempotent probabilities for place-dependent idempotent iterated function systems defined in compact metric spaces. Using well-known ideas from dynamical systems, such as the Mañé potential and the Aubry set, we provide a complete characterization of the densities of such idempotent probabilities. As an application, we provide an alternative formula for the attractor of a class of fuzzy iterated function systems.
MSC:
| 37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 37H10 | Generation, random and stochastic difference and differential equations |
| 08A72 | Fuzzy algebraic structures |
| 15A80 | Max-plus and related algebras |
| 28A33 | Spaces of measures, convergence of measures |
| 46E27 | Spaces of measures |
Keywords:
iterated function systems; Hutchinson measures; fuzzy sets; fixed points; idempotent measures; Maslov measuresReferences:
| [1] | Akian, M., Densities of idempotent measures and large deviations, Trans. Am. Math. Soc., 351, 11, 4515-4543, 1999 · Zbl 0934.28005 · doi:10.1090/S0002-9947-99-02153-4 |
| [2] | Akian, M., Quadrat, J.-P., Viot, M.: Duality between probability and optimization. In: Idempotency (Bristol, 1994), Publ. Newton Inst., vol. 11. Cambridge University Press, Cambridge, pp. 331-353 (1998) · Zbl 0906.60031 |
| [3] | Baraviera, A., Leplaideur, R., Lopes, A.: Ergodic optimization, zero temperature limits and the max-plus algebra. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro. \(29^{\rm o}\) Colóquio Brasileiro de Matemática. [29th Brazilian Mathematics Colloquium] (2013) · Zbl 1385.37011 |
| [4] | Baraviera, A.; Lopes, AO; Thieullen, P., A large deviation principle for the equilibrium states of Hölder potentials: the zero temperature case, Stoch. Dyn., 6, 1, 77-96, 2006 · Zbl 1088.60091 · doi:10.1142/S0219493706001657 |
| [5] | Bazylevych, L.; Repovš, D.; Zarichnyi, M., Spaces of idempotent measures of compact metric spaces, Topol. Appl., 157, 1, 136-144, 2010 · Zbl 1183.54009 · doi:10.1016/j.topol.2009.04.040 |
| [6] | Cabrelli, CA; Forte, B.; Molter, UM; Vrscay, ER, Iterated fuzzy set systems: a new approach to the inverse problem for fractals and other sets, J. Math. Anal. Appl., 171, 1, 79-100, 1992 · Zbl 0767.58024 · doi:10.1016/0022-247X(92)90377-P |
| [7] | Contreras, G.; Lopes, AO; Thieullen, P., Lyapunov minimizing measures for expanding maps of the circle, Ergod. Theory Dyn. Syst., 21, 5, 1379-1409, 2001 · Zbl 0997.37016 · doi:10.1017/S0143385701001663 |
| [8] | Conze, J.-P., Guivarc’h, Y., Propriété.: de droite fixe et fonctions propres des opérateurs de convolution. In: Séminaire de Probabilités, I (Univ. Rennes, Rennes, 1976). Université de Rennes, Département de Mathématiques et Informatique, Rennes, Exp. No. 4, 22 (1976) |
| [9] | 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(1), Paper No. 8, 11 (2023) · Zbl 07633374 |
| [10] | da Cunha, RD; Oliveira, ER; Strobin, F., Fuzzy-set approach to invariant idempotent measures, Fuzzy Sets Syst., 457, 46-65, 2023 · Zbl 07898087 · doi:10.1016/j.fss.2022.06.008 |
| [11] | Del Moral, P.; Doisy, M., Maslov idempotent probability calculus. I, Teor. Veroyatnost. i Primenen., 43, 4, 735-751, 1998 · Zbl 0952.60001 · doi:10.4213/tvp2029 |
| [12] | Diamond, P., Kloeden, P.: Metric spaces of fuzzy sets. World Scientific Publishing Co., Inc., River Edge (1994). Theory and applications · Zbl 0873.54019 |
| [13] | Fan, AH; Lau, K-S, Iterated function system and Ruelle operator, J. Math. Anal. Appl., 231, 2, 319-344, 1999 · Zbl 1029.37014 · doi:10.1006/jmaa.1998.6210 |
| [14] | Garibaldi, E.; Lopes, AO, On the Aubry-Mather theory for symbolic dynamics, Ergod. Theory Dyn. Syst., 28, 3, 791-815, 2008 · Zbl 1144.37003 · doi:10.1017/S0143385707000491 |
| [15] | Iturriaga, R.; Lopes, AO; Mengue, JK, Selection of calibrated subaction when temperature goes to zero in the discounted problem, Discrete Contin. Dyn. Syst., 38, 10, 4997-5010, 2018 · Zbl 1394.37040 · doi:10.3934/dcds.2018218 |
| [16] | Kolokol’tsov, VN; Maslov, VP, Idempotent analysis as a technique of the theory of control and optimal synthesis. I, Funktsional. Anal. i Prilozhen., 23, 1, 1-14, 1989 · Zbl 0692.49022 · doi:10.1007/BF01078568 |
| [17] | Kolokol’tsov, VN; Maslov, VP, Idempotent analysis as a technique of the theory of control and optimal synthesis. II, Funktsional. Anal. i Prilozhen, 23, 4, 53-62, 92, 1989 · Zbl 0709.49015 |
| [18] | Kolokoltsov, V.N., Maslov, V.P.: Idempotent analysis and its applications, Mathematics and its Applications, vol. 401. Kluwer Academic Publishers Group, Dordrecht (1997). Translation of Idempotent analysis and its application in optimal control (Russian), “Nauka” Moscow, [MR1375021 (97d:49031)], Translated by V. E. Nazaikinskii, With an appendix by Pierre Del Moral (1994) · Zbl 0857.49022 |
| [19] | Leplaideur, R., Mengue, J.: On the selection of subaction and measure for perturbed potentials, preprint (2024). arXiv:2404.02182v1 |
| [20] | Lewellen, GB, Self-similarity, Rocky Mt. J. Math., 23, 3, 1023-1040, 1993 · Zbl 0806.58033 · doi:10.1216/rmjm/1181072539 |
| [21] | Litvinov, GL; Maslov, VP, Idempotent mathematics: the correspondence principle and its computer realizations, Uspekhi Mat. Nauk 51, 6, 312, 209-210, 1996 · Zbl 0916.49019 |
| [22] | Mañé, R., Lagrangian flows: the dynamics of globally minimizing orbits, Bol. Soc. Brasil. Mat. (N.S.), 28, 2, 141-153, 1997 · Zbl 0892.58064 · doi:10.1007/BF01233389 |
| [23] | Mazurenko, N., Zarichnyi, M.: Invariant idempotent measures. Carpathian Math. Publ. 10(1), 172-178 (2018) · Zbl 1400.28009 |
| [24] | Mengue, J., Large deviations for equilibrium measures and selection of subaction, Bull. Braz. Math. Soc., 49, 1, 17-42, 2018 · Zbl 1384.60067 · doi:10.1007/s00574-017-0044-x |
| [25] | Mengue, J.; Oliveira, E., Duality results for iterated function systems with a general family of branches, Stoch. Dyn., 17, 3, 1750021, 2017 · Zbl 1362.37016 · doi:10.1142/S0219493717500216 |
| [26] | Mihail, A., The shift space of a recurrent iterated function system, Rev. Roumaine Math. Pures Appl., 53, 4, 339-355, 2008 · Zbl 1212.28025 |
| [27] | Mihail, A.; Miculescu, R., Applications of fixed point theorems in the theory of generalized IFS, Fixed Point Theory Appl., 11, 2008 · Zbl 1201.54037 · doi:10.1155/2008/312876 |
| [28] | Oliveira, ER; Strobin, F., Fuzzy attractors appearing from GIFZS, Fuzzy Sets Syst., 331, 131-156, 2018 · Zbl 1386.37015 · doi:10.1016/j.fss.2017.05.003 |
| [29] | Phelps, R.R.: Lectures on Choquet’s theorem, 2nd edn. Lecture Notes in Mathematics, vol. 1757. Springer, Berlin (2001) · Zbl 0997.46005 |
| [30] | Strobin, F.; Swaczyna, Ja, On a certain generalisation of the iterated function system, Bull. Aust. Math. Soc., 87, 1, 37-54, 2013 · Zbl 1282.54022 · doi:10.1017/S0004972712000500 |
| [31] | Zadeh, LA, Fuzzy sets, Inf Control, 8, 338-353, 1965 · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X |
| [32] | Zaitov, AA, On a metric on the space of idempotent probability measures, Appl. Gen. Topol., 21, 1, 35-51, 2020 · Zbl 1511.28012 · doi:10.4995/agt.2020.11865 |
| [33] | Zarichnyĭ, MM, Spaces and mappings of idempotent measures, Izv. Ross. Akad. Nauk Ser. Mat., 74, 3, 45-64, 2010 · Zbl 1220.18002 |
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