Abstract
In this paper we will consider the concept of \(\mathbb {P}\)-weakly hyperbolic iterated function systems on compact metric spaces that generalizes the concept of weakly hyperbolic iterated function systems, as defined by Edalat (Inf Comput 124(2):182–197, 1996) and by Arbieto, Santiago and Junqueira (Bull Braz Math Soc New Ser 2016) for a more general setting where the parameter space is a compact metric space. We prove the existence and uniqueness of the invariant measure of a \(\mathbb {P}\)-weakly hyperbolic IFS. Furthermore, we prove an ergodic theorem for \(\mathbb {P}\)-weakly hyperbolic IFS with compact parameter space.
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References
Arbieto, A., Junqueira, A., Santiago, B.: On weakly hyperbolic iterated functions systems. Bull. Braz. Math. Soc, New Series (2016)
Barnsley, M.F.: Fractal image compression. Notices Am. Math. Soc. 43(2), 657–662 (1996)
Barnsley, M.F., Elton, J.H., Hardin, D.P.: Recurrent Iterated Function Systems. Constructive Approximation. Springer, New York (1989)
Barnsley, M.F., Vince, A.: The chaos game on a general iterated function systems. Ergodic Theory Dyn. Syst. 31(4), 1073–1079 (2011)
Barnsley, M.F., Demko, S.G., Elton, J.H., Geronimo, J.S.: Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities. Ann. Inst. H. Poincaré Probab. Stat. 24(3), 367–394 (1988)
Billingsley, P.: Probability and Measure. Wiley, New York (1986)
Cong, N.D., Son, D.T., Siegmund, S.: A computational ergodic theorem for infinite iterated. Stoch. Dyn. 8(3), 365–381 (2008)
Díaz, L., Matias, E.: Non-hyperbolic iterated function systems: attractors and stationary measures (preprint). arXiv:1605.02752
Edalat, A.: Power domain and iterated function systems. Inf. Comput. 124(2), 182–197 (1996)
Elton, J.: An ergodic theorem for iterated maps. Ergodic Theory Dyn. Syst. 7, 481–488 (1987)
Hata, M.: On the structure of self-similar sets. Japan J. Appl. Math. 2(2), 381–414 (1985)
Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)
Kravchenko, A.S.: Completeness of the space of separable measures in the Kantorovich-Rubinshtein metric. Sib. Math. J. 47(1), 68–76 (2006)
Malicet, D.: Random walks on Homeo\((S^1)\) (preprint). arXiv:1412.8618
Máté, L.: The Hutchinson–Barnsley theory for certain non-contraction mappings. Period. Math. Hungar. 27, 21–33 (1993)
Mendivil, F.: A generalization of ifs with probabilities to infinitely many maps. Rocky Mt. J. Math. 28(3), 1043–1051 (1998)
Öberg, A.: Approxiation of invariant measures for random iterations. Rocky Mt. J. Math. 36(1), 273–301 (2006)
Stenflo, O.: Ergodic theorems for iterated function systems with time dependent probabilities. Theory Stoch. Process. 3–4, 436–446 (1997)
Williams, R.F.: Composition of contractions. Bol. Soc. Brasil. Mat. 2(2), 55–59 (1971)
Acknowledgements
The author would like to thank to Ermerson Araujo and Fernando Lenarduzzi for useful conversations during the preparation of this work. The author was supported by a CNPq-Brazil doctoral fellowship.
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Melo, Í. On \(\mathbb {P}\)-Weakly Hyperbolic Iterated Function Systems. Bull Braz Math Soc, New Series 48, 717–732 (2017). https://doi.org/10.1007/s00574-017-0042-z
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DOI: https://doi.org/10.1007/s00574-017-0042-z