Abstract
We study weakly hyperbolic iterated function systems on compact metric spaces, as defined by Edalat (Inform Comput 124(2):182–197, 1996), but in the more general setting of compact parameter space. We prove the existence of attractors, both in the topological and measure theoretical viewpoint and the ergodicity of invariant measure. We also define weakly hyperbolic iterated function systems for complete metric spaces and compact parameter space, extending the above mentioned definition. Furthermore, we study the question of existence of attractors in this setting. Finally, we prove a version of the results by Barnsley and Vince (Ergodic Theory Dyn Syst 31(4):1073–1079, 2011), about drawing the attractor (the so-called the chaos game), for compact parameter space.
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Notes
We shall use the convention that \(w_{\sigma _j}\circ \dots \circ w_{\sigma _1}(x)=x\), if \(j=0\).
References
Barnsley, M.F.: Fractals Everywhere. Academic Press, New York (1988)
Barnsley, M.F.: Fractal image compression. Notices Am. Math. Soc. 43(6), 657–662 (1996)
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. Statist. 24(3), 367–394 (1988)
Barnsley, M.F., Vince, Andrew: The chaos game on a general iterated function system. Ergodic Theory Dyn. Syst. 31(4), 1073–1079 (2011)
Barnsley, M.F., Vince, Andrew: The Conley attractor of an iterated function system. Bull. Aust. Math. Soc. 88(2), 267–279 (2013)
Cong, N.D., Son, D.T., Siegmund, S.: A computational ergodic theorem for infinite iterated function systems. Stoch. Dyn. 8(3), 365–381 (2008)
Edalat, A.: Power domains and iterated function systems. Inf. Comput. 124(2), 182–197 (1996)
Hata, M.: On the structure of self-similar sets. Jpn. J. Appl. Math. 2(2), 381–414 (1985)
Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)
Jachymski, J.R.: An iff fixed point criterion for continuous self-mappings on a complete metric space. Aequat. Math. 48, 163–170 (1994)
Jachymski, J.R., Lukawska, G.G.: The Hutchinson–Barnsley Theory for infinite iterated function systems. Bull. Austral. Math. Soc. 72, 441–454 (2005)
Kravchenko, A.S.: Completeness of the space of separable measures in the Kantorovich–Rubinshtein metric. Siberian Math. J. 47(1), 68–76 (2006)
Lewellen, G.B.: Self-similarity. Rocky Mt. J. Math. 23(3), 1023–1040 (1993)
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)
Walters, P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79. Springer, New York (1982)
Williams, R.F.: Composition of contractions. Bol. Soc. Brasil. Mat. 2(2), 55–59 (1971)
Acknowledgments
We would like to thank Katrin Gelfert and Daniel Oliveira for presenting us the paper of Kravchenko (2006).
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A.A. was supported by CNPq, FAPERJ, CAPES and PRONEX-DS. and André Junqueira A.J. was supported by CNPq and FAPERJ. and Bruno Santiago B.S. was supported by CNPq and FAPERJ.
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Arbieto, A., Junqueira, A. & Santiago, B. On Weakly Hyperbolic Iterated Function Systems. Bull Braz Math Soc, New Series 48, 111–140 (2017). https://doi.org/10.1007/s00574-016-0018-4
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DOI: https://doi.org/10.1007/s00574-016-0018-4