Contractive function systems, their attractors and metrization
KeywordsFractal, attractor, iterated function system, contracting function system, fixed point
AbstractIn this paper we study the Hutchinson-Barnsley theory of fractals in the setting of multimetric spaces (which are sets endowed with point separating families of pseudometrics) and in the setting of topological spaces. We find natural connections between these two approaches.
R. Atkins, M. Barnsley, A. Vince and D.C. Wilson, A characterization of hyperbolic affine iterated function systems, Topology Proc. 36 (2010), 189–2011.
T. Banakh, M. Nowak, A 1-dimensional Peano continuum which is not an IFS attractor, Proc. Amer. Math. Soc. 141 (2013), 931–935.
M.F. Barnsley, Fractals everywhere. Academic Press Professional, Boston, MA, 1993.
M. Barnsleya and K. Igudesman, Topological contracting systems, Lobachevskii Journal of Mathematics, 32 (2011), 220–223.
M. Barnsley and A. Vince, The eigenvalue problem for linear and affine iterated function systems, Linear Algebra Appl. 435 (2011), 3124–3138.
J. Diestel, Sequences and Series in Banach Spaces. Springer–Verlag, New York, 1984.
D. Dymitru, Attractors of topological iterated function system, Annals of Spiru Haret University: Mathematics-Informatics series, 8 (2012), 11–16.
A. Edalat, Power domains and iterated function systems, Inform. and Comput. 124 (1996), no. 2, 182–197.
M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74–79.
R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
M. Hata, On the structure of self-similar sets, Japan J. Appl. Math. 2 (1985), 381–414.
J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713–747.
J. Jachymski, Remetrization theorems for finite families of mappings, and hyperbolic iterated function systems, preprint.
J. Jachymski and I. Józwik, Nonlinear contractive conditions: a comparison and related problems, Banach Center Publ. Polish Acad. Sci., 77 (2007), 123–146.
A. Kameyama, Distances on topological self-similar sets and the kneading determinants, J. Math. Kyoto Univ. 40 (2000), 601–672.
B. Kieninger, Iterated Function Systems on Compact Hausdorff Spaces. Ph.D. Thesis, Augsburg University, Shaker–Verlag, Aachen 2002.
S. Leader, Equivalent Cauchy sequences and contractive fixed points in metric spaces, Studia Math. 76 (1983), 63–67.
J. Matkowski, Integrable solutions of functional equations, Diss. Math. 127 (1975) 68 pp.
R. Miculescu and A. Mihail, Alternative characterization of hyperbolic affine infinite iterated function systems, J. Math. Anal. Appl. 407 (2013) 56–68.
R. Miculescu and A. Mihail, On a question of A. Kameyama concerning self-similar metrics, J. Math. Anal. Appl. DOI: 10.1016/j.jmaa.2014.08.008.
A. Mihail, A topological version of iterated function systems, An. Stiint. Univ. Al. I. Cuza, Ia si, (S.N.), Matematica, Tom 58 (2012), 105–120.
M. Nowak and T. Szarek, The shark teeth is a topological IFS-attractor, Siberian Math. J. 55 (2014), 296–300.
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