Sums of convex compacta as attractors of hyperbolic IFS's

Valeriu Guţu



We prove that a finite union of convex compacta in $\mathbb{R}^n$ may be represented as the attractor of a hyperbolic IFS. If such a union is the condensation set for some hyperbolic IFS with condensation, then its attractor can be represented as the attractor of a standard hyperbolic IFS. We illustrate this result with the hyperbolic IFS with condensation, whose attractor is the well-known ``The Pythagoras tree'' fractal.


Attractor; iterated function system; convex set; Pythagoras tree

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E. Akin, The General Topology of Dynamical Systems, Amer. Math. Soc., Providence, RI, 1993.

T. Banakh and M. Nowak, A 1-dimensional Peano continuum which is not an IFS attractor, Proc. Amer. Math. Soc. 141 (2013), no. 3, 931–935.

M.F. Barnsley, Fractals Everywhere, Acad. Press Profess., Boston, 1993.

M.F. Barnsley and K. Leśniak, On the continuity of the Hutchinson operator, arXiv:1202.2485v1 [math.GN] 12 Feb 2012.

K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre, Fund. Math. 20 (1933), 177–190.

Ir.A.E. Bosman, Het wondere onderzoekingsveld der vlakke meetkunde, N.V. Uitgeversmaatschappij Parcival, Breda, 1957.

S. Crovisier and M. Rams, IFS attractors and Cantor sets, Topology Appl. 153 (2006), 1849–1859.

E. D’Aniello, Non-self-similar sets in [0, 1]N of arbitrary dimension, J. Math. Anal. Appl. 456 (2017), no. 2, 1123–1128.

E. D’Aniello and T.H. Steele, Attractors for iterated function schemes on [0, 1]N are exceptional, J. Math. Anal. Appl. 424 (2015), no. 1, 537–541.

E. D’Aniello and T.H. Steele, Attractors for iterated function systems, J. Fractal Geom. 3 (2016), no. 2, 95–117.

E. D’Aniello and T.H. Steele, Attractors for classes of iterated function systems, Eur. J. Math. 5 (2019), no. 1, 116–137.

P.F. Duvall and L.S. Husch, Attractors of iterated function systems, Proc. Amer. Math. Soc. 116 (1992), 279–284.

E. Friedman and D. Paterson, Covering squares with unit squares, Geombinatorics 15 (2006), no. 3, 130-137.

L. Gadomski, V. Glavan and V. Guţu, Playing Chaos game on the Pythagoras tree with CAS Mathematica, Computer Algebra Systems in Teaching and Research. Differential Equations, Dynamical Systems and Celestial Mechanics (L. Gadomski et al., eds.), Wydawnictwo Collegium Mazovia, Siedlce, 2011, 38–45.

V. Glavan and V. Guţu, On the dynamics of contracting relations, Analysis and Optimization of Differential Systems, (V. Barbu et al., eds.), Kluwer Academic Publishers, Boston, Dordrecht, London, 2003, 179–188.

L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Springer, Dordrecht, 2006.

V. Guţu, How to Construct Compact Convex Spots on the Plane Using CAS Mathematica, Computer Algebra Systems in Teaching and Research, Siedlce IV (2013), no. 1, 16–23.

M. Hata, On the structure of self-similar sets, Japan J. Appl. Math. 2 (1985), no. 2, 381–414.

J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747.

M. Kulczycki and M. Nowak, A class of continua that are not attractors of any IFS, Cent. Eur. J. Math. 10 (2012), no. 6, 2073–2076.

M. Kwieciński, A locally connected continuum which is not an IFS attractor, Bull. Pol. Acad. Sci. Math. 47 (1999), no. 2, 127–132.

R. McGehee, Attractors for closed relations on compact Hausdorff spaces, Indiana Univ. Math. J. 41 (1992), no. 4, 1165–1209.

V.S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Var. Anal. 6 (1998), 83–111.

S.B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 (1969), no. 2, 475488.

R.T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.

M.J. Sanders, Non-attractors of iterated function systems, Texas Project NexT J. 1 (2003), 1–9.

M.J. Sanders, An n-cell in Rn+1 that is not the attractor of any IFS on Rn+1 , Missouri J. Math. Sci. 21 (2009), no. 1, 13–20.


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