On the Continuity of the Hutchinson Operator

We investigate when the Hutchinson operator associated with an iterated function system is continuous. The continuity with respect to both the Hausdorff metric and Vietoris topology is carefully considered. An example showing that the Hutchinson operator on the hyperspace of nonempty closed bounded sets need not be Hausdorff continuous is given. Infinite systems are also discussed. The work clarifies and generalizes several partial results scattered across the literature.

[1]  J. Jachymski,et al.  Equivalence of some contractivity properties over metrical structures , 1997 .

[2]  Jean-Pierre Aubin,et al.  Differential Inclusions - Set-Valued Maps and Viability Theory , 1984, Grundlehren der mathematischen Wissenschaften.

[3]  Krzysztof Leśniak.  Stability and invariance of multivalued iterated function systems , 2003 .

[4]  W. Strother,et al.  Fixed points, fixed sets, and $M$-retracts , 1955 .

[5]  Keith R. Wicks,et al.  Fractals and Hyperspaces , 1992 .

[6]  Jan Andres,et al.  Metric and Topological Multivalued Fractals , 2004, Int. J. Bifurc. Chaos.

[7]  Krzysztof Leśniak.  Infinite Iterated Function Systems: A Multivalued Approach , 2004 .

[8]  Richard McGehee,et al.  Attractors for closed relations on compact Hausdorff spaces , 1992 .

[9]  Susumu Hayashi,et al.  Self-similar Sets as Tarski's Fixed Points , 1985 .

[10]  J. Aubin,et al.  Differential inclusions set-valued maps and viability theory , 1984 .

[11]  Richie Khandelwal,et al.  PATTERNS IN NATURE , 2005 .

[12]  E. Michael.  Topologies on spaces of subsets , 1951 .

[13]  The Hutchinson-Barnsley theory for certain non-contraction mappings , 1993 .

[14]  Gerald Beer,et al.  Topologies on Closed and Closed Convex Sets , 1993 .

[15]  Pablo G. Barrientos,et al.  On the chaos game of Iterated Function Systems , 2015, 1506.07101.

[16]  V. S. Melnik,et al.  On Attractors of Multivalued Semi-Flows and Differential Inclusions , 1998 .

[17]  Sam B. Nadler,et al.  Hyperspaces: Fundamentals and Recent Advances , 1999 .

[18]  L. Górniewicz.  Topological Fixed Point Theory of Multivalued Mappings , 1999 .

[19]  M. Barnsley,et al.  Iterated function systems and the global construction of fractals , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[20]  Shouchuan Hu,et al.  Handbook of multivalued analysis , 1997 .

[21]  M. Barnsley,et al.  Chaos game for IFSs on topological spaces , 2014, 1410.3962.

[22]  Andrew Vince,et al.  Real Projective Iterated Function Systems , 2010, 1003.3473.

[23]  M. Hata.  On some properties of set-dynamical systems , 1985 .

[24]  Efe A. Ok.  Fixed set theory for closed correspondences with applications to self-similarity and games☆ , 2004 .

[25]  John E. Hutchinson,et al.  V -variable fractals: Fractals with partial self similarity ✩ , 2008, 0802.0064.

[26]  Andrew Vince,et al.  The chaos game on a general iterated function system , 2010, Ergodic Theory and Dynamical Systems.

[27]  A. Arbieto,et al.  On Weakly Hyperbolic Iterated Function Systems , 2012, 1211.1738.

[28]  Ethan Akin,et al.  The general topology of dynamical systems , 1993 .