Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 085, 27 pages      arXiv:1803.06537      https://doi.org/10.3842/SIGMA.2018.085

Renormalization of the Hutchinson Operator

Yann Demichel
Laboratoire MODAL'X - EA3454, Université Paris Nanterre, 200 Avenue de la République, 92000 Nanterre, France

Received March 20, 2018, in final form August 10, 2018; Published online August 16, 2018

Abstract
One of the easiest and common ways of generating fractal sets in ${\mathbb R}^D$ is as attractors of affine iterated function systems (IFS). The classic theory of IFS's requires that they are made with contractive functions. In this paper, we relax this hypothesis considering a new operator $H_\rho$ obtained by renormalizing the usual Hutchinson operator $H$. Namely, the $H_\rho$-orbit of a given compact set $K_0$ is built from the original sequence $\big(H^n(K_0)\big)_n$ by rescaling each set by its distance from $0$. We state several results for the convergence of these orbits and give a geometrical description of the corresponding limit sets. In particular, it provides a way to construct some eigensets for $H$. Our strategy to tackle the problem is to link these new sequences to some classic ones but it will depend on whether the IFS is strictly linear or not. We illustrate the different results with various detailed examples. Finally, we discuss some possible generalizations.

Key words: Hutchinson operator; iterated function system; attractor; fractal sets.

pdf (2044 kb)   tex (1047 kb)

References

  1. Barnsley M.F., Fractals everywhere, 2nd ed., Academic Press, Boston, MA, 1993.
  2. Barnsley M.F., Demko S., Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A 399 (1985), 243-275.
  3. Barnsley M.F., Vince A., The eigenvalue problem for linear and affine iterated function systems, Linear Algebra Appl. 435 (2011), 3124-3138, arXiv:1004.5040.
  4. Barnsley M.F., Wilson D.C., Leśniak K., Some recent progress concerning topology of fractals, in Recent Progress in General Topology. III, Atlantis Press, Paris, 2014, 69-92.
  5. Berger M.A., Wang Y., Bounded semigroups of matrices, Linear Algebra Appl. 166 (1992), 21-27.
  6. Beyn W.-J., Elsner L., Infinite products and paracontracting matrices, Electron. J. Linear Algebra 2 (1997), 1-8.
  7. Bru R., Elsner L., Neumann M., Convergence of infinite products of matrices and inner-outer iteration schemes, Electron. Trans. Numer. Anal. 2 (1994), 183-193.
  8. Cox J.T., Durrett R., Some limit theorems for percolation processes with necessary and sufficient conditions, Ann. Probab. 9 (1981), 583-603.
  9. Daubechies I., Lagarias J.C., Sets of matrices all infinite products of which converge, Linear Algebra Appl. 161 (1992), 227-263.
  10. Edgar G.A., Measure, topology, and fractal geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1990.
  11. Elsner L., Friedland S., Norm conditions for convergence of infinite products, Linear Algebra Appl. 250 (1997), 133-142.
  12. Falconer K., Fractal geometry: mathematical foundations and applications, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003.
  13. Fisher G., Fractal image compression: theory and applications, Springer, New York, 1995.
  14. Fraser J.M., Inhomogeneous self-similar sets and box dimensions, Studia Math. 213 (2012), 133-156, arXiv:1301.1881.
  15. Gentil C., Neveu M., Mixed-aspect fractal surfaces, Comput.-Aided Des. 45 (2013), 432-439.
  16. Hartfiel D.J., Nonhomogeneous matrix products, World Scientific Publishing Co., Inc., River Edge, NJ, 2002.
  17. Hutchinson J.E., Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747.
  18. Le Gall J.-F., The topological structure of scaling limits of large planar maps, Invent. Math. 169 (2007), 621-670, math.PR/0607567.
  19. Mandelbrot B.B., The fractal geometry of nature, W.H. Freeman and Co., San Francisco, Calif., 1982.
  20. Marckert J.-F., Mokkadem A., Limit of normalized quadrangulations: the Brownian map, Ann. Probab. 34 (2006), 2144-2202, math.PR/0403398.
  21. Peres Y., Solomyak B., Problems on self-similar sets and self-affine sets: an update, in Fractal Geometry and Stochastics, II (Greifswald/Koserow, 1998), Progr. Probab., Vol. 46, Birkhäuser, Basel, 2000, 95-106.
  22. Richardson D., Random growth in a tessellation, Proc. Cambridge Philos. Soc. 74 (1973), 515-528.
  23. Seneta E., Non-negative matrices and Markov chains, Springer Series in Statistics, Springer, New York, 2006.
  24. Theys J., Joint spectral radius: theory and approximations, Ph.D. Thesis, University Catholique de Louvain, 2006.

Previous article  Next article   Contents of Volume 14 (2018)