Abstract
Stationary distributions for certain Markov chains of inverse branches of rational maps are put forward as the basis of an approximation theory for fractals. Results on existence and on computability of moments are proved.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. F. Barnsley, J. S. Geronimo, A. N. Harrington, "Orthogonal polynomials associated with invariant measures on Julia sets," Bulletin A.M.S. 7 (1982), 381–384.
—, "On the invariant sets of a family of quadratic maps," Comm. Math. Phys. (1983) 88, 479–501.
—, "Infinite dimensional Jacobi matrices associated with Julia sets," Proc. A.M.S. (1983) 88, 625–630.
—, "Geometry electrostatic measure and orthogonal polynomials on Julia sets for polynomials," to appear Journal of Ergodic Theory and Dynamical Systems (1982).
—, "Geometrical and electrical properties of some Julia sets," Ch. 1 of Quantum and Classical Models and Arithmetic Problems, (Dekker, 1984; edited by G. and D. Chudnovsky).
—, "Some treelike Julia sets and Padé approximants," Letters in Math. Phys. 7 (1983), 279–286.
—, "Almost periodic operators associated with Julia sets," Preprint (Georgia Institute of Technology, 1983), submitted to Comm. Math. Phys.
—, "Condensed Julia sets, with an application to a fractal lattice model Hamiltonian," submitted Trans. A.M.S. (1983).
M. F. Barnsley, A. N. Harrington, "Moments of balanced measures on Julia sets," to appear Trans. A.M.S. (1983).
D. Bessis, J. S. Geronimo, P. Moussa, "Mellin transforms associated with Julia sets and physical applications," Preprint (CEN-SACLAY, Paris, 1983).
D. Bessis, M. L. Mehta, P. Moussa, "Polynômes orthogonaux sur des ensembles de Cantor et iterations des transformations quadratiques," C. R. Acad. Sci. (Paris) 293 (1981), 705–708.
—, "Orthogonal polynomials on a family of Cantor sets and the problem of iteration of quadratic maps," Letters in Math. Phys. 6 (1982), 123–140.
D. Bessis, P. Moussa, "Orthogonality properties of iterated polynomial mappings," Comm. Math. Phys. 88 (1983), 503–529.
H. Brolin, "Invariant sets under iteration of rational functions," Arkiv för Matematik 6 (1965), 103–144.
L. Gaal, Classical Galois Theory with Examples, Markham, Chicago, 1971.
N. Dunford, J. Schwartz, Linear Operators (Part I), John Wiley, 1957.
R. Mañè, "On the uniqueness of the maximizing measure for rational maps," Preprint (Rio de Janeiro, 1982).
B. Mandelbrot, The Fractal Geometry of Nature, (W. H. Freeman, San Francisco, 1982).
T. S. Pitcher, J. R. Kinney, "Some connections between ergodic theory and iteration of polynomials," Arkiv för Matematik 8 (1968), 25–32.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1984 Springer-Verlag
About this paper
Cite this paper
Barnsley, M.F., Demko, S.G. (1984). Rational approximation of fractals. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072400
Download citation
DOI: https://doi.org/10.1007/BFb0072400
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13899-0
Online ISBN: 978-3-540-39113-5
eBook Packages: Springer Book Archive