A computer algorithm for determining the Hausdorff dimension of certain fractals

Author:
Lucy Garnett

Journal:
Math. Comp. **51** (1988), 291-300

MSC:
Primary 58F11; Secondary 30D05

DOI:
https://doi.org/10.1090/S0025-5718-1988-0942156-0

MathSciNet review:
942156

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Abstract: A fractal is a set which has nonintegral Hausdorff dimension. Computation of the dimension directly from the definition would be very time-consuming on a computer. However, the dimension can be computed using Newton's method if there exists a self-expanding map on the set. This technique is applied to compute the dimension of the Julia set of the quadratic mapping for small real values of *c*.

**[1]**Rufus Bowen,*Hausdorff dimension of quasicircles*, Inst. Hautes Études Sci. Publ. Math.**50**(1979), 11–25. MR**556580****[2]**Paul Blanchard,*Complex analytic dynamics on the Riemann sphere*, Bull. Amer. Math. Soc. (N.S.)**11**(1984), no. 1, 85–141. MR**741725**, https://doi.org/10.1090/S0273-0979-1984-15240-6**[3]**Adrien Douady and John Hamal Hubbard,*Itération des polynômes quadratiques complexes*, C. R. Acad. Sci. Paris Sér. I Math.**294**(1982), no. 3, 123–126 (French, with English summary). MR**651802****[4]**Benoit B. Mandelbrot,*The fractal geometry of nature*, W. H. Freeman and Co., San Francisco, Calif., 1982. Schriftenreihe für den Referenten. [Series for the Referee]. MR**665254****[5]**Benoit B. Mandelbrot,*Discussion paper: fractals, attractors, and the fractal dimension*, Bifurcation theory and applications in scientific disciplines (Papers, Conf., New York, 1977) Ann. New York Acad. Sci., vol. 316, New York Acad. Sci., New York, 1979, pp. 463–464. MR**556850****[6]**V. A. Norton, "Generation and display of geometric fractals in 3-D,"*Computer Graphics*, v. 16, 1982, pp. 61-67.**[7]**Edward Ott,*Strange attractors and chaotic motions of dynamical systems*, Rev. Modern Phys.**53**(1981), no. 4, 655–671. MR**629209**, https://doi.org/10.1103/RevModPhys.53.655**[8]**C. A. Rogers,*Hausdorff measures*, Cambridge University Press, London-New York, 1970. MR**0281862****[9]**David Ruelle,*Repellers for real analytic maps*, Ergodic Theory Dynamical Systems**2**(1982), no. 1, 99–107. MR**684247****[10]**D. Sullivan,*Seminar on Conformal and Hyperbolic Geometry*, Inst. Hautes Études Sci. Seminar notes, 1982, pp. 1-92.

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DOI:
https://doi.org/10.1090/S0025-5718-1988-0942156-0

Article copyright:
© Copyright 1988
American Mathematical Society