Similarity between the Mandelbrot set and Julia sets

@article{Lei1990SimilarityBT,
  title={Similarity between the Mandelbrot set and Julia sets},
  author={Tan Lei},
  journal={Communications in Mathematical Physics},
  year={1990},
  volume={134},
  pages={587-617}
}
  • Tan Lei
  • Published 1990
  • Mathematics
  • Communications in Mathematical Physics
The Mandelbrot setM is “self-similar” about any Misiurewicz pointc in the sense that if we examine a neighborhood ofc inM with a very powerful microscope, and then increase the magnification by a carefully chosen factor, the picture will be unchanged except for a rotation. The corresponding Julia setJc is also “self-similar” in the same sense, with the same magnification factor. Moreover, the two setsM andJc are “similar” in the sense that if we use a very powerful microscope to look atM andJc… Expand
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References

SHOWING 1-10 OF 16 REFERENCES
Scaling of Mandelbrot sets generated by critical point preperiodicity
Letz→fμ(z) be a complex holomorphic function depending holomorphically on the complex parameter μ. If, for μ=0, a critical point off0 falls after a finite number of steps onto an unstable fixed pointExpand
A complete proof of the Feigenbaum conjectures
The Feigenbaum phenomenon is studied by analyzing an extended renormalization group map ℳ. This map acts on functionsΦ that are jointly analytic in a “position variable” (t) and in the parameter (μ)Expand
Universal properties of maps on an interval
We consider itcrates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied.Expand
Fixed points of Feigenbaum's type for the equationfp(λx)≡λf(x)
Existence and hyperbolicity of fixed points for the mapNp:f(x) →λ−1fp(λx), withfpp-fold iteration and λ=fp(0) are given forp large. These fixed points come close to being quadratic functions, and ourExpand
The universal metric properties of nonlinear transformations
AbstractThe role of functional equations to describe the exact local structure of highly bifurcated attractors ofxn+1 =λf(xn) independent of a specificf is formally developed. A hierarchy ofExpand
On the existence of Feigenbaum's fixed point
AbstractWe give a proof of the existence of aC2, even solution of Feigenbaum's functional equation $$g{\text{(}}x) = - \lambda _0^{ - 1} g{\text{(}}g( - \lambda _0 x)),g{\text{(0) = 1,}}$$ whereg isExpand
Complex analytic dynamics on the Riemann sphere
Dichotomie dynamique de Fatou et Julia. Points periodiques. Consequences du theoreme de Montel. L'ensemble de Julia est la fermeture de l'ensemble des points periodiques repulseurs. ResultatsExpand
The science of fractal images
TLDR
Fractal Modelling of Real World Images and a Unified Approach to Fractal Curves and Plants are studied. Expand
The Beauty of Fractals
A can made of a steel sheet the surface of which is coated with a three-layered chromium coating, consisting of a metallic chromium coating, a crystalline chromium oxide coating and a non-crystallineExpand
Invariant Distributions and Stationary Correlation Functions of One-Dimensional Discrete Processes
Abstract The connection between one-dimensional dynamical laws generating discrete processes and their invariant densities as well as their stationary correlaton functions is discussed. In particularExpand
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