We consider the quadratic family of complex maps given by qc(z) = z 2 + c where c is the center of a hyperbolic component in the Mandelbrot set. Then, we introduce a singular perturbation on the corresponding bounded superattracting cycle by adding one pole to each point in the cycle. When c = −1 the Julia set of q−1 is the well known basilica and the perturbed map is given by fλ(z) = z 2 − 1 +...

1.1. Overview. We will describe recent developments in several intimately related problems of complex and real one-dimensional dynamics: rigidity of polynomials and local connectivity of the Mandelbrot set, measure of Julia sets, and attractors of quasi-quadratic maps. A combinatorial basis for this study is provided by the Yoccoz puzzle. The main problem is to understand the geometry of the pu...

Benoit Mandelbrot has produced a comprehensive, wellpresented review of essential topics related to Mandelbrot set theory and applications. The last part of the title "The Mandelbrot set and beyond" fully describes its potential allowing the reader to navigate through pictures, hard-tofind early papers and important and effective chapters on the historical background. All chapters are assembled...

Xue-Zao Ren, Zimo Yang, Bing-Hong Wang, and Tao Zhou School of Science, Southwest University of Science and Technology, Mianyang 621010, People’s Republic of China Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China Department of Modern Physics, University of Science and Technology of China, Hefei 230026, People’s Republic of...

A curious observation was made that the rank statistics of scientific citation numbers follows Zipf-Mandelbrot’s law. The same pow-like behavior is exhibited by some simple random citation models. The observed regularity indicates not so much the peculiar character of the underlying (complex) process, but more likely, than it is usually assumed, its more stochastic nature.

We study the combinatorics of distance doubling maps on the circle R/Z with prototypes h(β) = 2β mod 1 and h(β) = −2β mod 1, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where iterates f◦n of a distance doubling map f provide ‘distance doubling behavior’. The results include well-known statements for h relat...

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