We are interested in periodic points on the boundaries of rotation domains of rational functions R Perez-Marco showed that there are no periodic points on the boundaries of Siegel disks having Jordan neighborhoods with certain properties [12] In this paper, we consider periodic points on the boundaries of rotation domains under more weakly conditions

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Periodic points on the boundaries of rotation domains of some rational functions

Topology of the Julia sets of rational functions

The paper emphasized the importance of students’ creativity development based on the principle of variability and synergetic approach. The basic characteristics of the variability principle are also presented. The contribution of Russian teachers to studying the concepts of “mental flexibility” and the “principle of variability” is noted. This paper presents the ways of students’ mental flexibility development when elaborating the algorithms of constructing the Julia sets with numerous applications in various areas of the real world in different environments. The Julia sets of quadratic polynomials are considered first and the algorithms of constructing the Julia sets are proposed. During studying Chebyshev polynomials and elaborating their Julia sets, the possibility of developing students’ mental flexibility is substantiated on the specific problems. Attention to the filling Julia sets is given. The algorithms of constructing the Julia sets of rational functions are considered and the examples are given. A description of the algorithms is given with the aid of computer program. The parameter values, at which the Julia sets become smooth for quadratic polynomials, are specified. It is established that the segments of a real straight line are the Julia sets of Chebyshev polynomials. The examples of the Julia sets with a fractal structure are given. The relationship of the Julia sets with random maps is indicated. Construction of the Julia sets provides an opportunity to develop students’ mental flexibility—one of the most important creative qualities of personality.

Research Synergy of Multi-Stage Mathematical and Informational Tasks on Fractal Geometry

We are interested in biaccessible points in the Julia sets of rational functions. D. Schleicher and S. Zakeri studied which points can be biaccessible in the Julia sets of quadratic polynomials with irrationally indifferent fixed points [SZ, Za]. In this paper, we consider the two polynomial families fc(z) = zd + c, gθ(z) = e2πiθz + zd and the cubic rational family hθ,a(z) = e2πiθz2$\frac{z-a}{1-\bar{a}z}$.

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Mitsuhiko Imada

Papers

On biaccessible points in the Julia sets of some rational functions