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We study a family of renormalization transformations of generalized diamond hierarchical Potts models through complex dynamical systems. We prove that the Julia set (unstable set) of a renormalization transformation, when it is treated as a complex dynamical system, is the set of complex singularities of the free energy in statistical mechanics. We give a sufficient and necessary condition for the Julia sets to be disconnected. Furthermore, we prove that all Fatou components (components of the stable sets) of this family of renormalization transformations are Jordan domains with at most one exception which is completely invariant. In view of the problem in physics about the distribution of these complex singularities, we prove here a new type of distribution: the set of these complex singularities in the real temperature domain could contain an interval. Finally, we study the boundary behavior of the first derivative and second derivative of the free energy on the Fatou component containing the infinity. We also give an explicit value of the second order critical exponent of the free energy for almost every boundary point.

By the symmetry of the Julia set of a polynomial, we mean a Euclidean isometry preserving the Julia set. Each such symmetry is, in fact, a rotation about the centroid of the polynomial. This article conducts a survey of the symmetries of polynomial Julia sets. The Euclidean isometries, which preserve the Julia set of rational maps, are then considered. A rotation preserving the Julia set of a rational map is called a rotational symmetry of its Julia set. A sufficient condition is provided for a rational map to have rotational symmetries whenever the rational map has an exceptional point. Two classes of rational maps are provided whose Julia sets have rotational symmetries of finite orders. Using this, it is proved that z ↦ μ z where μ m + n = 1 is a rotational symmetry of the McMullen map z m + λ z n for all m ,  n with m ≥ 2 and λ ∈ C \\ { 0 } . Assuming that a normalized polynomial has a simple root at the origin, it is shown that the groups of the rotational symmetries of the polynomial coincide with that of its Newton’s method and Chebyshev’s method under certain assumptions.

Higher-dimensional hypercomplex fractal sets are getting more and more attention because of the discovery of more and more interesting properties and visual aesthetics. In this study, the attention was focused on generalized biquaternionic Julia sets and a generalization of classical Julia sets, defined by power and monic higher-order polynomials. Despite complex and quaternionic Julia sets, their biquaternionic analogues are still not well investigated. The performed morphological analysis of 3D projections of these sets allowed for definition of symmetries, limit shapes, and similarities with other fractal sets of this class. Visual observations were confirmed by stability analysis for initial cycles, which confirm similarities with the complex, bicomplex, and quaternionic Julia sets, as well as manifested differences between the considered formulations of representing polynomials.

In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map’s fixed points are explored. The existence of fractal Mandelbrot sets and Julia sets, as well as their fractal properties, are examined in detail. Several detailed simulations illustrate the effects of the fractional-order parameter, as well as the values of the map constant and exponent. In addition, complex domain controllers are constructed to control Julia sets produced by the proposed map or to achieve synchronization of two Julia sets in master/slave configurations. We identify the more realistic synchronization scenario in which the master map’s parameter values are unknown. Finally, numerical simulations are employed to confirm theoretical results obtained throughout the work.

In this paper, we achieve the control of spatial-alternated Julia sets and plane-alternated Julia sets by using the gradient control. And we achieve the synchronization of two different plane-alternated Julia sets using the gradient control. The simulations illustrate the effectiveness of the control methods.

Julia sets are considered one of the most attractive fractals and have wide range of applications in science and engineering. The strong physical meaning of Mandelbrot and Julia sets is broadly accepted and nicely connected by Christian Beck (Physica D 125(3–4):171–182, 1999 ) to the complex logistic maps, in the former case, and to the inverse complex logistic map, in the latter. Argyris et al. (Chaos Solitons Fractals 11(13):2067–2073, 2000 ) have studied the effect of noise on Julia sets and concluded that Julia sets are stable for noises of low strength, and a small increment in the strength of noise may cause considerable deterioration in the configuration of the Julia sets. It is well-known that the method of function iterates plays a crucial role in discrete dynamics utilizing the techniques of fractal theory. However, recently Rani and Kumar (J. Korea Soc. Math. Edu. Ser. D: Res. Math. Edu. 8(4):261–277, 2004 ) introduced superior iterations as a generalization of function iterations in the study of Julia sets and studied superior Julia sets. This technique is further utilized to study effectively new Mandelbrot sets and related properties (see, for instance, Negi and Rani, Chaos Solitons Fractals 36(2):237–245, 2008 ; 36(4):1089–1096, 2008 , Rani and Kumar, J. Korea Soc. Math. Edu. Ser. D: Res. Math. Edu. 8(4):279–291, 2004 ). The intent of this paper is to study certain effects of noise on superior Julia sets. We find that the superior Julia sets are drastically more stable for higher strength of noises than the classical Julia sets. Finally, we make a humble attempt to discuss some applications of superior orbit in discrete dynamics and of superior Julia sets in particle dynamics.

We survey the definition of the radial Julia set Jr(f)J_r(f) of a meromorphic function (in fact, more generally, any Ahlfors islands map), and give a simple proof that the Hausdorff dimension of Jr(f)J_r(f) and the hyperbolic dimension dimhyp⁡(f)\\dim _{\\operatorname {hyp}}(f) always coincide.

The aim of the present paper is to study the dynamics of a class of orbitally continuous non-linear mappings defined on the set of real numbers and to apply the results on dynamics of functions to obtain tests of divisibility. We show that this class of mappings contains chaotic mappings. We also draw Julia sets of certain iterations related to multiple lowering mappings and employ the variations in the complexity of Julia sets to illustrate the results on the quotient and remainder. The notion of orbital continuity was introduced by Lj. B. Ciric and is an important tool in establishing existence of fixed points.