Narrow Results

Properties of the different configurations of Julia sets J, generated by the complex map T_Z: z′ = z² - c, are revisited when c is a real parameter, -1/4 < c < 2. This is done from a detailed knowledge of the fractal bifurcation organization "box-within-a-box", related to the real Myrberg's map T: x′ = x² - λ, first described in 1975. Part I of this paper constitutes a first step, leading to Part II dealing with an embedding of T_Z into the two-dimensional noninvertible map . For γ = 0, is semiconjugate to T_Z in the invariant half-plane (y ≤ 0). With a given value of c, and with γ decreasing, the identification of the global bifurcations sequence when γ → 0, permits to explain a route toward the Julia sets. With respect to other papers published on the basic Julia and Fatou sets, Part I consists in the identification of J singularities (the unstable cycles and their limit sets) with their localization on J. This identification is made from the symbolism associated with the "box-within-a-box" organization, symbolism associated with the unstable cycles of J for a given c-value. In this framework, Part I gives the structural properties of the Julia set of T_Z, which are useful to understand some bifurcation sequences in the more general case considered in Part II. Different types of Julia sets are identified.

Part I of this paper has been devoted to properties of the different Julia set configurations, generated by the complex map T_Z: z′ = z² - c, c being a real parameter, -1/4 < c < 2. These properties were revisited from a detailed knowledge of the fractal organization (called "box-within-a-box"), generated by the map x′ = x² - c with x a real variable. Here, the second part deals with an embedding of T_Z into the two-dimensional noninvertible map ; y′ = γ y + 4x²y, γ ≥ 0. For is semiconjugate to T_Z in the invariant half plane (y ≤ 0). With a given value of c, and with γ decreasing, the identification of the global bifurcations sequence when γ → 0, permits to explain a route toward the Julia sets, from a study of the basin boundary of the attractor located on y = 0.