On Fatou sets concerning Yang-Lee theory

Properties of the different configurations of Julia sets J, generated by the complex map TZ: 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 TZ into the two-dimensional noninvertible map [Formula: see text]. For γ = 0, [Formula: see text] is semiconjugate to TZ 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 TZ, which are useful to understand some bifurcation sequences in the more general case considered in Part II. Different types of Julia sets are identified.

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Examples of Simply and Multiply Connected Fatou Sets for a Class of Meromorphic Functions

British Journal of Mathematics & Computer Science ◽

10.9734/bjmcs/2013/2507 ◽

2013 ◽

Vol 3 (2) ◽

pp. 108-114

Author(s):

P. Dom´ınguez ◽

A. Hern´andez

Keyword(s):

Meromorphic Functions ◽

Multiply Connected ◽

Fatou Sets

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Fatou Sets of Permutable Functions Satisfying Functional Equations

Proceedings of the Second ISAAC Congress - International Society for Analysis, Applications and Computation ◽

10.1007/978-1-4613-0271-1_13 ◽

2000 ◽

pp. 903-908

Author(s):

Yuefei Wang

Keyword(s):

Functional Equations ◽

Fatou Sets

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Total disconnectedness of Julia sets of random quadratic polynomials

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.148 ◽

2021 ◽

pp. 1-17

Author(s):

KRZYSZTOF LECH ◽

ANNA ZDUNIK

Keyword(s):

Dynamical System ◽

Julia Set ◽

Julia Sets ◽

Mandelbrot Set ◽

Quadratic Polynomials ◽

Large Disc ◽

Fatou Sets ◽

Autonomous Version ◽

Composition Of Functions ◽

Totally Disconnected

Abstract For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

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