{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:30:27Z","timestamp":1772292627507,"version":"3.50.1"},"reference-count":28,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p> Properties of the different configurations of Julia sets J, generated by the complex map T<jats:sub>Z<\/jats:sub>: z\u2032 = z<jats:sup>2<\/jats:sup> - c, are revisited when c is a real parameter, -1\/4 &lt; c &lt; 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\u2032 = x<jats:sup>2<\/jats:sup> - \u03bb, first described in 1975. Part I of this paper constitutes a first step, leading to Part II dealing with an embedding of T<jats:sub>Z<\/jats:sub> into the two-dimensional noninvertible map [Formula: see text]. For \u03b3 = 0, [Formula: see text] is semiconjugate to T<jats:sub>Z<\/jats:sub> in the invariant half-plane (y \u2264 0). With a given value of c, and with \u03b3 decreasing, the identification of the global bifurcations sequence when \u03b3 \u2192 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<jats:sub>Z<\/jats:sub>, which are useful to understand some bifurcation sequences in the more general case considered in Part II. Different types of Julia sets are identified. <\/jats:p>","DOI":"10.1142\/s0218127409022877","type":"journal-article","created":{"date-parts":[[2009,3,20]],"date-time":"2009-03-20T10:36:08Z","timestamp":1237545368000},"page":"281-327","source":"Crossref","is-referenced-by-count":12,"title":["FROM THE BOX-WITHIN-A-BOX BIFURCATION ORGANIZATION TO THE JULIA SET PART I: REVISITED PROPERTIES OF THE SETS GENERATED BY A QUADRATIC COMPLEX MAP WITH A REAL PARAMETER"],"prefix":"10.1142","volume":"19","author":[{"given":"CHRISTIAN","family":"MIRA","sequence":"first","affiliation":[{"name":"19 rue d'Occitanie, Fonsegrives, 31130 QUINT, and Istituto di Scienze Economiche, University of Urbino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"LAURA","family":"GARDINI","sequence":"additional","affiliation":[{"name":"Istituto di Scienze Economiche, University of Urbino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1142\/S021812740300762X"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4422-6"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1984-15240-6"},{"key":"rf5","first-page":"545","volume":"24","author":"Blockh A. 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