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- W2187965276 abstract "We analyse a farnily of discontinuous orientation - preserving maps of the circle onto itself. We obtain a nalytical expression for the location and for the lengthof each stepof the devil's staircase. We show that the staircase is complete and that the set of points corre - sponding to irrational winding numbers is a Cantor set of fractal di- mension -=O. Mappings have been used extensively to descri be dynamic systems that e xhibi t periodic as wel 1 as chaotic behavior. They arealso capable of describing the static properties of systems that possess modul ated stru~tures'-~. The standard map 4 which is studied in connection wi th the Frenkel -Kontorowa model of modul ated structures i s an exampl e of such maps. For maps of the circle onto itself which are continuous and in- vertible (homeomorphisms) the fol lowing resul ts have been obtained 5'6. The winding number w(R) is well def ined and is a continuous function of Q, the parameter that describes a family of mappings. For homeomorphisms that a re not pure rotations, w(Q) exhibi ts a p lateau for each rational value of W, that i s, the function w(Q) is a devil's staircase. If, in addition, the maps and their inverses are differentiable (diffeomor- phisms), then the devil's staircase is incomplete: the s et of pointscor- respondi ng to i rrational windi ng numbers has a nonzero measure. AI so, the orbit corresponding to an irrational winding number is topologically equivalent to a simple rotation what means that the orbit is one-dimen- sional. In this paper we analyse a family of orientation-preservingmaps of the circle onto itself which are not homeomorphisms. Nevertheless, some of the above results still hold. The winding number is well defined and consti t utes a devil's stai rcase. We obtain some exact r esul ts con- cerning t he devil's staircase including an analytical expression forthe" @default.
- W2187965276 created "2016-06-24" @default.
- W2187965276 creator A5073735567 @default.
- W2187965276 date "1988-01-01" @default.
- W2187965276 modified "2023-09-27" @default.
- W2187965276 title "Exact Calculation of the Devil's Staircase of a Discontinuous Linear Circle Map" @default.
- W2187965276 hasPublicationYear "1988" @default.
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