On finite limit sets for transformations on the unit interval

Metropolis, N.; Stein, M. L.; Stein, P. R.

doi:10.1016/0097-3165(73)90033-2

Cited by 591 publications

(308 citation statements)

References 4 publications

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“…Much understanding has been achieved for one-dimensional unimodal maps [64,65,66,67,68,69,70]. Based on the Sharkovskii theorem about the ordering of periodic orbits [64], Metropolis, Stein and Stein organized periodic windows in universal symbolic sequences (U-sequences) such that the sequence of next order is uniquely determined by the previous one [65]. Later Jacobson came up with the proof that chaotic parameter values in one-dimensional unimodal maps with a single maximum do have positive measure [66].…”

Section: Periodic Windowsmentioning

confidence: 99%

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

Korabel

Klages

2004

Physica D: Nonlinear Phenomena

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The nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter. We show that on a suitable coarse scale this map generates an oscillating parameterdependent diffusion coefficient, similarly to hyperbolic maps, whose asymptotic functional form can be understood in terms of simple random walk approximations. On finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. By using a Green-Kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations. Starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized Takagi functions obeying generalized de Rham-type functional recursion relations. We finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive, and that for normal diffusion it increases by increasing the parameter value.

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Section: Periodic Windowsmentioning

confidence: 99%

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

Korabel

Klages

2004

Physica D: Nonlinear Phenomena

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“…Antiharmonics have already been introduced by Metropolis Stein and Stein in 1973 [31]. Normally, it is thought that antiharmonics have no real existence.…”

Section: Autocompositionmentioning

confidence: 99%

“…. In 1973 MSS introduced the concept of harmonic in 1D unimodal maps [31]. By extension, we call "MSS-harmonics" of a hyperbolic component of the Mandelbrot set antenna to the set constituted by itself and the disks of its period doubling cascade.…”

Section: Mss-harmonics Of a Hyperbolic Componentmentioning

confidence: 99%

“…In a previous paper, we based on the Metropolis, Stein and Stein (MSS) harmonics [31] to introduce other harmonics what we called there Fourier harmonics [21] though later we have called them F-harmonics or simply harmonics. By using this harmonic tool we could calculate the symbolic sequences of all the structural components (and the Misiurewicz points that separates them) of the 1D quadratic maps; i. e., we could determine the structure of such maps [23].…”

Section: Introductionmentioning

confidence: 99%

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Operating with external arguments in the Mandelbrot set antenna

Pastor

Romera

Álvarez

et al. 2002

Physica D: Nonlinear Phenomena

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The external argument theory of Douady and Hubbard allows us to know both the potential and the field-lines in the exterior of the Mandelbrot set. Nonetheless, there are no explicit formulae to operate with external arguments, and the external argument theory is difficult to apply. In this paper we introduce some tools in order to obtain formulae to operate with external arguments in the Mandelbrot set antenna. Thus, we introduce the harmonic tool to calculate both the external arguments of the period doubling cascade hyperbolic components and the external arguments of the last appearance hyperbolic components. Likewise, we introduce composition rules applied to external arguments that, with the aid of the concept of heredity, allows the calculation of all the external arguments that constitutes the family tree of a given external argument. 1

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“…for some i, then the itinerary stops). We note that Igix) is either an infinite sequence of R 's and L 's or is a finite sequence of R 's and L 's followed by a C. If g is unimodal and k. G [0,1 ] is such that the orbit of k under the scaled map kg contains 1/2, then the finite sequence / gik) is referred to as an MSS sequence [2,9].…”

mentioning

confidence: 99%

Uniqueness of Aperiodic Kneading Sequences

Brucks¹

1989

Proceedings of the American Mathematical Society

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Abstract.The trapezoidal function fe[x) is defined for fixed e 6 (0,1/2) by fe(x) = (l/e)x for x e [0,e], fe{x) =1 for x e (e, 1 -e), and fe(x) = (l/e)(l-jf) for x€ [l-e, 1], Foragiven e and the associated one-parameter family of maps {Xfe(x)\X e [0,1]} , we show that if A is an aperiodic kneading sequence, then there is a unique A 6 [0, 1] so that the itinerary of A under the map Xfe is A . From this, we conclude that the "stable windows" are dense in [0,1] for the one-parameter family Xfe .This note is mainly concerned with those maps which are trapezoidal. The trapezoidal function, fe, is defined for e G (0,1/2) by feix) = x/e for x G [0,e], feix) = l for xGie,l-e),and feix) = (1 -x)/e for xe [l-e,l]. For g unimodal and k G [0,1], the intinerary of k under the map kg, I 8ik), is referred to as the kneading sequence of kg [6]. BMS show that / sik) is shift maximal in the parity-lexicographical order when g is unimodal and k G [0,1 ] (throughout this note, when comparing kneading sequences the paritylexicographical order is used). Furthermore, if B is a finite or infinite shift maximal sequence, then there is some unimodal map g and some k G [0,1] so that I gik) = B . Thus any kneading sequence is shift maximal, and any shift maximal sequence is a kneading sequence. We note that if g is unimodal and

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On finite limit sets for transformations on the unit interval

Cited by 591 publications

References 4 publications

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

Operating with external arguments in the Mandelbrot set antenna

Uniqueness of Aperiodic Kneading Sequences

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