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- W2079561663 abstract "This paper studies the behavior under iteration of the maps T ab (x,y) = (F ab (x) − y, x) of the plane ℝ2, in which F ab (x) = ax if x ≥ 0 and bx if x < 0. These maps are area-preserving homeomorphisms of ℝ2 that map rays from the origin to rays from the origin. Orbits of the map correspond to solutions of the nonlinear difference equation x n+2 = 1/2(a − b)|x n+1|+1/2(a+b)x n+1 − x n . This difference equation can be rewritten in an eigenvalue form for a nonlinear difference operator of Schrödinger type − x n+2+2x n+1 − x n +V μ(x n+1)x n+1 = Ex n+1, in which μ = (1/2)(a − b) is fixed, and V μ(x) = μ(sgn(x)) is an antisymmetric step function potential, and the energy E = 2 − 1/2(a+b). We study the set Ω SB of parameter values where the map T ab has at least one bounded orbit, which correspond to l ∞-eigenfunctions of this difference operator. The paper shows that for transcendental μ the set Spec∞[μ] of energy values E having a bounded solution is a Cantor set. Numerical simulations suggest the possibility that these Cantor sets have positive (one-dimensional) measure for all real values of μ." @default.
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- W2079561663 date "2005-12-01" @default.
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- W2079561663 title "Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra" @default.
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- W2079561663 doi "https://doi.org/10.1080/10236190500273184" @default.
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