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Resonance and fractal geometry

Resonance and fractal geometry

From gaps to

From gaps to tonguesuniversal geometry from gaps to tongueswith ε extra parametercollapse theory of gapswith Singularity Theory A 2k−1quasi-periodic analogueẍ + (a + εf(t))x = 0with f(t) = F(ω 1 t,ω 2 t,...,ω n t) whereF : T n → Rgeometry per tongue as beforeglobally fractal geometry with infinite regressH.W. Broer, H. Hanßmann, Á. Jorba, J. Villanueva and F.O.O. Wagener, Normal-internalresonances in quasi-periodically forces oscillators: a conservative approach. Nonlinearity16 (2003) 1751-1791

Quasi-periodic Schrödingertongues → gaps spectrum Schrödinger operator(H εq x) (t) = −ẍ(t) − εf(t)x(t)with potential εf, for x = x(t) ∈ L 2 (R) Cantor spectrum and ...J. Moser and J. Pöschel, An extension of a result by Dinaburg and Sinai on quasi-periodicpotentials. Comment. Math. Helvetici 59 (1984) 39-85L.H. Eliasson, Floquet solutions for the one-dimensional quasi-periodic Schrödingerequation. Commun. Math. Phys. 146 (1992) 447-482H.W. Broer, J. Puig and C. Simó, Resonance tongues and instability pockets in thequasi-periodic Hill-Schrödinger equation. Commun. Math. Phys. 241 (2003) 467-503

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