Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra Export

@misc{citeulike:186783, abstract = {This paper studies the behavior under iteration of the maps T\_{ab}(x,y) = (F\_{ab}(x)- y, x) of the plane R^2, in which F\_{ab}(x)= ax if x\>0 and bx if x\<0. These maps are area-preserving homeomorphisms of the plane that map rays from the origin into 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 written in an eigenvalue form for a nonlinear difference operator of Schrodinger type, in which \mu= 1/2(a-b) is viewed as fixed and the energy E=2- 1/2(a+b). The paper studies the set of parameter values where T\_{ab} has at least one nonzero bounded orbit, which corresponds to an l\_{\infty} eigenfunction of the difference operator. It shows that the for transcendental \mu the set of allowed energy values E for which there is a bounded orbit is a Cantor set. Numerical simulations suggest that this Cantor set may have positive one-dimensional measure, when \mu is held fixed.}, archivePrefix = {arXiv}, author = {Lagarias, Jeffrey C. and Rains, Eric}, citeulike-article-id = {186783}, citeulike-linkout-0 = {http://arxiv.org/abs/math.DS/0505103}, citeulike-linkout-1 = {http://arxiv.org/pdf/math.DS/0505103}, day = {6}, eprint = {math.DS/0505103}, keywords = {2d, area-preserving, discrete}, month = {May}, posted-at = {2005-05-09 08:46:33}, priority = {2}, title = {Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra}, url = {http://arxiv.org/abs/math.DS/0505103}, year = {2005} }

TY - ELEC ID - citeulike:186783 L3 - citeulike-article-id:186783 N2 - This paper studies the behavior under iteration of the maps T_{ab}(x,y) = (F_{ab}(x)- y, x) of the plane R^2, in which F_{ab}(x)= ax if x>0 and bx if x<0. These maps are area-preserving homeomorphisms of the plane that map rays from the origin into 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 written in an eigenvalue form for a nonlinear difference operator of Schrodinger type, in which \mu= 1/2(a-b) is viewed as fixed and the energy E=2- 1/2(a+b). The paper studies the set of parameter values where T_{ab} has at least one nonzero bounded orbit, which corresponds to an l_{\infty} eigenfunction of the difference operator. It shows that the for transcendental \mu the set of allowed energy values E for which there is a bounded orbit is a Cantor set. Numerical simulations suggest that this Cantor set may have positive one-dimensional measure, when \mu is held fixed. TI - Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra KW - 2d KW - area-preserving KW - discrete AU - Lagarias, Jeffrey AU - Rains, Eric PY - 2005/05/06/ UR - http://arxiv.org/abs/math.DS/0505103 ER -