Dynamics towards the Feigenbaum attractor.pdfVIP

Dynamics towards the Feigenbaum attractor A. Robledo Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, Apartado postal 20-364, M´exico 01000 D.F., M´exico L. G. Moyano Departamento de Matem´aticas and Grupo Interdisciplinar de Sistemas Complejos, 7 Universidad Carlos III de Madrid, 28911 Legan´es, Madrid, Spain 0 0 We expose at a previously unknown level of detail the features of the dynamics of trajectories that 2 either evolve towards the Feigenbaum attractor or are captured by its matching repellor. Amongst these features are the following: i) The set of preimages of the attractor and of the repellor are n embedded (dense) into each other. ii) The preimage layout is obtained as the limiting form of u the rank structure of the fractal boundaries between attractor and repellor positions for the family J of supercycle attractors. iii) The joint set of preimages for each case form an infinite number of 9 families of well-defined phase-space gaps in the attractor or in the repellor. iv) The gaps in each 2 of these families can be ordered with decreasing width in accord to power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions. v) The ] power law with log-periodic modulation associated to