Hausdorff dimension of repellors in low sensitive systems.pdf

a r X i v : n l i n / 0 0 1 0 0 5 3 v 1 [ n l i n .C D ] 3 0 O c t 2 0 0 0 Hausdorff dimension of repellors in low sensitive systems A. E. Mottera and P. S. Letelierb Departamento de Matema?tica Aplicada - IMECC Universidade Estadual de Campinas, Unicamp 13081-970 Campinas, SP, Brazil Abstract Methods to estimate the Hausdorff dimension of invariant sets of scattering systems are presented. Based on the levels’ hierarchical structure of the time delay function, these techniques can be used in systems whose future-invariant-set codimensions are approximately equal to or greater than one. The discussion is illustrated by a nu- merical example of a scatterer built with four hard spheres located at the vertices of a regular tetrahedron. PACS numbers: 05.45.Df Keywords: chaotic scattering, fractal, Hausdorff dimension, repellor. aE-mail: motter@ime.unicamp.br. bE-mail: letelier@ime.unicamp.br. 1 Usually, in a scattering process almost all particles outcome asymptoti- cally free after interacting with a potential. Some particles, however, remain in the scattering region bouncing forever under the action of the potential. The nature of these trapped orbits is in general very intricate, including fractal dimensions and chaotic behaviour. The dimension seems to be the most fundamental quantity characterising this set. When the dimension is large enough, scattered particles may present sensitive dependence on ini- tial conditions reminiscent of the chaotic dynamics of the trapped orbits. This phenomenon allows the computation of the uncertainty dimension of the bounded orbits, defined as follows. For a scattering system, colour by black initial conditions of particles scattered, for instance, to the right-hand side of the scattering centre and by white initial conditions corresponding to particles scattered to the left-hand side. An initial condition of a given colour at a distance less than ε of initial conditions of a different colour is labelled ε-uncertain. The fi