Mass and dimension of Feigenbaum attractors

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Abstract:

Attractor structure on the real line is represented as a mass function M(x) defined as the fraction of attractor points to the left of x. For Feigenbaum attractors, M is a devils staircase which possesses both orbit and mass scaling features. The attractors scrAp for maps with a pth-order extremum are discussed in terms of this scaling, which leads to a hierarchy of bounds for the corresponding mass Mp. Both scrAp and Mp transform in a characteristic way under power-law conjugacy; this gives rise to two kinds of global scaling. A further intermediate scaling for the quartic attractor which underlies vector scaling in two-extremum maps is discussed in this context. As p tends to infinity, the capacity dimension D tends to 1 and the information dimension to 0. The effects of this trend on the appearance and observability of attractors are discussed. © 1985 The American Physical Society.