Rigorous bounds on the Hausdorff dimension of Feigenbaum attractors

TL;DR

This paper provides rigorous bounds on the Hausdorff dimension of Feigenbaum attractors at the period-doubling accumulation point for various critical map families, using interval arithmetic and renormalisation techniques.

Contribution

It introduces a method to calculate precise bounds on attractor dimensions for quadratic, cubic, and quartic maps via renormalisation and interval arithmetic.

Findings

Dimensions for quadratic maps: (0.5370,0.5392)

Dimensions for cubic maps: (0.6040,0.6091)

Dimensions for quartic maps: (0.6395,0.6474)

Abstract

We calculate rigorous bounds on the Hausdorff dimension of the attractor at the accumulation of the period-doubling cascade for families of maps with quadratic, cubic, and quartic critical point. To do this, we express the attractors as the limit sets of appropriate Iterated Function Systems constructed using rigorous bounds on the corresponding renormalisation fixed point functions. We use interval arithmetic with rigorous directed rounding modes to show that the respective dimensions lie in subintervals of the intervals (0.5370,0.5392), (0.6040,0.6091), and (0.6395,0.6474).