Keith Briggs - PhD thesis (University of Melbourne 1997)

Feigenbaum scaling in discrete dynamical systems - abstract

In this thesis I study some generalizations of Feigenbaum's discovery of scaling in families of nonlinear discrete dynamical systems.

I first make a precise computation of the Feigenbaum constants for unimodal real maps. This allows me to study number-theoretic properties of these quantities. I then make a detailed study of the asymptotic limits of the feigenvalues as the degree of the maximum of the unimodal map goes to infinity. The result is the first precise computation of the asymptotic feigenvalues.

I also generalize the Feigenbaum scaling law to compute corrections to scaling in real maps. In the next chapter I consider scaling in complex analytic maps. The results here include a complete classification of the possible feigenvalues up to degree eight. I construct a complex version of the thermodynamic formalism, which allows a computation of the Hausdorff dimension of attractors of universal functions. I also study scaling in the area of hyperbolic components of the Mandelbrot set.

The next chapter concerns circle maps, including scaling on the boundaries of Siegel disks of complex analytic maps. I compute corrections to scaling in circle maps, and the asymptotic limit of the Feigenbaum-Kadanoff-Shenker scaling constant, followed by a discussion of Manton-Nauenberg scaling and corrections to scaling.

I next move to higher dimensions with a study of scaling in two-torus maps. This requires some introductory discussion of cubic number field theory. This chapter contains the first evidence for scaling in two-torus maps. I give next a detailed discussion of iteration of quaternion maps. This is an attempt to see which properties of complex analytic maps continue in the four-dimensional space of quaternions. In particular, I classify all regularly iterable linear quaternion maps.

A final chapter describes the design of algorithms for the solution of functional equations which were used throughout the thesis.