Title:

Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation and periodic orbits for dissipative twist maps

I study two quite different problems in this thesis. Both have been written up as papers, with their own detailed introductions. The thesis essentially consists of these two papers, although the first paper has been extended for this purpose to include extra explanatory material. In this summary I give a less formal introduction to the two papers. The aim is to give the reader some idea as to how these papers came into being. 1. Symbolic Dynamics for the Renormalization of a Quasiperiodic Schrödinger Equation. The subject of dynamical systems first captured my imagination on a reading of Feigenbaum’s renormalization theory of universality in period doubling. David suggested that I work on a renormalization theory for a onedimensional Schrödinger equation with quasiperiodic potential, there being very little known about the problem. These equations are very interesting for applications, e.g. stability analysis of Saturn’s rings, theory of an electron in a two dimensional crystal with superimposed magnetic field, and electronic structure of quasicrystals, to mention but a few. David pointed out a paper by Kadanoff concerning a particularly simple example of a quasiperiodic Schrödinger equation, for which the renormalization map is twodimensional. He suggested that if the map could be shown to possess a Smale horseshoe, then the spectrum of the Schrödinger equation must contain a Cantor set. It was relatively easy to deduce that this picture was correct by performing some computer experiments on the renormalization map. In fact I found more to be true: the dynamics of the map could be completely described using six symbols. Strings of these six symbols can then be used to label the spectrum. The proof of these facts is geometrical in spirit. It is a lengthy exercise in applying standard techniques developed for proving the existence of invariant Cantor sets in nonlinear maps. The fact that precisely six symbols are required for the description of the renormalization map seems to be the consequence of the occurrence of this symbolic dynamics in an "exactly solvable" map, related to the problem, which I describe. An interesting question then arises: what scaling properties of the spectra of the optimally approximating periodic equations can be deduced from the global dynamics of the renormalization map? It is, of course, well known that the existence of a fixed point in a renormalization map leads to a scaling law, with exponent governed by the expanding eigenvalue of the linearized map at the fixed point A remark by Newhouse, that the theory topological pressure would be relevant to the problem, was very helpful at this point I deduced the existence of a "global" scaling exponent, describing the total measure of the bands of the optimally approximating periodic systems. The exponent is obtained by taking a certain average of eigenvalues at all the periodic points of the renormalization map. Using the multiplicative ergodic theorem, I deduced the existence of an "ergodic" scaling exponent, which measures how the length of a "typical" band in the spectrum of a periodic system decreases as the period increases. Finally, I applied a theorem of McCluskey and Manning to deduce bounds on the Hausdorff dimension of the spectrum of the quasiperiodic equation in terms of the two exponents mentioned above. 2. Periodic Orbits for Dissipative Twist Maps. Periodic orbits can be proved to exist in area preserving maps of a cylinder by an elegant variational approach. The orbits are obtained as minima of a real valued function of many variables (the number of variables being equal to the period), subjected to periodic boundary conditions. Given that periodic orbits exist, from results of Hall and Katok one can deduce the existence of quasiperiodic orbits (with irrational rotation number), using only the twist hypothesis. David suggested that I look for periodic orbits in dissipative twist maps, via a variational approach devised by him. However the function that it was suggested I minimize was highly inhomogeneous in its variables, and it was inappropriate to apply periodic boundary conditions. Nevertheless, I explored the minimization problem on a computer, and found that results could be obtained using rigid boundary conditions, fixed by a parameter which is later varied. This lead naturally to a more powerful topological approach for deducing the existence of periodic orbits, which exploits the geometry of the twist and the topology of the cylinder in a particularly simple way. I thus obtained a theorem on the existence of periodic (and hence quasi periodic) orbits in one parameter families of dissipative twist maps. Using this topological approach, I also obtained results on the allowed periodic motions which can occur on an attractor of an individual dissipative twist map. The result relies heavily on a pioneering paper of Birkhoffs. A key step is to introduce the concept of an "attractor with the intersection property", which is a generalization of a strange attractor. After I had written up this result, I found that P.le Calvez had obtained a closely related result a few months previously, using a similar topological criterion for the existence of periodic orbits. The paper presented here was rewritten to take this into account.
