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Abstract
Area preserving maps provide the simplest non-trivial class of conservative systems. It has been known from the time of Poincare that they can exhibit structure on arbitrarily small scales. This thesis describes numerical studies showing that there are features in area perserving maps which are repeated asymptotically exactly on smaller scales. Moreover, the forms of self-similarity appear to be universal. The key to understanding this is renormalisation, which means looking at the behaviour of the system on successively longer time scales and smaller spatial scales.
I discuss two phenomena in area preserving maps, exhibiting self-similarity. The first is the breakup of invariant circles, in particular, those with noble rotation number. Invariant circles are important for questions of stability and confinement. In the critical case when a circle is about to break up, it loses smoothness, and gains structure on arbitrarily small scales, of universal self-similar form. I explain this in terms of a fixed point of a renormalisation, to which I find a good approximation. The same renormalisation also has a simple fixed point which I show to be attracting, giving a new proof of persistence of noble circles, a result of KAM theory.
The second phenomenon is the disintegration of the stable region associated with an initially stable periodic orbit. It is accompanied by an infinite sequence of successive period doublings, starting from the initial periodic orbit. The accumulation point of this sequence corresponds to a fixed point of a renormalisation. Beyond the accumulation point there are only very small regions of stability.
The value of renormalisation explanations is that if the small scale behaviour is known to be universal and the universal behaviour is known, then it is necessary to measure only a few parameters for a system to be able to deduce the whole behaviour locally. So properties of the critical fixed point for breakup of invariant circles lead to a criterion for existence of invariant circles, and properties of the fixed point for period doubling lead to a criterion for disintegration of the stable region associated with a periodic orbit.
