Broadly
speaking, the subject of my thesis was dimension theory in dynamical
systems and thermodynamic formalism. I studied systems with
certain topological dynamical properties (specification properties)
which are frequently exhibited by systems which resemble uniformly
hyperbolic or expanding systems in some aspects of their behaviour (see
my thesis for a precise
list of examples). In particular, I gave a detailed application of my
results to the beta-transformation.
The beta-transformation is the rather innocent looking transformation f(x) =
beta x (mod 1), where beta is some real number greater than 1. Interest
in the study of the beta-transformation arises from its connection with
number theory and its special role as a model example of a
one-dimensional expanding dynamical system which admits
discontinuities. The beta-shifts, which are the natural coding spaces
for the beta-transformations, give a very natural and rich class of
symbolic spaces which, while being very explicit, do not have a finite
Markov structure.
One of the catchier results from my thesis is as follows. Fix a
beta-transformation. For a continuous function on the unit interval,
consider the set of points for which the Birkhoff average with respect
to the beta-transformation diverges. I showed that this set either has
full Hausdorff dimension (i.e. Hausdorff dimension 1) or is empty. The first case is generic and I
gave very explicit conditions on the function which describe which of the two cases applies.
This contrasts sharply with the measure-theoretic point of view in
which, by Birkhoff's theorem, this set is always a null set.
A version of this result (using topological entropy as the dimension
characteristic) applies to a much broader class of systems which
includes ergodic toral automorphisms.