Finite and infinite ergodic theory for linear and conformal dynamical systems

Abstract

The first main topic of this thesis is the thorough analysis of two families of piecewise linear maps on the unit interval, the α-Lüroth and α-Farey maps. Here, α denotes a countably infinite partition of the unit interval whose atoms only accumulate at the origin. The basic properties of these maps will be developed, including that each α-Lüroth map (denoted Lα) gives rise to a series expansion of real numbers in [0,1], a certain type of Generalised Lüroth Series. The first example of such an expansion was given by Lüroth. The map Lα is the jump transformation of the corresponding α-Farey map Fα. The maps Lα and Fα share the same relationship as the classical Farey and Gauss maps which give rise to the continued fraction expansion of a real number. We also consider the topological properties of Fα and some Diophantine-type sets of numbers expressed in terms of the α-Lüroth expansion. Next we investigate certain ergodic-theoretic properties of the maps Lα and Fα. It will turn out that the Lebesgue measure λ is invariant for every map Lα and that there exists a unique Lebesgue-absolutely continuous invariant measure for Fα. We will give a precise expression for the density of this measure. Our main result is that both Lα and Fα are exact, and thus ergodic. The interest in the invariant measure for Fα lies in the fact that under a particular condition on the underlying partition α, the invariant measure associated to the map Fα is infinite. Then we proceed to introduce and examine the sequence of α-sum-level sets arising from the α-Lüroth map, for an arbitrary given partition α. These sets can be written dynamically in terms of Fα. The main result concerning the α-sum-level sets is to establish weak and strong renewal laws. Note that for the Farey map and the Gauss map, the analogue of this result has been obtained by Kesseböhmer and Stratmann. There the results were derived by using advanced infinite ergodic theory, rather than the strong renewal theorems employed here. This underlines the fact that one of the main ingredients of infinite ergodic theory is provided by some delicate estimates in renewal theory. Our final main result concerning the α-Lüroth and α-Farey systems is to provide a fractal-geometric description of the Lyapunov spectra associated with each of the maps Lα and Fα. The Lyapunov spectra for the Farey map and the Gauss map have been investigated in detail by Kesseböhmer and Stratmann. The Farey map and the Gauss map are non-linear, whereas the systems we consider are always piecewise linear. However, since our analysis is based on a large family of different partitions of U , the class of maps which we consider in this paper allows us to detect a variety of interesting new phenomena, including that of phase transitions. Finally, we come to the conformal systems of the title. These are the limit sets of discrete subgroups of the group of isometries of the hyperbolic plane. For these so-called Fuchsian groups, our first main result is to establish the Hausdorff dimension of some Diophantine-type sets contained in the limit set that are similar to those considered for the maps Lα. These sets are then used in our second main result to analyse the more geometrically defined strict-Jarník limit set of a Fuchsian group. Finally, we obtain a “weak multifractal spectrum” for the Patterson measure associated to the Fuchsian group.