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In this paper, we introduce and study the alpha-Farey map and its associated jump transformation, the alpha-Luroth map, for an arbitrary countable partition alpha of the unit interval with atoms which accumulate only at the origin. These maps represent linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called alpha-sum-level sets for the alpha-Luroth map. Similar results have previously been obtained for the Farey map and the Gauss map by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the alpha-Farey map and the alpha-Luroth map in terms of the thermodynamical formalism. We show how to derive these spectra and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partition alpha. Publisher PDF

In this paper we introduce and study the $\alpha$-Farey map and its associated jump transformation, the $\alpha$-L\"uroth map, for an arbitrary countable partition $\alpha$ of the unit interval with atoms which accumulate only at the origin. These maps represent linearised generalisations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic-theoretic properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called $\alpha$-sum-level sets for the $\alpha$-L\"uroth map. Similar results have previously been obtained for the Farey map and the Gauss map, by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the $\alpha$-Farey map and the $\alpha$-L\"uroth map in terms of the thermodynamical formalism. We show how to derive these spectra, and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partition $\alpha$., Comment: 29 pages, 16 figures

In this note we employ infinite ergodic theory to derive estimates for the algebraic growth rate of the Poincar\'e series for a Kleinian group at its critical exponent of convergence., Comment: 8 pages

This thesis deals with the complexity inherent in the long-term behavior of both chaotic and non-chaotic dynamical systems. Thereby, two particular examples form the starting point of our work. The first example is a simple model for the occurrence of a so-called strange non-chaotic attractor. We study fractal aspects of this attractor by determining several associated dimensional quantities. Interestingly, the considered model system shows a complex long-term behavior despite having zero topological entropy. It is hence natural to ask whether there exists another topological invariant which is able to detect this inherent complexity. This question is the origin for the investigation launched in the second part of the thesis where we introduce the notion of amorphic complexity. After examining basic properties of this new quantity, we study its applicability to almost sure 1-1 extensions of equicontinuous systems with the particular focus on Sturmian subshifts, Denjoy homeomorphisms on the circle and regular Toeplitz subshifts. The second motivating example of this thesis is closely related to a parameter family of sets of bounded orbits associated with the classical Farey map. This family of sets was recently studied as a generalization of the sets of bounded continued fraction expansions where several topological and dimensional properties were considered. In particular, it was shown that a natural associated bifurcation set plays a central role in the understanding of this family of sets. In the last part of the present dissertation, we extend these results to parameter families of sets of bounded orbits associated with more general continuous interval maps and thereby focus on topological aspects.

We introduce and study the class of geometric Schottky groups which includes many infinitely generated groups. We define a natural coding of the limit set and characterise in terms of this coding important subsets of the limit set. Further, we study a certain notion of entropy for the geodesic flow on the quotient manifold. We show that for geometric Schottky groups it is equal to the convex core entropy of the group, which for finitely generated groups agrees with the Poincar'e exponent. Subsequently, we discuss a subclass of the geometric Schottky groups called the P-class. For groups in this class, the geodesic flow is ergodic with respect to the Liouville-Patterson measure and the Liouville-Patterson measure is finite. We present proofs of these facts and deduce some consequences. Lastly, we provide methods of constructing groups in the P-class and then show that an infinite-index normal subgroup of a finitely generated geometric Schottky group cannot belong to the P-class.