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We study the asymptotics of iterates of the transfer operator for non-uniformly hyperbolic $��$-Farey maps. We provide a family of observables which are Riemann integrable, locally constant and of bounded variation, and for which the iterates of the transfer operator, when applied to one of these observables, is not asymptotic to a constant times the wandering rate on the first element of the partition $��$. Subsequently, sufficient conditions on observables are given under which this expected asymptotic holds. In particular, we obtain an extension theorem which establishes that, if the asymptotic behaviour of iterates of the transfer operator is known on the first element of the partition $��$, then the same asymptotic holds on any compact set bounded away from the indifferent fixed point.

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 establish a Fr��chet law for maximal cuspidal windings of the geodesic flow on a Riemannian surface associated with an arbitrary finitely generated, essentially free Fuchsian group with parabolic elements. This result extends previous work by Galambos and Dolgopyat and is obtained by applying Extreme Value Theory. Subsequently, we show that this law gives rise to an Erd��s-Philipp law and to various generalised Khintchine-type results for maximal cuspidal windings. These results strengthen previous results by Sullivan, Stratmann and Velani for Kleinian groups, and extend earlier work by Philipp on continued fractions, which was inspired by a conjecture of Erd��s.

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

The main result of this paper is to show that if $\H$ is a normal subgroup of a Kleinian group $G$ such that $G/\H$ contains a coset which is represented by some loxodromic element, then the Hausdorff dimension of the transient limit set of $\H$ coincides with the Hausdorff dimension of the limit set of $G$. This observation extends previous results by Fern\'andez and Meli\'an for Riemann surfaces., Comment: 11 pages, 3 figures

We employ infinite ergodic theory to show that the even Stern-Brocot sequence and the Farey sequence are uniformly distributed mod 1 with respect to certain canonical weightings. As a corollary we derive the precise asymptotic for the Lebesgue measure of continued fraction sum-level sets as well as connections to asymptotic behaviours of geometrically and arithmetically restricted Poincar\'e series. Moreover, we give relations of our main results to elementary observations for the Stern-Brocot tree., Comment: 9 pages

In this paper we introduce and study the $��$-Farey map and its associated jump transformation, the $��$-L��roth map, for an arbitrary countable partition $��$ 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 $��$-sum-level sets for the $��$-L��roth 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 $��$-Farey map and the $��$-L��roth 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 $��$., 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

In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the Lebesgue measure of these level sets decays to zero, for the level tending to infinity. The second and third main result then give precise asymptotic estimates for this decay. The proofs of these results are based on recent progress in infinite ergodic theory, and in particular, they give non-trivial applications of this theory to number theory. The paper closes with a discussion of the thermodynamical significance of the obtained results, and with some applications of these to metrical Diophantine analysis., Comment: 1 figure

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

In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynamic on the associated surface admits a canonical symbolic representation by a finitely irreducible shift space. We then use this representation to derive an `almost complete' multifractal description of the higher--dimensional level sets arising from Manin--Marcolli's limiting modular symbols., Comment: 20 pages, 1 figure

We study relations between $(H,\beta)$--KMS states on Cuntz--Krieger algebras and the dual of the Perron--Frobenius operator $\mathcal{L}_{-\beta H}^{*}$. Generalising the well--studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one--one correspondence between $(H,\beta)$--KMS states and eigenmeasures of $\mathcal{L}_{-\beta H}^{*}$ for the eigenvalue 1. We then consider representations of Cuntz--Krieger algebras which are induced by Markov fibred systems, and show that if the associated incidence matrix is irreducible then these are $\ast$--isomorphic to the given Cuntz--Krieger algebra. Finally, we apply these general results to study multifractal decompositions of limit sets of essentially free Kleinian groups $G$ which may have parabolic elements. We show that for the Cuntz--Krieger algebra arising from $G$ there exists an analytic family of KMS states induced by the Lyapunov spectrum of the analogue of the Bowen--Series map associated with $G$. Furthermore, we obtain a formula for the Hausdorff dimensions of the restrictions of these KMS states to the set of continuous functions on the limit set of $G$. If $G$ has no parabolic elements, then this formula can be interpreted as the singularity spectrum of the measure of maximal entropy associated with $G$., Comment: 30 pages, minor changes in the proofs of Theorem 3.9 and Fact 9

In this paper we investigate aspects of rigidity and flexibility for conformal iterated function systems. For the case in which the systems are not essentially affine we show that two such systems are conformal equivalent if and only if in each of their Lyapunov spectra there exists at least one level set such that the corresponding Gibbs measures coincide. We then proceed by comparing this result with the essentially affine situation. We show that essentially affine systems are far less rigid than non--essentially affine systems, and subsequently we then investigate the extent of their flexibility., Comment: 15 pages, 1 figure

In this paper we consider rational functions $f\colon \oc \to \oc$ with parabolic and critical points contained in their Julia sets $J(f)$ such that $$ \sum_{n=1}^\infty|(f^n)'(f(c))|^{-1}0 $$ and which are well-approximable by backward iterates of the parabolic periodic points of $f$.

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.