We generalize the mass redistribution principle and apply it to prove the Bishop–Jones relation for limit sets of metrically proper isometric actions on real infinite-dimensional hyperbolic space. We also show that the Hausdorff and packing measures on the limit sets of convex-cobounded groups are finite and positive and coincide with the conformal Patterson measure, up to a multiplicative constant.

The mini-workshop Differentiable Ergodic Theory, Dimension Theory and Stable Foliations brought together experts in thermodynamical formalism, hyperbolic dynamics and dimension theory from several countries. The geographic representation was broad, from Europe, USA and Japan. All participants gave interesting 1-hour talks, and there was organized also an open problem session, where directions for future work and many open problems were discussed. Among the topics presented/discussed in the workshop, there were ones related to dimension theory and probability measures on fractals, various types of hyperbolicity, systems with overlaps, complex dynamics and iterated function systems.

We study the asymptotics of iterates of the transfer operator for non-uniformly hyperbolic α-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 α. 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 α, then the same asymptotic holds on any compact set bounded away from the indifferent fixed point.

Dennis Sullivan, in his IHÉS Seminar on Conformal and Hyperbolic Geometry [40] that ran during the late 1970’s and early ’80s, indicated a possibility of developing the theory of discrete groups acting by hyperbolic isometries on the open unit ball of a separable infinite dimensional real Hilbert space. Later in the early ’90s, Misha Gromov lamented the paucity of results regarding such actions in his seminal lectures Asymptotic Invariants of Infinite Groups [19, 6A.III] where he encouraged their investigation in memorable terms: “The spaces like this [infinite dimensional symmetric spaces] . . . look as cute and sexy to me as their finite dimensional siblings but they have been for years shamefully neglected by geometers and algebraists alike”. Gromov’s lament had not fallen to deaf ears and the geometry and representation theory of infinite dimensional hyperbolic space H∞ and its isometry group have been studied in the last decade by a handful of mathematicians. See, for example, the work by Burger-Iozzi-Monod [3], Delzant-Py [12], and Monod-Py [31]. However, infinite dimensional hyperbolic space has come into prominence most spectacularly through the recent resolution of a longstanding conjecture in

For a hyperbolic map f on a saddle type fractal Lambda with self-intersections, the number of f- preimages of a point x in Lambda may depend on x. This makes estimates of the stable dimensions more difficult than for diffeomorphisms or for maps which are constant-to-one. We employ the thermodynamic formalism in order to derive estimates for the stable Hausdorff dimension function delta^s on Lambda, in the case when f is conformal on local stable manifolds. These estimates are in terms of a continuous function on Lambda which bounds the preimage counting function from below. As a corollary we obtain that if delta^s attains its maximal possible value in Lambda, then the stable dimension is constant throughout Lambda, whereas the preimage counting function is constant on at least an open and dense subset of Lambda. In particular, this shows that if at some point in Lambda, the stable dimension is equal to the analogue of the similarity dimension in the stable direction at that point, then f behaves very much like a homeomorphism on Lambda. Finally we also obtain results about the stable upper box dimension for these type of fractals. We end the paper with a discussion of two explicit examples.

Abstract In this paper we establish a Fréchet 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 Erdős–Philipp law and to various generalized 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 Erdős.

In this paper we derive a Diophantine analysis for Julia sets of parabolic rational maps. We generalise two theorems of Dirichlet and Jarnik in number theory to the theory of iterations of these maps. On the basis of these results, we then derive a ‘weak multifractal analysis’ of the conformal measure naturally associated with a parabolic rational map. The results in this paper contribute to a further development of Sullivan’s famous dictionary translating between the theory of Kleinian groups and the theory of rational maps. 1 Statement of main results In this paper we derive a Diophantine analysis for Julia sets J(T ) of parabolic rational maps T : Ĉ → Ĉ . We generalise two classical number theoretical theorems of Dirichlet and Jarnik to the theory of iterations of rational maps. We then show that these results embed in the concept of conformal measures, where they admit a ‘weak multifractal analysis’ of the dimH(J(T )) -conformal measure which is naturally associated with the dynamical system (J(T ), T ) . Also, a combination of the results in this paper with those for Kleinian groups obtained in [10], [19], [22] and [24] adds another interesting chapter to Sullivan’s famous ‘Julia-Klein dictionary’ [25] (see also [14], [23]). Recall that for parabolic rational maps it is well-known that J(T ) = Jr(T )∪Jp(T ) , i.e. the Julia set J(T ) admits a disjoint decomposition into the radial Julia set Jr(T ) and the countable set of pre-parabolic points Jp(T ) := ⋃ ω∈Ω ⋃ n∈N T −n(ω) , where Ω denotes the set of rationally indifferent periodic points ([27], [23]). For each ω ∈ Ω , we fix a standard neighbourhood B(ω, rω) and consider, roughly speaking, all its holomorphic, inverse iterates B(c(ω), rc(ω)) . We call these balls canonical balls (see section 2, for the precise definition). A major aim of this paper will be the fractal analysis of the Jarnik-Julia sets. For ω ∈ Ω and σ > 0 , these sets are ‘ lim sup sets’ which are defined by J ω σ (T ) := ⋂

Abstract In this paper, we introduce and study the α-Farey map and its associated jump transformation, the α-Lüroth map, for an arbitrary countable partition α 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 α-sum-level sets for the α-Lüroth 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 α-Farey map and the α-Lüroth 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 α.

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 results 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.

Thermodynamic formalism and all its branches and applications in conformal dynamics, probability theory, stochastics and fractal geometry represent highly important fields in modern Mathematics, which are currently very active and rapidly growing. The workshop brought leading experts in these fields together with junior researchers to provide them with the opportunity to exchange their knowledge and experience. It led to various new insights as well as promising new research collaborations.

AbstractIn this paper, we introduce and study theα-Farey map and its associated jump transformation, theα-Lüroth map, for an arbitrary countable partitionαof 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 calledα-sum-level sets for theα-Lüroth 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 theα-Farey map and theα-Lüroth 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 partitionα.

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

This survey is dedicated to S. J. Patterson’s 60th birthday in recognition of his seminal contribution to measurable conformal dynamics and fractal geometry. It focuses on construction principles for conformal measures for Kleinian groups, symbolic dynamics, rational functions and more general dynamical systems, due to Patterson, Bowen-Ruelle, Sullivan and Denker-Urbański.

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.

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.