Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk.If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with the hitting measure is the Poisson boundary.

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In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the $\alpha$-continued fraction transformations $T_\alpha$ and unimodal maps. This correspondence identifies bifurcation parameters in the two families, and allows one to transfer topological and metric properties from one setting to the other. As an application, we recover results about the real slice of the Mandelbrot set, and the set of univoque numbers

We construct a countable family of open intervals contained in (0,1] whose endpoints are quadratic surds and such that their union is a full measure set. We then show that these intervals are precisely the monotonicity intervals of the entropy of α-continued fractions, thus proving a conjecture of Nakada and Natsui.

Let G X be a non-elementary action by isometries of a hyperbolic group G on a (not necessarily proper) hyperbolic metric space X. We show that the set of elements of G which act as loxodromic isometries of X has density one in the word metric on G. That is, for any finite generating set of G, the proportion of elements in G of word length at most n, which are X-loxodromics, approaches 1 as n → ∞. We also establish several results about the behavior in X of the images of typical geodesic rays in G; for example, we prove that they make linear progress in X and converge to the Gromov boundary ∂X. We discuss various applications, in particular to mapping class groups, Out(FN ), and right-angled Artin groups. and its limit in a suitable compactification (when it exists) is usually called a Patterson-Sullivan (PS) measure. In the second case, for each n one has the convolution measure μ n := μ · · · μ n times , which is precisely the distribution of the nth step of the random walk. In this case, the limit measure in a suitable compactification is the hitting measure for the random walk.The two types of counting, however, need not yield the same notion of genericity; indeed, an underlying theme of much research has been the question: Are Patterson-Sullivan measures also hitting measures for a certain random walk?This is for instance the main question of [64]. As Furstenberg showed [28], for a semisimple Lie group G, the appropriate boundary is B = G/P , where P is a minimal parabolic subgroup. Furthermore, the analogue of the Patterson-Sullivan measure is the unique measure ν on B which is invariant by a maximal compact subgroup. In fact, he proved that for any lattice Λ in G there exists a measure on Λ whose hitting measure is ν. Moreover, for groups acting properly and cocompactly on δ-hyperbolic manifolds, Connell and Muchnik [21] showed that there exists a certain random walk which produces the Patterson-Sullivan measure on the boundary. They also extended this result to arbitrary Gibbs measures in the case the space is CAT(−1) [20].However, for actions of hyperbolic groups G on spaces X, if the harmonic measure and PS measure are in the same measure class, then the metrics on G and on X are quasi-isometric [10]. Interestingly, for hyperbolic groups Gouëzel, Mathéus, and Maucourant [30] recently proved that, when considering a word metric on G, harmonic measures for symmetric random walks of finite support are never in the same measure class as Patterson-Sullivan measures, unless the group is virtually free. One should compare this to Ledrappier's result that for compact hyperbolic surfaces the hitting measure for Brownian motion coincides with the PS measure if and only if the surface has constant curvature [49]. There are many examples of harmonic measures which are singular with respect to the PS-type measure, see for example, [39].In this paper, we will use random walk methods to prove results about genericity with respect to counting in balls in the word metric on G. In particular, we consider the case of G a w...

We give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic-like properties. As an application, we prove sublinear tracking in Teichmüller distance for random walks on mapping class groups, and on Cayley graphs of a large class of finitely generated groups.

To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed that the visual boundary of non-positively curved (CAT(0)) groups is not well-defined, since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. For any sublinear function κ, we consider a subset of the visual boundary called the κ-Morse boundary and show that it is QI-invariant and metrizable. This is to say, the κ-Morse boundary of a CAT(0) group is well-defined. In the case of Right-angled Artin groups, it is shown in the Appendix that the Poisson boundary of random walks is naturally identified with the ( √ t log t)-boundary.

We study the dynamics of a family Kα of discontinuous interval maps whose (infinitely many) branches are Möbius transformations in SL(2, Z), and which arise as the critical-line case of the family of (a, b)continued fractions.We provide an explicit construction of the bifurcation locus E KU for this family, showing it is parametrized by Farey words and it has Hausdorff dimension zero. As a consequence, we prove that the metric entropy of Kα is analytic outside the bifurcation set but not differentiable at points of E KU , and that the entropy is monotone as a function of the parameter.Finally, we prove that the bifurcation set is combinatorially isomorphic to the main cardioid in the Mandelbrot set, providing one more entry to the dictionary developed by the authors between continued fractions and complex dynamics.