Suppose we have two finitely supported, admissible, probability measures on a hyperbolic group $\Gamma $. In this article we prove that the corresponding two Green metrics satisfy a counting central limit theorem when we order the elements of $\Gamma $ according to one of the metrics. Our results also apply to various other metrics including length functions associated to Anosov representations and to group actions on hyperbolic metric spaces.

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In 2000, Margulis proved that any group of homeomorphisms of the circle either preserves a probability measure on the circle or contains a free subgroup on two generators, which is reminiscent of the Tits alternative for linear groups. In this article, we prove an analogous statement for groups of locally monotonic homeomorphisms of a compact subset of $\mathbb{R}$. The proof relies on dynamical properties of random walks on the group, which may be of independent interest.

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We study the distribution of the angles between Oseledets subspaces and their log-integrability, focusing on dimension $2$. For random i.i.d. products of matrices, we construct examples of probability measures on $\operatorname{GL}_2(\mathbb{R})$ with finite first moment where the Oseledets angle is not log-integrable. We also show that for probability measures with finite second moment the angle is always log-integrable. We then consider general measurable $\operatorname{GL}_2(\mathbb{R})$-cocycles over an arbitrary ergodic automorphism of a non-atomic Lebesgue space, proving that no integrability condition on the matrix distribution ensures log-integrability of the angle. In fact, the joint distribution of the Oseledets spaces can be chosen arbitrarily. A similar flexibility result for bounded cocycles holds under an unavoidable technical restriction.

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Many results of smooth hypoellipticity are available for scalar equations. Much remains to be done for systems and/or at different levels of regularity and in particular for $L^1$-hypoellipticity. In this article we provide some examples and counter-examples.

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We show that there is a rank $1$ transformation that is mildly mixing but does not have minimal self-joinings, answering a question of Thouvenot.

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