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We study skew-products of the form , where is a nonuniformly expanding map on a space , preserving a (possibly singular) probability measure , and is a function. Under mild assumptions on and , we prove that such a map is exponentially mixing and satisfies both the central limit and local limit theorems. These results apply to a random walk related to the Farey sequence, thereby answering a question of Guivarc'h and Raugi [GR, Section 5.3]
We investigate the Dirichlet solution for a semianalytic continuous function on the boundary of a semianalytic bounded domain in the plane. We show that the germ of the Dirichlet solution at a boundary point with angle greater than zero lies in a certain quasi-analytic class used by Ilyashenko – in his work on Hilbert's 16th problem. With this result we can prove that the Dirichlet solution is definable in an o-minimal structure if the angles at the singular boundary points of the domain are irrational multiples of
Let be a connected reductive group. Recall that a homogeneous -space is called spherical if a Borel subgroup has an open orbit on . To one assigns certain combinatorial invariants: the weight lattice, the valuation cone, and the set of -stable prime divisors. We prove that two spherical homogeneous spaces with the same combinatorial invariants are equivariantly isomorphic. Further, we recover the group of -equivariant automorphisms of from these invariants