We study skew-products of the form $(x,\omega )\mapsto (\mathrm{Tx},\omega +\phi (x))$, where $T$ is a nonuniformly expanding map on a space $X$, preserving a (possibly singular) probability measure $\tilde{\mu }$, and $\phi :X\mapsto {\mathbb{S}}^1$ is a $C^1$ function. Under mild assumptions on $\tilde{\mu }$ and $\phi$, 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 [21]–[23] 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 $\pi$

Let $G$ be a connected reductive group. Recall that a homogeneous $G$-space $X$ is called spherical if a Borel subgroup $B\subset G$ has an open orbit on $X$. To $X$ one assigns certain combinatorial invariants: the weight lattice, the valuation cone, and the set of $B$-stable prime divisors. We prove that two spherical homogeneous spaces with the same combinatorial invariants are equivariantly isomorphic. Further, we recover the group of $G$-equivariant automorphisms of $X$ from these invariants

We prove a Łojasiewicz-Simon gradient inequality for the Willmore functional near Willmore surfaces, and we apply this inequality to exclude compact blowups for the Willmore flow