We establish sufficient conditions on the two weights w and v so that the Beurling-Ahlfors transform acts continuously from $L²\left(w^{-1}\right)$ to L²(v). Our conditions are simple estimates involving heat extensions and Green’s potentials of the weights.

A holomorphic family $f_z$, |z|<1, of injections of a compact set E into the Riemann sphere can be extended to a holomorphic family of homeomorphisms $F_z$, |z|<1, of the Riemann sphere. (An earlier result of the author.) It is shown below that there exist extensions $F_z$ which, in addition, commute with some holomorphic families of holomorphic endomorphisms of $ℂ̅̅̅̅̅̅f_z\left(E\right)$, |z|<1 (under suitable assumptions). The classes of covering maps and maps with the path lifting property are discussed.

Extending previous results of H. Salas we obtain a characterisation of hypercyclic weighted shifts on an arbitrary F-sequence space in which the canonical unit vectors $\left(e_n\right)$ form a Schauder basis. If the basis is unconditional we give a characterisation of those hypercyclic weighted shifts that are even chaotic.

Applying the “exact WKB method” (cf. Delabaere-Dillinger-Pham) to the stationary one-dimensional Schrödinger equation with polynomial potential, one is led to a multivalued complex action-integral function. This function is a (hyper)elliptic integral; the sheet structure of its Riemann surface above the plane of its values has interesting properties: the projection of its branch-points is in general a dense subset of the plane, and there is a group of symmetries acting on the surface. The distribution...